UFDC Home myUFDC Home  |   Help
<%BANNER%>

# Statistical Mechanics and Linear Response for a Granular Fluid

PAGE 4

PAGE 5

page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii ABSTRACT .................................... viii CHAPTER 1INTRODUCTION .............................. 1 1.1GranularMaterialsasObjectsofTheoreticalStudy ......... 1 1.2GranularFluids ............................. 3 1.3TheoreticalDevelopmentsintheStudyofGranularFluids ..... 5 1.4ObjectiveofthisWork ......................... 8 1.5StructureofthePresentation ..................... 10 2STATISTICALMECHANICSOFAGRANULARFLUID ........ 12 2.1MicroscopicModelsforGranularFluids ................ 13 2.2GeneratorsofDynamicsforInelasticHardSpheres ......... 15 2.3IsolatedHomogeneousSystem ..................... 20 2.4StationaryRepresentationandTimeCorrelationFunctions ..... 23 2.5FluctuationandResponseinaGranularFluid:SomeObservations 28 3PHENOMENOLOGICALHYDRODYNAMICS .............. 34 3.1NonlinearNavier-StokesHydrodynamicEquations .......... 35 3.2LinearizedHydrodynamics ....................... 40 3.3HydrodynamicModesandStability .................. 43 4LINEARRESPONSE ............................ 48 4.1GeneralProcedureofLinearResponse ................ 49 4.2SpecialInitialPreparation ....................... 54 4.2.1SpecialHomogeneousSolutiontotheLiouvilleEquation ... 54 4.2.2LocalHomogeneousCoolingStatePreparation ........ 56 4.2.3SimplifyingPropertiesoftheLHCSPreparation ....... 57 4.3k-ExpansionoftheTransportMatrix ................. 60 4.3.1KFatEulerOrder ....................... 65 4.3.2KFatNavier-StokesOrder ................... 66 4.3.3KSatEulerOrder ........................ 70 v

PAGE 6

................... 71 4.4SummaryofResults .......................... 72 5TRANSPORTCOEFFICIENTS ...................... 74 5.1HomogeneousOrder:TheCoolingRateh 75 5.2EulerOrderTerms ........................... 76 5.2.1Pressure ............................. 77 5.2.2EulerTransportCoecientU 80 5.3Navier-StokesTransportCoecients ................. 83 5.3.1ShearViscosity ......................... 84 5.3.2BulkViscosity .......................... 86 5.3.3ThermalConductivity ..................... 87 5.3.4TheCoecient ........................ 90 6KINETICTHEORYOFTIMECORRELATIONFUNCTIONS ..... 95 6.1GeneralFormalism ........................... 96 6.2Illustration:EnskogKineticTheoryforElasticHardSpheres .... 101 6.3DerivingtheKineticTheoryforInelasticHardSpheres ....... 106 6.4StructureoftheTransportCoecients ................ 112 6.5ComparisonwithChapman-EnskogResults:ShearViscosity .... 115 7DISCUSSIONANDOUTLOOK ...................... 122 7.1SummaryofPrimaryResults ..................... 122 7.2ContextandScopeofthisWork .................... 124 APPENDIX AGENERATORSOFDYNAMICSFORINELASTICHARDSPHERES 128 BSTATIONARYREPRESENTATIONOFTHEDYNAMICS ....... 136 CMICROSCOPICCONSERVATIONLAWS ................. 144 DSPECIALSOLUTIONTOTHELIOUVILLEEQUATION ........ 152 ECONSERVATIONLAWSINTHESTATIONARYREPRESENTATION 155 FDETAILSINTHEkEXPANSIONOFK(k;s) .............. 159 GDERIVATIONOFTHEFORMSOFTRANSPORTCOEFFICIENTS 168 HELASTICHARDSPHERES:SOMEDETAILS .............. 179 IENSKOGKINETICTHEORY ....................... 185 REFERENCES ................................... 207 BIOGRAPHICALSKETCH ............................ 211 vi

PAGE 7

Table page A{1HardSphereCollisionOperators ...................... 135 C{1MicroscopicBalanceEquations ....................... 150 C{2FormsoftheForwardFluxesandSource .................. 151 C{3FormsoftheBackwardFluxesandSource ................. 151 vii

PAGE 8

Theobjectiveofthisstudyistoexploretheapplicationofmethodsfromnon-equilibriumstatisticalmechanicstothestudyofgranularuids.Thisexplorationiscarriedoutinthespeciccontextofobtainingahydrodynamicdescriptionfortheseuidsthatisbasedunambiguouslyonanunderlyingmicroscopictheory.Aparticularmodelofagranularuidamenablefortheoreticalstudy,namelyinelastichardspheres,isconsidered.Theisolatedhomogeneoussystemisstudiedandtheassociatedensemblecharacterized.Then,smallperturbationsawayfromthisreferenceensembleareconsideredwiththeaimofisolatingthehydrodynamicresponseofthesystemandhenceobtainingmicroscopicexpressionsforthevarioushydrodynamictransportcoecients.TheprimaryoutcomesofthisprojectareexactGreen-KuboandHelfandformsforallthehydrodynamictransportcoecients.Theseexpressionsarearstoftheirkindinthattheydonothaveanyapriorilimitationsonthedomainoftheirvalidity,unlikethoseobtainedfromkinetictheorythatexistintheliteraturesofar.Theseexactformsareawelldenedstartingpointforfurtheranalyticandnumericalanalysisinordertoobtainusefulinsightintothenatureoftransportinthis viii

PAGE 9

ix

PAGE 10

1 2 ],ithasbeenestimatedthat40%ofthepotentialcapacityofindustrialplantsiswastedduetodicultiesinhandlingandtransportinggranularmaterials.Thesematerialshavebeenwidelystudiedintheengineeringcommunityforalongtime[ 3 4 ]andareofactiveinteresteventodayasanyimprovementintheunderstandingofthedynamicsofthesesystemstranslatesintocostreductionintheseindustries. Granularmaterialsarestudiedindierentregimes.Someoftheinterestingphenomenaexhibitedbythesesystemsaremediatedbytheinterstitialuidpresentinthem,eitherthroughmediatingcohesionbetweengrainsorthroughaStokesiancouplingofthegrainstotheirow.Thesesystemsarecalledwetgranularmedia.Granularsystemsinwhichtheinterstitialuiddoesnotplayanimportantroleinthephenomenologyofthematerialarecalleddrygranularmedia.Thelatteristheclassofsystemsofinteresthere.Theyareinterestingfromthepointofviewofatheoreticalphysicistforanumberofreasons.Drygranularmaterialsexhibitawidevarietyofphenomena.Apileofsandislikeasolidinthesensethatitcanwithstandapplicationofnormalstress(i.e.,ifyoupoursandandletitcometo 1

PAGE 11

rest,itsitsinaheap,bearingitsownweight).Butitisunlikeasolidinthatitsstressresponseishighlyinhomogeneouswithafewparticlescarryingthebulkoftheload[ 5 ].Whenactivated,itowslikeauid,forexample,inanhourglassoragrainhopper.Buttheowischaracterizedbydierentphenomenathaninthecaseofanormaluid,suchasjamming,novelinstabilitiesandpatternformation[ 6 7 ].Further,modelsystemswithsimpleprescriptionsforinter-graininteractionsexhibitallofthephenomenologydescribedabove.InelastichardspheresordissipativeHertziancontactforcestogetherwithsimplerealizationsoftangentialfrictionconstituteveryaccuratemodelsforcapturingthephysicsofthissystem[ 11 ].Also,thenumericaltoolofMolecularDynamics(MD)simulationcanbebroughttobearonthissystemtosupplementandelaborateexperimentalobservations,andtoverifypredictionsandstimulatetheformulationoftheoreticalanalysis(forexamplesee[ 12 13 ]forMDstudiesofvariousaspectsofthephysicsofthissystem).Hence,therichphenomenologyassociatedwiththisclassofsystemscanbestudiedinatheoreticallytractablesetting. Theoreticalinterestinthesesystemsisalsomotivatedbythefollowingconsiderations.Agranularmaterialcanbetreatedasaprototypicalnon-equilibriumsysteminthefollowingsense.Firstly,noticethatmost\microscopic"modelsusedtodescribegranularmaterialshavetheconstituentgrainsastheirsmallestentities.But,thesegrainsthemselvesarelargeparticlesandhencethethermalenergykBT(kBbeingtheBoltzmannconstant)isverysmallcomparedtothegravitationalpotentialenergymghofthegrainsatroomtemperatureandunderterrestrialconditions.Soconventionalthermodynamicsplaysnoroleindescribingthephenomenaexhibitedbygranularsystems.Theseareexplainedbasedonpurelydynamicalconsiderations.Forexample,amixtureoftwogranularmaterialsofdierentgrainsizessegregateswhenactivatedbyvibrationorrotation[ 8 9 10 ].Butfortheaboveobservationabouttheirrelevanceofthethermodynamic

PAGE 12

temperatureinthedynamicsofthissystem,theabovephenomenawouldappeartobeaviolationofthesecondlawofthermodynamics.Secondly,thespectrumoftheoreticaltoolsthatcanbefruitfullyappliedtostudygranularmaterialsinvariousregimesisverywide.Forexample,thewellknownphenomenaofcompactionofgranularmaterialsundertappingcanbemodelledtheoreticallybyintroducingapseudothermodynamicswheretheroleofenergyinconventionalthermodynamicsistakenbythefreevolumeinthesystemandtheentropyisnowgivenacongurationalinterpretation[ 14 15 ].Thebroadlyappliedtheoryofselforganizedcriticality,usedtoexplaintheubiquitousoccurrenceof\1=fnoise"indisorderednon-equilibriumsystemswasrstformulatedinthecontextofavalanchesinasandpileattheangleofmaximumstability[ 16 ].Agranularsysteminthejammedstatecanbedescribedusingthetheoreticallanguageapplicabletodescribeglassydynamics[ 17 18 ].Asuciently\uidized"granularmaterialcanbestudiedusingthetoolsapplicabletonormaluidsinnon-equilibriumstates.Thusdrygranularmaterialsprovidetheopportunityforthedevelopmentandapplicationsofawidevarietyofmethodsusedinthestudyofnon-equilibriumphenomena.

PAGE 13

thisuctuatingvelocityofthegrainandexpectthatastatisticaldescriptionintermsofacoursegrainednumberdensityeld,owvelocityeldandtemperatureeldwillberelevantinunderstandingthedynamicsofthesystem.Hencetheactivationmustbeinawaysoastoinduceacollisionalowinthemedium.Suchanactivatedgranularmaterialiscalleda\granularuid".Mostgranularowsofinterest(forexamplevibrationinducedphenomena,gravitationallyinducedowdownabumpyincline,rapidshearinducedows)fallunderthisregime.Underthesecircumstances,thestandardtoolsusedinthedescriptionofnormaluidscanbeappliedtostudythegranularsystem. 19 ].Theenergylosttotheseinternaldegreesoffreedomdoesnotcontributetotransportintheuid.So,inmosttheoreticalmodelsofagranularuidthegrainistakentobethemicroscopicentityinthesystem,withtheinternaldegreesoffreedomofthegrainsactingasa\blackbox"sinktothekineticenergyofthegrains.Therefore,thetotalenergyofthesystemisnolongeraconservedquantityinthetheoreticalmodelsandthisservesasatechnicalcomplicationintheanalysisofthedynamicsofthesystem.Butmoreimportantly,therateatwhichkineticenergyofthegrainsislosttotheinternalmodesgivesrisetoaninternaltimescaletotheuidapartfromthetimebetweencollisionsandthisplaysanimportantroleinthephysicsoftheseuids.

PAGE 14

[ 20 ].ThisiscalledBagnoldscalingintheliterature.Sincetheresponseofthesystemunderashearstrainisratedependent,theuidissaidtoberheologicaloracomplexuid.Granularuidsexhibitrheologicalbehaviorwhenevertheexternaldrivingoftheuidissuchthatitprobestheintrinsictimescaleintheuid,namelythetimescalesetbytherateoflossofkineticenergyduetotheinelasticcollisionsbetweengrainsmentionedabove.Thisbehaviormanifestsitselfinshearowsandisanimportantpropertytofoldintothetheoreticaldescriptionofagranularuidundershear.But,whentheactivationofthegranularmaterialisinsuchawaythattheexternaldrivingiswellremovedfromthisinternaltimescale,forexampleinthecaseofactivationthroughvibrationattheboundary,thegranularuidbehaveslikeaNewtonianuid.Thislatterregimewillbetheoneforwhichtheresultsobtainedinthisworkcanbeapplieddirectly. AtheoreticalmodelwidelyusedasanidealizedrepresentationofagranularmaterialinitsuidizedstateisthatofNsmoothhardspheresthatcollidepair-wiseinelasticallysothatthecollisionsconservemomentumbutthereisafractionallossinthekineticenergyofthepair.Theenergylossischaracterizedby

PAGE 15

asingleparameter,thecoecientofrestitution,with=1correspondingtotheelasticlimitofnoenergyloss.ThisisthegeneralizedversionoftheHardSpheremodelfornormaluidsthathasbeenextensivelyusedinthetheoreticalstudyofequilibriumuids. Thetheoreticaldevelopmentinthecontextofthismodelgranularuidhasfollowedinthesamespiritasthatofnormaluidsahundredyearsago.Inthediluteregime,thedynamicsofthisgranularuidcanbeconsideredtobegovernedbyaBoltzmannkineticequation[ 21 22 ].Inthisregimethegranularmaterialisreferredtoasagranulargas.Extensivestudieshavebeencarriedoutonthekinetictheoryofthissystem[ 23 ].Theseinclude,butarebynomeanslimitedtocharacterizationoftheisolatedhomogeneousstateassociatedwiththisequation[ 24 ]andcarryingoutasystematicsmallgradientsexpansionusingtheChapman-Enskogprocedure,therebydeterminingthehydrodynamictransportcoecientssuchastheviscosityandthermalconductivityfromtheunderlyingkinetictheory(forexamplesee[ 25 26 ]).Alsotheeectofdierentdrivingforcesonthenatureoftheresultingreferencesteadystates[ 27 28 ]hasbeenstudied.Further,numericaltoolssuchasMolecularDynamicssimulations[ 29 ]andDirectSimulationMonteCarlomethod[ 30 ]forsolvingtheBoltzmannequationhavebeenusedtostudystatesthatarefarfromhomogeneity,suchastheUniformShearFlowstate,whichistheprototypeshearstateforthissystemthatmanifestsitsrheologicalproperties[ 31 32 ].Thesenumericaltoolshavealsobeenusedtoevaluatethedierenttransportcoecientsobtainedfromkinetictheory.Intheprocess,potentialstumblingblocksforcarryingoutnumericalsimulationsofthesesystems,suchasinelasticcollapsewereidentiedandcharacterized[ 33 ].Thelowdensityhydrodynamicsthatresultsfromtheaboveanalysishasbeenextensivelystudiedanalyticallyandnumericallywithspecicapplicationstomanyexperimentalscenarios[ 26 34 35 ].Theinstabilitiesintheseequations

PAGE 16

havebeenanalyzedtofruitfullypredictpatternformationsthatoccurundersomeexperimentalconditions[ 36 ]. ThenextsystematicstepforwardinunderstandingthehydrodynamicdescriptionwastakenintheformoftryingtoestablishthatthehydrodynamicmodesareindeedpresentinthespectrumoftheBoltzmanncollisionoperatorandthattheyaretheslowestmodes.AsimilaranalysiscarriedoutfornormaluidsintermsoftheelastichardsphereBoltzmanncollisionoperatorgaveapositiveresult,establishingthattherealwaysexistsalengthandtimescalesuchthatthehydrodynamicdescriptionoftheuidiscomplete[ 37 ].Forthecaseofgranularuids,itwasestablishedthatthehydrodynamicmodesareindeedpartofthespectrumofthecollisionoperator[ 38 ].Buttheissueoftimescaleseparationbetweenthesemodesandthefaster\microscopic"modeswasleftunresolvedduetothecomplicatednatureoftheinelastichardsphereBoltzmanncollisionoperator.Furtherprogressinthisdirectionwasmadepossiblebytheuseofkineticmodelsforthecollisionoperatorthatretainedtheessentialfeaturesofthetruekineticequation,butwereanalyticallytractableenoughtocarryoutexactcalculations(forexamplesee[ 39 ]). Alloftheaboveanalysisisvalidonlyinthelimitofarbitrarilysmalldensities.But,mostexperimentalrealizationsdonotfallintothisdomain.TherststepinthedirectionofincorporatingdensityeectsinthehydrodynamictransporttheoryforagranularuidwastakenintheformofstudyingaRevisedEnskogTheoryforinelastichardparticles.ThisisaBoltzmann-likekineticequationthatincorporatesdensityeectsphenomenologicallyintwoways.1)Ittakesintoaccountthenitesizeofthehardparticle(whichistreatedasapointparticleintheBoltzmanntheory).2)Thecollisionfrequencyisnowweightedbythepaircorrelationfunctionatcontactthatamountstoincorporatingthreeparticleeectsinameaneldlikeapproximation,whichfurtheremphasizesthedensityeects.Thiskinetic

PAGE 17

equationwasanalyzedforstatesclosetohomogeneityandhydrodynamictransportcoecientsidentiedusingaChapmanEnskogprocedure[ 40 ].Inthecaseofnormaluids,thesetransportcoecientsarefoundtobeaccurateuptorelativelyhighdensities.Asimilardomainofvaliditywasexpectedforthesecoecientsingranularuidsaswell.Further,thehydrodynamicequationsthusobtainedweresolvedforaparticularboundarydrivenscenarioandtheresultinghydrodynamicproleswerecomparedwiththoseobtaineddirectlyfromexperimentandverygoodagreementwasfound[ 41 ]. Alloftheabovementionedtheoreticaldevelopmentshavetheirbasisinkinetictheoryandhencehavelimitationsonthedomainindensityinwhichtheycanbeexpectedtoberelevant.Inthecaseofnormaluids,furtherdevelopmentofthetheoryoftransportwasstimulatedbytheapplicationoftheexactmethodsofnon-equilibriumstatisticalmechanics,namely,linearresponseandthetimecorrelationfunctionmethod.Thismethodyieldsexactexpressionsforthetransportcoecientsthathavenoapriorilimitationsontheirdomainofvalidity.Thesystematicstudyofgranularuidtransportisatexactlythispoint.Thegeneralnon-equilibriumstatisticalmechanicsformalismisinplace[ 42 ].Firststepsinitsapplicationhavebeentakeninthecontextoftheprototypicaltransportprocessinauid,namelydiusion[ 43 44 ].Theworkhereaimstoapplythismethodextensivelytoenergyandmomentumtransportprocessesinthegranularuidaselaboratedbelow.

PAGE 18

systemconsistingofsmoothsphericalhardparticlesisconsideredandalinearresponseanalysisisformulatedandcarriedouttogiveexactexpressionsforthehydrodynamictransportcoecientsofthisuid.TheyhavetheformofHelfandandGreen-Kuboexpressionsthataretimecorrelationfunctionsoverthereferencehomogeneousensemble.Theseresultsarecomparedwiththeanalogousresultsfornormaluidsandtheroleplayedbythenon-conservationofenergyinthetransportprocessesofthissystemisexplicated.Further,anapproximateevaluationoftheseexactexpressionsiscarriedoutandcomparedtoknownresultsobtainedfromkinetictheory. Forthepurposesofclarication,consideranormaluidthatconsistsofatomsthatinteractthroughacontinuouspotential.Then,alinearresponseanalysisgivestheexactexpressionfortheshearviscosityofthisuidintheform whereHxyisthevolumeintegratedmicroscopicmomentumuxandhieqdenotestheequilibriumensembleaverageoverthecanonicalensemble.Inotherwords,theshearviscosityofanormaluidisthetimeintegralofthemomentumux-momentumuxautocorrelationfunctioncalculatedovertheequilibriumensemble.Thisworkpresentsaderivationoftheanalogousresultforagranularuidfortheshearviscosityandalltheotherhydrodynamictransportcoecients. Theimportantconsequencesofthisworkcanbesummarizedasfollows. 1.Thisistherstsystematicandextensiveapplicationofthemethodsofnon-equilibriumstatisticalmechanicstogranularuids(exceptfortheprototypeprocessofdiusion[ 44 ]andpreliminaryconsiderationsofGreen-Kuboformulafortheshearviscosity[ 45 ]). 2.Thetransportcoecientswhoseexpressionsareobtainedherearethesameonesstudiedusingkinetictheory(see[ 26 ]andotherscitedabove)andusedinthehydrodynamicequationstoexplainandinterpretexperimentalresults.Buttheexpressionsavailableintheliteratureuptothistimearelimitedby

PAGE 19

thefactthattheyarevalidonlyatlowormoderatedensities.Theexpressionsobtainedhereareapplicableforalldensitiesandinelasticities. 3.Intheprocessofobtainingtheaboveexpressions,notionssuchastheOnsagerregressionhypothesisandtheFluctuation-Dissipationtheoremarerevisitedandareinterpretationprovidedinthecontextofthisinherentlynon-equilibriumsystem. 4.Fornormaluids,fruitfulinsightintothenatureofenergyandmomentumtransportintheuidwereobtainedbynumericalevaluationsusingMolecularDynamicssimulations[ 46 47 ]andapproximateanalyticalevaluationsusingextensivelydevelopedmethodssuchasmodecoupling[ 48 ].Thisworksetsthestageforasimilarexplorationintotransportprocessesingranularuids. 5.Experimentalevidenceindicatesthatahydrodynamicdescriptionforthedynamicsofagranularuidworkswellinthevibro-uidizedregimeinaconsiderableregionoftheparameterspaceofdensityandcoecientofrestitution.Inthedomainthatitfails,itisunclearifthehydrodynamictheoryfailsortheestimatedvaluesforthetransportcoecientsarenolongervalid.Numericalevaluationofthetransportcoecientsobtainedherecanshedlightonthisproblem,astheexpressionsarevalidforallvaluesofdensityandcoecientofrestitution. 1.InChapter2,theingredientsnecessarytodostatisticalmechanicsforagranularuidofNinelasticsmoothhardspheres,namely,thegeneratorsofthedynamics,thecharacteristicreferenceensembleandtheformulationoftimecorrelationfunctionsoverthisensemblearegiven. 2.InChapter3,startingfromthemacroscopicbalanceequationsforthehydrodynamicvariables,usingtheuidsymmetryandtheassumptionthatthegradientsinthehydrodynamiceldsaresmall,asetofnonlinearhydrodynamicequationsdescribingagranularuidareobtainedphenomenologically.Then,thesearelinearizedaboutthehomogeneousstateandthelinearizedequationsareanalyzedtoidentifythehydrodynamicmodesandtheassociatedeigenfunctionsforthisgranularuid.Thelinearequationsobtainedhereareparameterizedbyunknowncoecientssuchasthepressureandviscosity.Thus,thisservesasanidenticationoftheprecisetargetsofthemicroscopicanalysisthatfollows. 3.InChapter4,theproblemoflinearresponseanditsuseinidentifyingthehydrodynamicdescriptionoftheuidisformulatedandsolvedtogiveaformalexpressionforthehydrodynamictransportmatrixasamatrixoftime

PAGE 20

PAGE 21

Thefocusofthischapterofthepresentationisthesettingupofthestatisticalmechanicsofagranularuid.Statisticalmechanicsistheframeworkthatallowstheextractionofthemacroscopicobservablesofasystemasstatisticalaveragesovertheassociatedsetofmicroscopicstatesthatcharacterizethesystem.Inparticular,non-equilibriumstatisticalmechanicsisamanybodytoolthatallowstheidenticationofthemicroscopicbasisoftransportphenomenainasystem.Also,aswillbeseenintherestofthiswork,timecorrelationfunctionsplaythesameroleintransportphenomenaasensembleaveragesdointhermodynamics.Thesetimecorrelationfunctionsforagranularuidaredenedandcharacterizedinthischapter. Thelayoutofthischapterisasfollows.Theprimaryingredientinthemicroscopicprescriptionofthesystemaretheinteractionsbetweenitsconstituentparticles.ThisspecicationallowsthecalculationoftheNparticletrajectoryforagiveninitialconditionthatspeciesthevalueofthepositionandvelocitiesofallthegrains.Someofthemodelsforgrain-graininteractionarerstdiscussed.Next,thedynamicsofthesystemischaracterizedinthecasewhentheinitialstateisamacrostatebyidentifyingthegeneratorsassociatedwiththetrajectoriesandphasespacedistributions.Then,thecharacteristicensembleassociatedwithahomogeneousisolatedgranularuidisidentiedandcharacterized.Further,theessentialmachineryforstudyingperturbationsandresponseinnon-equilibriumstatisticalmechanics,namelytimecorrelationfunctions,arecharacterized. 12

PAGE 22

PAGE 23

PAGE 24

Herebisaunitvectordirectedfromthecenterofparticlejtothecenterofparticleithroughthepointofcontact.Theparameter(thecoecientofnormalrestitution)ischosenaprioriintherange0<1andremainsxedforagivensystem.Asnotedabove,thevalue=1correspondstoelastic,energyconservingcollisions,while<1describesaninelasticcollisionwithancorrespondingenergylossforthepairgivenby Thecenterofmassvelocity(vi+vj)=2isunchangedsothatthetotalmassandmomentumofthepairsareconservedforallvaluesof.Subsequenttothechangeinrelativevelocityforthepairi;jthefreestreamingofallparticles

PAGE 25

continuesuntilanotherpairisatcontact,andthecorrespondinginstantaneouschangeintheirrelativevelocitiesisperformed.Thesequenceoffreestreamingandbinarycollisionsdeterminesauniquetrajectoryinphasespace,t,forgiveninitialconditions.Thecollisionruleisinvertiblesothetrajectorycanbereversed,althoughwiththeinvertedcollisionrule(\restituting"collisions). Thestatisticalmechanicsforthissystem[ 42 ]iscomprisedofthedynamicsjustdescribed,amacrostatespeciedintermsofaprobabilitydensity(),andasetofobservables(measurables)denotedbyfA()g.Theexpectationvalueforanobservableattimet>0forastate()givenatt=0isdenedby FortheoreticalanalysisthedynamicsdescribedabovecanberepresentedintermsofageneratorLdenedby SuchageneratorforthedynamicsisidentiedinAppendixA,withonlytheresultsquotedhere.TherearetwocomponentstothegeneratorL,correspondingtothetwostepsoffreestreamingandvelocitychangesatcontact, 2NXi=1NXj6=iT(i;j):(2{9) wherethebinarycollisionoperatorisgivenby Hereqijistherelativepositionvectorofthetwoparticles,istheHeavisidestepfunction,andbijisasubstitutionoperator

PAGE 26

whichchangestherelativevelocitygijintoitsscatteredvalueg0ij,givenbyEq.( 2{5 ).Thethetafunctionanddeltafunctionin( 2{10 )assurethatacollisiontakesplace,i.e.thepairisatcontactanddirectedtowardeachother. ThedynamicscanbetransferredfromtheobservableA()tothestate()bydeninganadjointgenerator L()A():(2{12) Theformgeneratedby 2{12 )sinceallacceptableprobabilitydensities()mustexcludethedomainofanypairoverlapping.However,therightsideof( 2{12 )doesnothavethatrestrictiononthedomainofintegrationandconsequentlythegeneratorforLiouvilledynamicsisnotthesameasthatforobservables(asinthecaseofcontinuouspotentials).Instead,directanalysisof( 2{12 )leadstotheresult(seeAppendixA) 2NXi=1NXj6=i withthenewbinarycollisionoperator Hereb1ijistheinverseoftheoperatorbijin( 2{11 ) (bgij)b:(2{15) Insummary,thedynamicsofphasefunctionsisgivenby (@tL)A(;t)=0;(2{16)

PAGE 27

andtheLiouvilleequationassociatedwithdistributionfunctionsis Asanaside,notethattheformofthegeneratorLandcorrespondingbinarycollisionoperatorT(i;j)doesnotdependonthedetailsofthecollisionrulegivenbytheoperatorbij,andthereforeappliesmoregenerallytoaclassofhardcollisions[ 51 ].Inparticular,itisformallythesameforbothelasticandinelasticcollisions.Incontrast,thegeneratorforLiouvilledynamicsisobtainedbyachangeofvariablesthatintroducestheJacobianofthetransformationbetweenthevariablesgijandbijgij.Henceitdependsexplicitlyontherestitutioncoecient. NotethatfromtheLiouvilleEquationin( 2{17 ),asetofequationsanalogoustotheBBGKYhierarchycanbeobtainedthatwouldcharacterizethedynamicsofthereduceddistributionfunctionsintheform wherethereduceddistributionfunctionisdenedas and (@t+v1rq1)f(1)(q1;p1;t)=Zdq2dp2 Inthisformitiseasytoseethattheconnectiontokinetictheoryofagranularuidistobemadebyexploringpossiblefunctionalrelationshipsofthetwoparticledistributionf(2)totheoneparticledistributionf(1).Ifsomesuchfunctionalrelationshipisidentiedorpostulated,theaboveequationbecomesaclosedequationforthedynamicsoftheoneparticledistributionfunctionandhence

PAGE 28

PAGE 29

toguaranteethatalltrajectoriesinvolvingthreeparticleinteractionswillalsohavezeromeasure.But,forinelastichardspheres,thereisanadditionaldynamicaleectcalled\collapse"[ 33 ].Itispossibleforagroupofparticlestoundergoinnitelymanybinarycollisionsamongthemselvesinanitetime,soastoloosealltheirkineticenergyandcometorest.Whensuchcollapsehappens,thenthereexistsapossibilityofathreebodyinteractionoccurringsubsequently.Thestatisticalweightassociatedwithsuchevents,thatis,completionofcollapsethatleavestwoormoreoftheparticlesinvolvedatrelativerestandincontact,isunknown.Forthepurposesathand,itisassumedthatsucheventshavezeroweightinthesensethatacollapsingpairisbrokenupbeforethecompletionofthecollapseandthatbinarycollisionsalonearesucienttogenerateawelldenedtrajectoryforalllatertimesandthisissuewillnotbeconsideredfurtherinthiswork. 2mv2iofthesystem.Thedynamicalequationgoverningthetimeevolutionofitsensembleaverageis @tDbEE=Zd()@ @tbE(t)=m

PAGE 30

wheretherighthandsideismanifestlynegative.Deneagranulartemperaturethroughtheequation 3N*NXi=11 2mv2i(t)+:(2{22) Usingtheequationabove,thetemperatureisfoundtobemonotonicallydecreasingwith where(t)>0isthe\cooling"rateduetoinelasticcollisionsgivenby Thisshowsthatthereisno\approachtoequilibrium"foragranularuidsincethereisnosuchstationaryequilibriumstate.Howeverthereexistsevidenceontheoretical(atthelevelofkinetictheory,foranillustrationinthecontextofakineticmodelsee[ 52 ])andnumericalgrounds[ 53 ]thatawideclassofinitialhomogeneousstatesrapidlyapproachauniversalstatethatisspatiallyhomogeneous(translationallyinvariant)andallofitstimedependenceoccursthroughtheaveragespeedT(t):ThisisknownastheHomogeneousCoolingSolution(HCS).TheequationgoverningitsdynamicscanbeobtainedbyeliminatingthetimederivativeintheLiouvilleequationEq.( 2{17 )usingthetimeevolutionequationofthetemperatureEq.( 2{23 )abovetogive withthedenition 2h(t)T(t)@X @T(t)+ Sincethesystemathandishardspheres,theonlyenergyscaleintheproblemisthekineticenergyoftheparticles.Hence,theonlywaythistemperaturedependencecanoccuristhroughascalingofthevelocities.Thisallowsthe

PAGE 31

homogeneouscoolingstatetobewrittenasascalingstateintheform wherevh(t)isthethermalvelocityand`isamicroscopiclengthscale(forexamplethemeanfreepath)chosentonon-dimensionalizethedistancesintheproblem. Firstofall,usingthescalingformoftheHCSinthedeningequationforthecoolingrate( 2{24 )abovegivesdirectly wherehisnowadimensionlessconstantthatdependsontheparametersofthemodelandthereduceddensityofthehomogeneousuid.Next,usingthisresultinthetemperatureequation( 2{23 )above,gives 2`ht2!2m`2 Alsogivenisthelimitingbehavioratlongtimes,showingthatthetemperaturebecomesindependentoftheinitialconditions.ThisformisknownastheHa'slawforafreelycoolinggranularuid[ 54 ]andisthesignatureofaHCSthatisreadilyrecognizedinnumericalsimulations.Also,withthisscalingformthe 2h(t)NXi=1rvi((viu)X)+ Thisequationmustbesolvedself-consistentlywith( 2{24 )forh(t)=h(tjh)whichisalinearfunctionalofh.Itiseasilyseenbydirectcalculationthatnoneoftheequilibriumensemblesfornormaluidsaresolutionstothisequation,evenwithageneralizationtothescalingform( 2{25 ). OneimmediateconsequenceofthescalingnatureoftheHCSensembleisdescribedbelow.Letf(P)beanarbitrarydierentiablefunctionofthe

PAGE 32

momentum.Bymomentumconservation 0=Zdh(t)Lf(P)=Zd 2h(t)ZdNXi=1rvi((viu)h(t))f(P): Integrationbypartsthengivesnally Sincef(P)isarbitrarythisimpliesthattheHCSdistributionissharpwithrespecttothetotalmomentum Insummary,thecharacteristicensembleofanisolatedhomogeneousgranularuidistheHCSensemble.Itisatimedependentnon-equilibriumensemblethathasitsentiretimedependencethroughthecoolingtemperature.Thecharacteristictimescaleassociatedwithitisthecoolingrateh:Also,thisensembleissharpwithrespecttothetotalmomentum,i.e.,eachtrajectoryinthisensembleoriginatesfrominitialpointsthatallhavethesamemomentumP.Thiscompletesthecharacterizationofthehomogeneousreferencestateofthissystem.ThisHCSisgoingtoplaythesameroleinthestatisticalmechanicsofthegranularuidthattheequilibriumGibbsstatedoesforanormaluid,aswillbeseeninthesubsequentchapters.

PAGE 33

newtimescaleinsuchawaythatthehomogeneouscoolingstateisthestationarystateofthisdynamics.Toseethis,deneasetofdimensionlessvariables And,foranydistribution,deneanassociateddimensionlessformas IntermsofthesevariablesthedimensionlessLiouvilleequationbecomes withthenewgeneratorforLiouvilledynamics 2hNXi=1rVi(Vi); L=` vh(t) Itisreadilyveriedthat 2{29 )and( 2{34 )tobe 2`ht;vh(t)=e1 2hsvh(0):(2{38) ThisdimensionlessformfortheLiouvilleequation,( 2{36 ),supportstheHCSasastationarysolution Consequently,inthefollowingitisreferredtohereasthestationaryrepresentationoftheLiouvilleequation.Thisrepresentationessentiallyinvolvespartitioningthedynamicsofanydistributionintotwoparts,oneduetothecoolingTh(t)andtheotheriseverythingelse.ThehomogeneouscoolingofthetemperatureisgeneratedbythescalingoperatorS=1 2hPNi=1rVi(Vi).Since,inthe

PAGE 34

HCSthisistheonlydynamics,itisthestationarystateofthedynamicsinthisrepresentation.TheusefulnessofthisstationaryrepresentationliesinthedenitionoftimecorrelationfunctionsovertheHCSensemble,whichisconsideredbelow. ForobservablesAandBandamacrostate,thetwo-timecorrelationfunctionisoftheform Rewritingtheaboveexpressionintermsofthegeneratorsidentiedearliergives, orequivalently,usingtheadjointrepresentationofthedynamics L(()B()):(2{42) Moregenerally,correlationfunctionsatthreetimescanbedenedbuttherepresentationintermsofgeneratorscanbeambiguous.Forexample,itiseasilyseenthatL(AB)6=(LA)B+A(LB)andconsequently Inthecaseofelasticsystems,whentimecorrelationfunctionsovertheequilibriumstateareconsidered,timetranslationinvarianceandthestationarityoftheGibbsstateareusedtomapthethree-timeproblemontooneofcalculatingatwotimecorrelationfunctionas ItturnsoutthatthestationaryrepresentationforthedynamicsdescribedaboveallowsthesamemanipulationstobecarriedoutinthecaseoftimecorrelationfunctionsovertheHCSensemble.TheresultsarestatedhereandthedetailsofobtainingtheseresultsaregiveninAppendixB.

PAGE 35

Firstconsidertwo-timecorrelationfunctionsovertheHCS.Thesecanbewrittenas L(h()B())=cB(v(0))ZdA(f`qi;v(t)vig)es (2{45) InthelastequalityachangeofvariablestothedimensionlessformhasbeenmadeandcA;BaretheappropriatefunctionsofthescalingvariablesthatarerequiredtomaketheobservablesAandBdimensionless.Asidefromapossibleexplicitdependenceontimethroughv(t);dependingonthespecicchoiceoffunctionA,thetimedependenceisnowgivenbytheLiouvilledynamicsinthestationaryrepresentation.Here,sisunderstoodtobes(t;0).Forhomogeneousfunctionsofvelocityaswillbeconsideredintherestofthepresentation, forsomeaandbandhence 2asva+b(0);(2{47) whichamountstoanadditionalsimpletimedependenceassociatedwiththecooling.Also,thereexistsanequivalentrepresentationforthestationarystatecorrelationfunctiongiveninEq.( 2{45 )aboveas where 2hNXi=1VirVi(2{49)

PAGE 36

Further,threetimecorrelationfunctionscanbewrittenas wherethesisnowunderstoodtobes(t;t0):ThederivationoftheseresultsisgiveninAppendixB.Itcanbeinferredfromtheresultsabovethatitissucienttoconsideraclassofdimensionlesscorrelationfunctionsoftheform (2{51) =ZdA()es Finally,aspecialpropertyofhardspheredistributionscanbeusedtotoextractthe\backwarddynamics"generatorforphasefunctions.ItisshowninAppendixAthatforanyhardspheredistribution(i.e.,onewithanexcludedvolumefactormultiplyingit),thereexistsanoperatorLsuchthat,foranyphasefunctionB; L(B)=( wherethenewoperatorLisfortheform 2NXi=1NXj6=iT(i;j)(2{54) InthedimensionlessvariablesEq.( 2{53 )becomes where 2hNXi=1virvi;L=` vh(t)L:(2{57)

PAGE 37

Inparticular,whenthedistributionistheHCSh,therstterminEq.( 2{56 )vanishes.Usingthisfactinthestationaryrepresentationfortimecorrelationfunctionsgives Thisrepresentationwillproveusefulwhencomparisonwithresultsobtainedforelastichardspheresissoughtinthesubsequentsections. Insummary,inthissection,anewrepresentationforthedynamicsofthesystemhasbeenidentiedsuchthatthereferencehomogeneousensemble,namelytheHCSisstationaryinthisrepresentation.Further,thishasbeenextendedtotimecorrelationfunctionsovertheHCSsothattheybecomestationarystatetimecorrelationfunctions.Thisisatechnicalsimplicationthatallowsatimedependentnon-equilibriumproblemtobemappedontoastationarystateproblem.Thismappingallowsforseveralformalmanipulationsofthetimecorrelationfunctionsthatwouldotherwisebecomplicatedandalsofacilitatescomparisonwithknownresultsfornormaluids.Thisconcludesthesettingupofthemachineryrequiredtodonon-equilibriumstatisticalmechanicsforagranularuid. 55 ]anditsmanifestationintheformofuctuationdissipationrelationshipsbetweentimecorrelationfunctionsandresponsefunctions.Therehasbeenconsiderableattentiongiventosuchuctuationdissipationrelationsinthecontextofgranularuidsintherecentliterature[ 56 57 58 59 60 ].Buttheconventionalresultsassociatedwiththeuctuationdissipationtheoremareinherentlytiedtothespecialpropertiesassociatedwith

PAGE 38

PAGE 39

thatcouplesintothemicroscopicdensityfunction,forexampleagravitationalforce. Theinitialpreparationmentionedabovegivesamacrostateoftheform where^y(r)isanasyetunspeciedphasefunctionandthedenotesitsdeviationfromitsaveragevalueinthe0statewhichguaranteesthat i.e.,theinitialpreparationisproperlynormalizedand whichensuresthat i.e.,theinitialpreparationindeedhastheprescribednumberdensityeld. TheexternalforceentersasacorrectiontotheLiouvilleoperatorintheform where with^n(r)=Pi(rqi)andf;gbeingthePoissonbracket @qi@B @pi@A @pi@B @qi:(2{66)

PAGE 40

TheperturbingpartoftheLiouvilleoperatorcanberewrittenas @pi!=ZdrF(r)Xi@ @pi(rqi) (2{67) SolvingtheLiouvilleequation torstorderinperturbationintheexternalforce,withtheinitialconditiongivenbyEq.( 2{60 )above,resultsin wherethestationarityofthe0statewithrespecttotheunperturbedLiouvilleoperatorhasbeenused.Nowconsidertheaverageofthenumberdensityinthisstate, (2{70) where

PAGE 41

isthecorrelationfunctionthatcharacterizestherelaxationofthesystemduetoaninitialperturbationand istheresponsefunctionthatcharacterizestheresponseofthesystemtoanexternalforce.Thisisthemostgeneralformthatcanbeobtainedforageneralhomogeneousstationarystate. NoticethattherstpartoftheOnsagerregressionhypothesiscomesoutnaturallyfromthelinearresponseprocedureifrestatedastherelaxationfromaninitialperturbationrelaxesinthesamewayasaspontaneousuctuationinthereferencestate,fortheresponsetosuchaperturbationisgivenbyatimecorrelationfunctioninthereferencestate.But,therelationshipbetweencorrelationfunctionsandresponsefunctionisnotsostraightforward.Inordertoseethis,rstconsidertheabovetwoexpressionswhenthereferencestateistheequilibriumstate.Firstnoticethatifthephasefunction^yischosentobe^nthenusingtheconservationlawforthenumberdensity,then,g1justbecomestheinverseofstaticstructurefactorSinthereferencestate0(seeAppendixH)andtheresponseinthedensitytakestheform (2{73) Or,equivalently,thiscanbegivenaFourierrepresentationas Further,when0=eqthen,

PAGE 42

andonecanidentifythesecondtermintheresponsefunctionas where^|n(r;t)isthenumberuxinthecontinuityequation Hencetheuctuationdissipationrelationshipisrecoveredinthefamiliarform ThepointoftheabovedemonstrationisthatthefamiliarformoftheuctuationdissipationrelationshiphingesonthefactthattheGibbsstateisoftheformeHwhereHisthegeneratorofthedynamicsofthesystem.Andhencethisparticularformoftherelationshipdoesnotingeneralapplyforotherstationarystatesofasystem.Further,otherformsoftheuctuationdissipationrelationshipknowntoholdforequilibriumsystems,likethatrelatingthediusioncoecienttothemobilitycoecientareknowntotakedierentformsforgranularuids[ 61 ].Hence,eventhoughalargebodyofthemethodsdevelopedforthestudyofnormaluidscanbeextendedandfruitfullyappliedtogranularuids,resultsobtainedfromthesetoolscannotbeborrowedandusedwithoutcriticalexaminationofthederivationassociatedwiththem. Inwhatfollows,attentionisrestrictedtotheuseofthesetoolsestablishedaboveforthespecicproblemofderivingaclosedhydrodynamicdescriptionforagranularuidandhenceobtainingexpressionsforthetransportcoecientsassociatedwithsuchadescriptionintermsoftimecorrelationfunctions.TheseexpressionsareexactandstillhavetheNbodyprobleminthem,butatthesametimeareamenabletocontrolledanalyticapproximationsandspecicnumericalevaluations.

PAGE 43

PAGE 44

PAGE 45

thersttwoeldsarelocallyconservedquantities,buttheenergydensityisnot.Thestartingpointforderivingsuchadescriptionarethemacroscopicbalanceequationsassociatedwiththeseeldsgivenby (3{2) (3{3) (3{4) wherehijisthemomentumux,sistheenergyuxandw(r;t)isthefunctioncharacterizingthelocallossinenergyduetothedissipativenatureofthecollisionbetweenthegranularparticles.Fromapurelymacroscopicpointofview,theseequationsaretakentodenetheuxesandthesourceterm.Hencethespecicationoftheuxesisarbitraryuptotermsofzerogradientandtheseparationintosourceanduxintheenergyequationisnotunique.ButiftheseequationsareviewedwithinthecontextofstatisticalmechanicsasensembleaveragesofthecorrespondingmicroscopicconservationlawsgiveninTableC-1inAppendixC,thentheuxesandthesourceareunambiguouslydened.Thiswillbetheviewpointtakenhere. AsfornormaluidsitisusualtoreplacethemomentumdensitybytheowvelocityU(r;t)denedthroughtherelation IfalocalrestframeisdenedfortheuidateachpointrthroughaGalileantransformationr!rU(r;t)t,then,atemperatureT(r;t)isdenedusingtheenergydensityoftheuidinthelocalrestframethroughthedenition 2mn(r;t)U2(r;t)e0(n(r;t);T(r;t))(3{6)

PAGE 46

Heree0(n;T)issomespeciedfunctionofnandT.Thetwomostcommonchoicesaree0(n;T)=3nT=2,ore0(n;T)=ee(n;T)thethermodynamicfunctionforthecorrespondingequilibriumuid.Theformeriscommoninapplicationsofcomputersimulations,whilethelatteristhehistoricalchoiceinmostformulationsofhydrodynamics.Forbothnormalandgranularuids,thechoicemadeconstitutesadenitionoftemperaturefornon-equilibriumstatesandhasnoapriorithermodynamicimplications.Withthesedenitions,theabovebalanceequationscanberecastintheform whereDt=@t+Uristhematerialderivativethatisthetimederivativeinthelocalrestframe,Pijisthemomentumuxinthisreferenceframe,calledthepressuretensor andqisthecorrespondingheatux,identiedas 2nmU2UiPijUj:(3{11) Further,intherestofthepresentation,thechoicee0(r;t)3 2n(r;t)T(r;t)ismade.Thisgivesthetemperatureequationintheform (Dt+)T+2 3n(Pij@jUi+rq)=0;(3{12) withthecoolingratedenedthroughtherelation 3n(r;t)w(r;t):(3{13)

PAGE 47

Theseequationshavethesameformasthoseforanormaluid,exceptforthecoolingratethatrepresentsthedissipativenatureoftheinteractionbetweentheparticlesofthisuid. Hydrodynamicsisacloseddescriptionintermsofthehydrodynamicelds.Butinspectionofthemicroscopicanaloguesoftheseuxestellsusthattheselocalrestframeuxesareingeneralcomplicatedfunctions.Theclosureoftheabovebalanceequationswithrespecttothehydrodynamiceldsismadebyassumingthattheuxesbecomenormalfunctionalsofthehydrodynamiceldsafterashortagingtime,i.e.,thepressuretensorPtakestheform anditdependsonspaceandtimeonlythroughthehydrodynamicelds.Similarformsarepostulatedfortheheatuxandthecoolingrateaswell.ThisclosureassumptioneectivelyconvertsthebalanceequationEqs.( 3{7 )-( 3{8 )intohydrodynamicequationsfortheveeldschosentobethehydrodynamicvariablesofinterest.Thisassumptionisthephenomenologyinvolvedinthisderivationofhydrodynamics. Onecangiveaphysicalinterpretationforthebasisofthisassumptionasfollows.Considerthegranularuidinaninhomogeneousstate.Visualizethesystemasbeingpartitionedintocellssuchthateachcellhasenoughparticlessothatmultiplecollisionscanoccurintimesshortcomparedtothetimetakenbyatypicalparticletotraversethesizeofthecell,butthesizeofthecellissmallcomparedtothewholesystem.Now,noticethatthehydrodynamicvariablesherearethecoursegrainedversionsofpreciselythosemicroscopicquantitiesthatprescribethecollisionrulebetweengrains.Henceastatisticalaverageofthesequantitiesoveronlytheparticleswithinthecellwillnotchangeexceptthroughboundaryeectsthatincludecollisionswithparticlesacrossthedeningwallsof

PAGE 48

PAGE 49

PAGE 50

rate0.Considerthecoolingequationinparticular.Noticethattheunderlyinguidisoneofhardspheres.Thisisasystemwithnocharacteristicinternalenergyscale.So,theonlyenergyscaleintheproblemisthetemperature.This,togetherwiththefactthatthecoolingrateisanormalfunctionsoftime,leadstothefactthat,ondimensionalgrounds, Eq.( 3{19 )isthemacroscopicanalogueofEq.( 2{28 )inthepreviouschapter.HencethesolutiontoEq.( 3{18 )aboveisindeedtheHomogeneousCoolingSolutionconsideredinthepreviouschapterinthecontextofstatisticalmechanics.ThishomogeneousdynamicsofthetemperatureisthesameasidentiedinEq.( 2{29 )inthepreviouschapter 2`ht2:(3{20) Thusthehomogeneouscoolingstateisthesolutiontothehomogeneoushydrodynamicequations.Thisistherstdistinctionfromthehydrodynamicsofnormaluidsinthatthehomogeneousreferencestatethatemergesisinherentlytimedependent. Next,theequations( 3{7 )through( 3{9 )arelinearizedaboutthishomogeneousstate.Withoutlossofgenerality,picktheconstanthomogeneousowvelocityUhabovetobezero.Forthispurpose,introducedimensionlessvariablesoftheform ThedimensionlessspaceandtimescalesarethesameasthoseintroducedinthepreviouschapterthroughEq.( 2{34 ), InthisformthecoecientsinthelinearizedNavier-Stokesequationsareconstants,independentofspaceandtime.Thesubsequentanalysisismostconvenientlydone

PAGE 51

intermsofaFourierrepresentationwithrespecttospace wherethesetfygarechosentobe Herebe1andbe2areunitvectorsorthogonaltoeachotherandtobk,deningthetransverseoweldcomponents.ThelinearizedNavier-Stokesequationsarethenidentiedas wherethetransportmatrixKhydisfoundtobeblockdiagonalwitha\longitudinal"partcorrespondingto;=1;2;3;givenby 3p+Uik1 2p@lnph 2pikh 3+k21CCCCA:(3{26) The\transverse"componentsdecouplefromthelongitudinaldegreesoffreedomintheaboveequationandaregivenby Thedimensionlesstransportcoecientsaredenedby vhh;U=U;T=Th ThiscompletesthederivationofthelinearizedNavier-Stokesequationsforsmalldeviationsfromthehomogeneousstate.Noticethatthetransportcoecients

PAGE 52

PAGE 53

identiedaboveisthedynamicsassociatedwithhomogeneousperturbationsofthereferencestate.Thisisatrivialdynamicsinthesensethatitisjustthecoolingofthenewhomogeneousstateasdescribedinthevariablesinwhichtheoriginalreferencehomogeneousstateisstationary.Furthernoticethatinthecaseofthehardspheregranularuidconsideredhere,theanalogueoftheproperty@t/rrthatcharacterizesthehydrodynamicvariablesfornormaluidsis@sI+Khyd(0)/rr,i.e.,thehydrodynamicvariablesherearethosewhosecharacteristictimeofapproachtothisresidualdynamicsdivergesinthelongwavelengthlimit.Itisinthissensethatthetemperaturecanbetreatedonthesamefootingastheowvelocityandthenumberdensity. Next,furtherunderstandingofthecontentofthehydrodynamicdescriptionandthefactthatthehydrodynamicresponseofagranularuidisverydierentfromthatofanelasticuidcanbeobtainedbyconsideringthehydrodynamicmodesforthissystem.Theeigenvaluesandeigenvectorsofthegeneratorforthisdynamics,Khyd,denestheveNavier-Stokesorderhydrodynamicmodes TheeigenvaluesofthematrixKhydaredeterminedbythecubicequation324 3+k2+(T)k22h 3+k21 2p2 3p+Uk21 2p@lnp @lnn+k2ph1 2@lnh 4@lnp @lnn=0 (3{32) andthedecoupledshearmodesaresolutionstotheequation

PAGE 54

Ifthelimit!1istakenforthisequation,then,itssolutiontoorderk2givethefamiliarhydrodynamicmodesassociatedwithnormaluids,namelythetwopropagatingsoundmodes,theheatmodeandthetwotransverseshearmode[ 37 ].But,whenthesolutiontotheaboveequationisconsideredfor6=1,toorderk2theyare @lnnhlnph 2hT+p 2hp @lnnhk2 +4 3+k2; (4;5)(k)=1 2h+1 2k2(3{37) Noticethatalltheeigenvaluesarerealandhencetherearenopropagatingmodesinthesystem,andthe!1limitofthesemodesdonotcorrespondtothefamiliarhydrodynamicmodesofanormaluid.Thedrasticdierenceinthenatureofthehydrodynamicmodesobtainedastheelasticlimitoftheaboveeigenvaluesisduetothenon-analyticityoftheeigenvaluesandeigenvectorsaboutthepoint=1andk=0.Closetotheelasticlimit,hwhichgoesas(12)(seeEq.( 2{24 )inthepreviouschapter)andkaresmallparametersandthetypeofmodesobtaineddependsonhowtheseparametersapproachzero[ 26 ].Thisisanindicationofthefactthattheinelasticity,evenwhensmall,givesrisetodrasticallydierenttransportintheuid.But,forthepurposesathand,attentionisrestrictedtothe6=1formsofthesemodes. NextobservethatthereexistsacriticalwavelengthkcSdenedby

PAGE 55

suchthatfork
PAGE 56

perturbationatconstanttemperatureanddensity.Thelasttwoaretheresponsetoatransversevelocityperturbation,againatconstanttemperatureanddensity. Thisconcludesouranalysisoftheresponseofthehomogeneousstatetosmallspatialperturbations.Recallthattheunknownquantitiesinthehydrodynamicequationsabovearethepressurep,thecoolingrate,thetransportcoecientsshearviscosity,thebulkviscosity,thethermalconductivity,thecoecient,andthesourcetransportcoecientsU,Tandn.Thesubsequentchaptersfocusonidentifyingthesetransportcoecientsexactlyfromtheunderlyingmicro-dynamicsintheformoftimecorrelationfunctionsovertheHCSensemble.Further,notethatalthoughthisisdoneinthecontextoflinearhydrodynamics,thesetransportcoecientsarethesamefunctionsofthedensityandtemperatureinthenonlinearequationsandhencetheresultscanbeusedinthenonlinearequationsaswell.

PAGE 57

Hydrodynamictransportcoecientsarethosequantitiesthatcharacterizetheresponseoftheuidtogradientsinthehydrodynamiceldsinternaltotheuid.Fornormaluids,themethodoflinearresponsehasbeensuccessfullyusedtogetexactexpressionsforthesetransportcoecientsstartingfromthemicroscopictheory[ 37 ]intheformoftimecorrelationfunctionsoverthehomogeneousreferencestate.Thus,non-equilibriumstatisticalmechanicsallowstheidenticationoftransportcoecientsintermsofthetimedependenceofuctuationsintheequilibriumensemblejustasequilibriumstatisticalmechanicsallowstheidenticationofthermodynamicquantitiessuchasthespecicheatintermsofthe\amplitude"ofstaticuctuationsintheequilibriumensemble.Forexample,theviscosityofauidisdeterminedbythetimeintegraloftheautocorrelationfunctionofthemomentumuxatatimetwiththemomentumuxatthetimet=0(seeEq.( 1{1 )earlier):ThekeyideathatmakessuchidenticationspossibleisOnsager'shypothesisthatperturbationsrelaxinthesamewayasaspontaneousuctuationinthesystem. AswasshowninChapter2earlier,theOnsagerregressionhypothesis,whensuitablyreformulated,worksforgranularuidsaswell.Hence,itisexpectedthatthereexistrepresentationsforthehydrodynamictransportcoecientsofagranularuidintermsoftimecorrelationfunctionsofuxesoverthereferencestate.Identifyingthegeneralmethodforobtainingsuchanexpressionforthetransportcoecientsistheobjectiveofthischapter. 48

PAGE 58

Thestructureofthepresentationisasfollows.First,theformalprocedureofusinglinearresponsetoextractthehydrodynamicresponseofauidandhenceobtainamicroscopicrepresentationforthetransportcoecientsisoutlined.Then,sometechnicalsimplicationsthatrenderthisformalprocedureanalyticallytractablearepointedout,intheformofidentifyingaspecialinitialpreparationthatexcitesonlythehydrodynamicmodesinthelongwavelengthlimit.Finally,aformalexpressionforthehydrodynamictransportmatrixthatwasidentiedonphenomenologicalgroundsinthepreviouschapterisgivenintheformofamatrixoftimecorrelationfunctions.Furtherunfoldingofthismatrixtoidentifythedetailedformsofeachofthetransportcoecientsisreserveduntilalaterchapterinthepresentation. 2`ht2:(4{1) ThisistheHomogeneousCoolingState.TheNparticlestatisticalensemblethatcorrespondstothismacrostateistheHCSensembleidentiedandcharacterizedinChapter2inEqs.( 2{25 )-( 2{33 ).Next,notethatthelinearizedhydrodynamicequationsgivenbyEq.( 3{25 )characterizetheresponseoftheuidtoweakinhomogeneitiesinthehydrodynamiceldswithrespecttotheHCS.Thisresponsecanbecapturedatthelevelofstatisticalmechanicsbythefollowingprocedure.Lettheinitialensemblebeaweaklyinhomogeneousensemble,withits

PAGE 59

inhomogeneitiesparameterizedbythehydrodynamicelds,thatis with where isthedeviationofquantityxfromitsvalueintheHCS.Thusthey'sarepreciselythevariableswhosedynamicswasgiveninEq.( 3{25 ).Further,thefunctionsb'saresuchthattheinitialdistributionisnormalizedanddoesindeedhavetheprescribedhydrodynamicelds.Thatis, Therstconditionabovemakesproperlynormalized.Inthesecondconditionabove,thea'saregivenby where 2mp2i+1 2Pj6=iu(qij)pi1CCCCA(rqi);(4{7) arerespectivelythemicroscopicnumberdensity,momentumdensityandenergydensity,and withe0beingthechosenfunctionthatdenesthetemperatureforthesystem(seeEq.( 3{6 )inChapter3earlier).Thesearepreciselythephasefunctionswhose

PAGE 60

ensembleaveragearethelinearizedhydrodynamiceldsy's,thatis, SothesecondconditioninEq.( 4{5 )guaranteestherequirementthattheinitialensembledoesindeedhavetheprescribedvaluesforthehydrodynamicelds.Further,aswiththehydrodynamicequationsearlier,thechoicee0=3 2nTismadetogive 3nhbe(;r)3 2Thbn(;r);1 Thespecicchoiceforfb(r)gisleftarbitraryatthispoint. AtalatertimettheensembleisobtainedasaformalsolutiontotheLiouvilleequationEq.( 2{17 ),intheform Nowtakingaverageswiththesetoffunctionsfagovertheaboveensemble,aresponseequationisobtainedintheform withtheresponsefunctionCbeinggivenby TheresponsefunctionisatimecorrelationfunctionovertheHCSensemble,liketheonesdenedandcharacterizedinChapter2. Beforefurtheranalysis,itisusefulrewritetheaboveequationinthestationaryrepresentationintroducedearlierinEq.( 2{34 )bytransformingtothedimensionlessvariables

PAGE 61

Thehydrodynamiceldsarenon-dimensionalizedwiththeirvaluesintheHCSensemble Thephasefunctionsfa(r)gthataretheobservablesofinterestinthedimensionlessformare 3be3 2bn;bg;(4{16) wherethesuperscriptdenotesthenon-dimensionalquantitywith (bn;be;bg)=1 Also,itisusefultotakeadvantageofthetranslationalinvarianceinheritedfromthehomogeneityoftheHCStointroduceaFourierrepresentationthroughthedenition Usingthese,Eq.( 4{12 )abovecanberewrittenas with Lseb(k):(4{20) whereexplicitusehasbeenmadeofthefactthatthetimecorrelationfunctionCisonlybeafunctionofrr0;duetothehomogeneityoftheHCS.Thisdimensionlessresponseequation( 4{19 )willbethefocusofstudyintherestofthischapter. Toidentifythemacroscopichydrodynamicequationsfromthisexpression,itisusefulrsttorewritetheresponseequation( 4{19 )intheformofatransport

PAGE 62

equationsimilarto( 3{25 )inthepreviouschapter, (@s+K(k;s))ey(k;s)=0;;=1:::5:(4{21) AformalexpressionforK(k;s)isreadilyidentiedfrom( 4{19 )as Asarguedinthepreviouschapter,oneexpectsthatthehydrodynamicdescriptionidentiedinEq.( 3{25 )isthecompletedescriptionofthedynamicsofthesysteminthelimitofwavelengthslongcomparedtothemeanfreepathandattimeslongcomparedtothemeanfreetimeoftheparticlesintheuid.Hence,thehydrodynamicmatrixgiveninEqs.( 3{26 )and( 3{27 ),whenitexists,followsfromthisformalresultforsmallk(longwavelengths)andlongtimes, (4{23) =lims!1K(0;s)+ikbkrkK(k;s)k=0+::: Comparisonofthisexpressionwiththeforms( 3{26 )and( 3{27 )notonlyprovidesa\derivation"ofthelinearhydrodynamicequations,butalsogivesthecoecientsofthoseequationsintermsoftheresponsefunctions. Thatcompletestheformalprescriptionofextractingthelinearhydrodynamictransportmatrixfromthefullmicroscopicdynamics.Notethattheabovederivationofhydrodynamicshasbeenaccomplishedwithoutanyconstraintsonthefunctionsbthatcharacterizetheperturbationchosenintheinitialstate,beyondnormalizationandthemomentconditionsassociatedwiththehydrodynamicvariables.ItturnsoutthatthetractabilitytoanalyticalandnumericalanalysisoftheresultsobtainedinEq.( 4{24 )aboverestsonusingthedegreeoffreedomaordedbytheexibilityinthechoiceofbtosimplifytheprocessofthelongwavelengthexpansiondescribedearlier.Inthefollowingsection,aspecialchoice

PAGE 63

ofb'sismadeandthesimplifyingpropertiesofthischoiceintheanalysisofthetransportmatrixpointedout. 37 ].Thissectionidentiestheanalogoussimplifyinginitialpreparationforagranularuid.Thisisdoneinthefollowingway.First,aspecialhomogeneoussolutiontotheLiouvilleequationisidentiedthatisrelatedtotheHCSinasimplemanner.Then,thishomogeneoussolutionisgeneralizedforweakinhomogeneitiesandthespecialinitialpreparationrequiredforthelinearresponseproceduretobeappliedtoagranularuidisidentied.Also,theparticularsimplifyingfeaturesofsuchachoiceareelucidated. where 2hrvmvm+ ThisequationisusedinAppendixDtoidentifyasetoffunctions()suchthattheyhavetheproperty whereKhydT(0)isthetransposeofthegeneratorofthehydrodynamicequationsinthehomogeneouslimitidentiedinEq.( 3{30 )and'sarethefunctions (;s)=(`vh(t))3Nyh(t)@h(;t)

PAGE 64

PAGE 65

4{2 )ischosensothatinthehomogeneouslimit,itreducestothespecialsolutiontotheLiouvilleequationcharacterizedinEq.( 4{30 )above.Then,thesimplifyingpropertiesofsuchachoiceareenumerated.Forthispurpose,rstdeneaninitiallocalHCSdistribution.ThisdistributionissimilartothelocalequilibriumdistributionforelasticcollisionsandrepresentsasystemthathastheHCSformlocallybutwithapointwisevaryingtemperature,density,andowvelocityelds.Formally,thelocalHCSisobtainedfromtheHCSofEq.( 2{27 )inChapter2bythetransformation (`vh)3N!NYi=1(`vh(qi))3;viU so wherethedependenceonaninhomogeneousdensityeldn(qi;0)hasnowbeenmadeexplicit.Clearly,theinitialHCSisrecoveredforspatiallyhomogeneoustemperature,density,andowelds.Then,ifthechoicefortheb'sinEq.( 4{2 )ismadesothat then,inthehomogeneouslimitthiswillcorrespondtothechoicethatgivesthespecialsolutiontotheLiouvilleequationcharacterizedabove,thatis Supposethischoiceismade,thenthetransportmatrixdenedinEq.( 4{22 )abovebecomes,

PAGE 66

with Lse(k):(4{39) Thisistheformofthetransportmatrixandresponsefunctionsthatwillbeconsideredintherestofthepresentation. 4{30 )andEq.( 4{37 )abovethatatk=0thetransportmatrixinEq.( 4{38 )reducesto thatis,thetransportmatrixispurelyhydrodynamicatalltimesinthehomogeneouslimit. Nextobservethat,inordertogofromthefulltransportmatrixtothehydrodynamictransportmatrix,twolimitshavetobecarriedout,onethatoflongwavelengths(i.e.,k!0)andthatoflongtimes(i.e.,s!1)asstatedinEq.( 4{23 )earlier.Thespecialinitialpreparationissuchthatthetimederivativeofboththefunctionsea(k)ande(k)haveanintrinsickorderingthatcanbeusedtokorderthetransportmatrixsothatthelongtimelimitcanbetakeninasimplemanner.Thiscanbeseenasfollows.First,itisshowninAppendixDthatthephasefunctionsea(k)obeyadynamicalequationoftheform with 3s^g;h;(4{42)

PAGE 67

where^gisthemomentumdensityandhencethenumberux,sistheheatux,histhemicroscopicmomentumuxandlisgivenby 3ew(k;s)+3 2h2 3bee(k;s)ben(k;s)+h@lnh Intheaboveequation,ewisthedimensionlessformofthesourceintheenergybalanceequationgiveninTableC-2inAppendixC.AsshowninAppendixD,thehomogeneouspartofthesourcetermhastheproperty 3ew(0):(4{44) TheoperatorPintheaboveequationisaprojectionontothesetoffunctionsneo'sgivenby (4{45) Thus,thesourcetermisorthogonaltothehomogeneouspartoftheinitialpreparation.So,ifthisbalanceequationisusedtoobtainadynamicalequationforthecorrelationfunctioneCthen, with and

PAGE 68

PAGE 69

onboththedirectandadjointfunctionsturnsouttobeproportionaltoikandhencecanbeusedtoexposethenaturalkorderingpresentinthetransportmatrixpriortothehydrodynamiclimitbeingtakenand3)thisinitialpreparationmakesthelongtimelimitofthekexpandedtransportmatrixpatentlywelldenedforallthetimedependentquantitiesturnouttobeorthogonaltothegeneratorofthedynamics.Thissectionisconcludedbyremarkingthatthesearepreciselythepropertiesthatareassociatedwiththelocalequilibriumpreparationofnormaluidsthatlendsthetransportmatrixinthehydrodynamiclimittobeexpressedinatractableform. 4{38 )above.Here,threerepresentationsforthevarioushydrodynamicparametersaregiven.Eachrepresentationisformallyequivalenttotheothertwo.Eachinturnissuitableforeitherinterpretationofstructureandcontent,developingdierentanalyticalapproximationschemesornumericalevaluationschemes.Inordertosimplifythenotationandexposetherelevantstructurewithoutdistractions,alltheGreekindicesassociatedwiththedierenthydrodynamiceldsaresuppressedinthefollowing. Asarststep,themostdirectmethodofcarryingoutthekexpansionisgiven.Forthispurpose,dene

PAGE 70

and whereeC(1),K(1),andsoondependontheunitvectorinthelongitudinaldirection^k,butthisdependenceisleftimplicithereinordertosimplifythenotation.AshasbeenestablishedaboveK(0)=Khyd(0)atalltimes.Thenthetransportmatrix,toEulerandNavier-Stokesorderturnsouttobe,respectively, whereexplicitusehasbeenmadeoftheorderk=0results.ThematrixelementsinEq.( 4{54 )containtheexpressionforthehydrodynamicparametersthatincludethepressurepandthetransportcoecientatEulerorder,namelyU.ThematrixelementsinEq.( 4{55 )containtheexpressionsfortheshearviscosity,bulkviscosity,andthetransportcoecientsassociatedwithheattransport,namely,,nandT.Theseareformalexpressionsdenedinturnthroughtermsinthekexpansionoftheelementsinthebi-orthogonalsetthatconsistsofthefunctionsfag'sandfg'sdenedearlier.TheaboverouteisthemostdirectwaytoobtainexplicitformsfortheelementsofthetransportmatrixupthroughtheNavier-StokesorderandresultinHelfandformsforthevarioustransportcoecientsinaformmostsuitablefornumericalevaluation.But,theseexpressionsarenottransparentwithrespecttotheformalstructureoftheresult.Inordertoillustratethecontentoftheseexpressionsandinterpretthestructure,analternateroutethatmakesexplicituseoftheconservationlawsandbalanceequationsassociatedwiththesevariablesisdescribedbelow. Thekeyideaincarryingoutakexpansionofthetransportmatrixinsuchawaythattheresultsareamenabletointerpretationandtheoreticalanalysisintermsofapproximateevaluationsisthefollowingrecognition.Thehydrodynamic

PAGE 71

variablesinthesystemarethosewhichhavethepropertythat@sI+Khyd(0)/ik.Themicroscopicprecursorsofthehydrodynamicelds,namelythephasefunctionsfag'sandtheadjointfunctionsneo'salsohavethispropertywithinthecorrelationfunctionsassociatedwiththespecialinitialpreparationdetailedabove.NoticethattheexpressionforthetransportmatrixinEq.( 4{38 )hasonetimederivative.Hence,theprocedureusedtokorderthetransportmatrixwouldbetointroduceasecondtimederivativeusinganidentityoftheform onthetimedependentquantitiesinK,andeliminateeachofthetimederivativesinfavorofagradientusingthespecialpropertiesassociatedwiththebi-orthogonalsetfag'sandfg'sthatoccurintheresponsefunctioneC.Detailsofcarryingoutthisprescriptionareoutlinedbelow. Asarststep,onecanuseEq.( 4{46 )abovetorewritethetransportmatrixinEq.( 4{38 )intheform whichessentiallyinvolvesusingthedirectconservationlawstoevaluatethersttimederivative.Notethatthek=0formofthetransportmatrixhasbeenextracted.Beforeproceedingfurther,itturnsouttobeusefultointroduceanintegratingfactorexpKhydT(0)sintheadjointconservationequationthatabsorbsthek=0dynamicsofe,thatis,dene Then,Eq.( 4{50 )abovebecomes

PAGE 72

with Next,Eq.( 4{39 )aboveisreexpressedintermsofcorrelationfunctionsinvolving where Lse(k)eKhyd(0)s=1 Similarly, and Further,usingformoftheadjointconservationlawsinEq.( 4{59 )aboveforthedynamicsof where

PAGE 73

NowconsiderthetransportmatrixasgiveninEq.( 4{61 ).UseanidentityoftheforminEq.( 4{56 )towrite andsimilarlyfor 4{65 )-( 4{67 )toeliminatethesetimederivativestoobtainthetransportmatrixas Thiscompletestheformalimplementationoftheprescriptiondescribedearlierwhichexposestheinherentkorderpresentinthefulltransportmatrix.Noticethatanyperturbativeexpansionisyettobedone,andallthemanipulationsdonesofarareexact.ThisintrinsickorderingispresentinKduetothespecialnatureofthedynamicsofthebi-orthogonalsetfag'sandfg'sasmanifestedbythedirectandadjointconservationlawsinEq.( 4{41 )andEq.( 4{59 )above. Intheremainderofthischapter,theexpression( 4{72 )isconsideredexplicitlyatEulerorderandatNavier-Stokesordertoidentifythevariousparametersinthehydrodynamicmatrixandtoelucidatethestructureoftheresult.Also,intheprocesstwonewformsforthehydrodynamicparametersareidentiedthatareequivalenttotheformsinEq.( 4{54 )andEq.( 4{55 )earlier. Inordertosimplifythepresentationoftheresultsthatfollow,introducethenotation

PAGE 74

wherethesuperscriptFdenotesthetermscomingfromtheuxesinthedirectconservationlawsandhencecorrespondtotheparametersintheconstitutiverelationsfortheuxesgiveninEqs.( 3{15 )and( 3{16 )inChapter3,namelythepressure,theshearandbulkviscosity,thethermalconductivityandthecoecientintheheatuxterm.ThesuperscriptSdenotesthetermscomingfromthesourceinthedirectconservationequations( 4{41 )andhencecorrespondtotheparametersintheconstitutiverelationinEq.( 3{17 ),namelythetransportcoecientsU,nandT.Eachoftheabovepartsofthetransportmatrixareconsideredinturn,rstatEulerorderandthenatNavier-Stokesorder,andthestructureoftheresultsobtainedisdiscussed. 4{72 ),itcanberecognizedthattheuxpartofthetransportmatrixKFatEulerorderis (4{74) Inordertobetterinterpretthecontentoftheaboveexpression,itisusefultorewritetheexpressioninlaboratoryvariables.ThedetailsofdoingthisaregiveninAppendixG.Butforthepurposeshere,itissucienttorecognizethatithastheform wheredisapre-factordeterminedbythedimensionsofthequantitiesefand,dependingontimethroughthetemperature.Thetemperaturedependenceof

PAGE 75

hasbeenmadeexplicitinthisnotation.Also,recallthat ThisallowsEq.( 4{75 )toberewrittenas @yh(t)Dbkef(0);tEh(4{77) Thus,theEulerordertermsintheconstitutiverelations( 3{15 )and( 3{16 )arederivativeswithrespecttothehomogeneoushydrodynamiceldsoftheensembleaveragesofthevolumeintegratedmicroscopicuxesovertheHCSensembleatatimet.Forthecaseofnormaluids,thesetermsaredeterminedbythederivativesoftheensembleaverageoftheuxesovertheequilibriumstate[ 37 ].Hencethesamestructureisretainedhere,withtheHCSstatetakingtheplaceoftheequilibriumstate.Theonlytimedependenceisthatofnormaltimedependencethroughthecoolinghomogeneoustemperature,whichispreciselyaswasfoundinthecaseofthelinearizedhydrodynamicequations.Notethatthisistrueforalls,thatisnolongtimelimitneededtobetakentomakethisEulertermpurelyhydrodynamic.Explicitphasefunctionsinvolvedandfurtherinterpretationoftheseresultsforspecictermssuchasthepressurearegiveninthenextchapter.

PAGE 76

Thecontentofthisexpressionsisasfollows.PuttinginthevariouscorrelationfunctionsdenedabovethroughEqs.( 4{63 )-( 4{70 )givestheresult(seeAppendixFfordetails) (4{79) where 4{45 ),givenas andthegeneratoroftimedependenceoftheadjointux 4{79 ),resultsinthefollowingidentity(SeeAppendixFforthedetails), Hencethetimegeneratore Summarizing,thetransportcoecientsinEqs.( 3{15 )and( 3{16 )areidentiedthroughtherelation

PAGE 77

wherethelimabovedenotesthethermodynamiclimit,namelyV!1andN!1suchthatn=N=Visaconstantandthelongtimelimitofs!1:Thus,theexpressionforthetransportcoecientsthatoccurintheconstitutiverelationsassociatedwiththehydrodynamicequationsconsistsofaux-uxcorrelationfunction.NoticethatthepropertyinEq.( 4{81 allowstheintroductionofaprojectionoperatorthatsubtractsoutthecomponentsoftheuxalongtheinvariantsofthedynamics,ensuringthatalongtimelimitexists.ThisisknownastheGreen-Kuboformofthetransportcoecients. Thetimeindependentrsttermintheaboveexpressionremainstobeinterpreted.Forthispurpose,observethatthedynamicalequationassociatedwith 4{59 )yieldstheidentity (1)j(0;s) Hence,Eq.( 4{79 )canbewritteninthealternateform (1)j(0;s0) Thisformallowstherecognitionofthersttermasthevalueoftheintegralinthesecondtermatthelowerlimit.Inthecaseofnormaluidswheretheparticlesinteractthroughcontinuouspotentials,thistermiszeroandtheGreen-Kuboexpressionforthetransportcoecientreducestojusttheux-uxtimecorrelationfunctioncharacterizedearlier.Thereasonthatsuchatermispresenthereistechnical.Itisassociatedwiththefactthattheformofthecorrelationfunctionsfors0+ands0aredierent.Therearetwocausesforthisdierence.Oneisthatthesystemconsistsofhardparticlesandthedynamicsisdiscontinuousabouts=0becauseofinstantaneousmomentumtransferpresentinthecollisionmodel.Ananalogoustermexistsforelastichardspheretransportcoecientsaswell.The

PAGE 78

PAGE 79

momentumtransferassociatedwithhardspheremodelsandpartlyduetotheinelasticityofthegraincollisions.Finally,aHelfandformofthetransportcoecientshasbeenidentiedinEq.( 4{86 )above. 4{72 ),groupingtermsthatariseduetothesourceinthedirectconservationlaws,upthroughEulerorder,onegets Substitutingfortheformsofthesecorrelationfunctionsgives(AppendixFhassomeintermediatedetails) Recallthatthetermel(0)hasthepropertythat 3ew(0)(4{89) Hence,theaboveexpressioncanbewrittenintheequivalentform 3ew(0)1P> 3ew(0)1P> ComparisonofEq.( 4{90 )withEq.( 4{82 )showsthatthelongtimelimitofthiscorrelationfunctioniswelldenedasthetimedependentquantityisorthogonaltotheinvariantsofthegeneratorofthisdynamics.Further,asshowninAppendixG,sphericalsymmetryoftheHCSgivesthattheonlynonzeroentryofthetypeaboveisfor=3.ExaminingthephenomenologicaltransportmatrixinEq.( 3{26 )inthe

PAGE 80

previouschapter,itcanbeconcludedthat 3ew(0)1P> 3ew(0)1P> ThisistheGreenKuboexpressionforthetransportcoecientoccurringatEulerorder,againthelimitbeingtakenasdescribedearlier. Further,asearlier,thetimeintegralcanbecarriedouttoget 3ew(0)1P> ThisistheHelfandformforthistransportcoecient.Notethatinthiscasealongtimelimithastobetakeninordertoobtainthehydrodynamicform,unliketheEulertermsthatcomefromtheuxpartofthetransportmatrix.Inthecaseofnormaluids,theEulerlevelhydrodynamicsisentirelynon-dissipative,thatis,involvesnoentropyproducingprocesses.ThecollisionallossinenergyinthecaseofgranularuidsgivesrisetothistransportprocessatEulerorder. 4{72 )thispartofthetransportmatrixcanbeidentiedas

PAGE 81

PAGE 82

correlationfunctionsovertheHCSensemblehasbeencarriedout.Thesearetheprimaryresultsofthiswork.Thereforethischapterisconcludedbyenumeratingthekeyresultsinit.Thegeneralmethodtoextracthydrodynamicresponseofauidfromthesolutiontoanappropriateinitialvalueproblemwasdescribed. 1.AspecialhomogeneoussolutiontotheLiouvilleequationwasidentiedandcharacterized.Thissolutionhasthepropertythatitsdynamicsiscompletelygivenbythelinearizedhydrodynamicequationsinthehomogeneouslimit.Equivalently,itcanbestatedthatthedynamicsofthisstateiscompletelyspeciedbythehydrodynamicmodesoftheuid. 2.Aspecialinitialstatewaschosensuchthat,inthehomogeneouslimit,itreducedtothespecialsolutionabove.Thiswasthelocalhomogeneouscoolingstate.Then,thepropertiesofthisinitialstatethatresultinthesimplicationofthesubsequentlinearresponseprocedureandtheprocessofformallyidentifyingthehydrodynamiclimitofthetransportmatrixwereidentied. 3.TheformalresultforthehydrodynamictransportmatrixwasderivedattheEulerandNavier-Stokesorderanditwasshownthatthehydrodynamiclimitiswelldenedineachcase,thatis,thelongtimelimitexisted. Inthenextchapterofthepresentation,thespecicphasefunctionsthatenterintotheformalresultsaboveforeachofthetransportcoecientsisidentiedandtheresultingexpressioncomparedwiththeknownresultsfornormaluidstoshedlightontheirphysicalcontentandhencetheimplicationstotransportinthesesystems.Also,inChapter6,akinetictheoryoftimecorrelationfunctionsisdevelopedthatmakescontactwiththevariousexpressionsforthesetransportcoecientsthathavebeenobtainedfromBoltzmannandEnskogkinetictheoriesintheliterature.

PAGE 83

Inthepreviouschapter,theprimaryresultsofthisworkwerederived.TheyareexactexpressionsforalltheparametersinthephenomenologicalhydrodynamicequationsthatweregiveninChapter3.Forthetransportcoecients,theseexactexpressionsareoftheHelfandandGreen-Kuboforms,timecorrelationfunctionsovertheHCSensemble.Thetechnicalcontentoftheseexpressionswerepartiallydiscussedandthegeneralstructureoftheresultwaselucidated. Inthischapter,furtherunfoldingoftheseexactresultsisgivenbymakingexplicitthedierentphasefunctionsinvolvedandbycomparingtheresultswiththeircorrespondinganalogsintheelasticlimit,whereapplicable.RecallthatthelinearhydrodynamictransportmatrixwasidentiedinEq.( 3{26 )inChapter3andhadtheform 3p+Uik1 2p@lnph 2pikh 3+k21CCCCA;(5{1) forthelongitudinaldegreesoffreedomcorrespondingto;=1;2;3,andthedecoupledtransversepartwas Theunknownparametersinthisequationarethecoolingrateh,thepressurep;theEulerordertransportcoecientU,theshearandbulkviscosityand,thethermalconductivityanddiusivityand,andthetwotransportcoecients 74

PAGE 84

fromthesourceatNavier-Stokesorder,nandT.EachofthesequantitiesisidentiedfromthecorrespondingformalexpressioninthepreviousChapteranddiscussedinturn. 2{24 )as Inparticular,whentheensembleinquestionistheHCS,thiscanberewritteninthestationaryrepresentationdenedinEq.( 2{34 )as vh(t)h(t)=(12)N Further,sincethecoolingrateisdeterminedbyonlyatwoparticlefunction,usecanbemadeofthehierarchyofdistributionfunctionsassociatedwiththeHCSensemble,asgiventhroughEq.( 2{18 )earlierandwrittenas 6(12)nh`32Zd12(g12bq12)3(g12bq12)(q12)f(2)h(q12;v1;v2)(5{5) where andthenotationgivingf(2)asafunctionofq12makesuseofthefactthattheHCSmustbeinvariantundertranslations.Lastlytheintegraloverthetwoparticlephasespacecanberecastintointegralsovertherelativeandcenterofmasscoordinates.Usingthe-functionintheaboveexpressionthatputsthetwo

PAGE 85

particlesatcontacttodopartoftherelativecoordinateintegral,oneobtains 6(12)nh`322Zdbdg12(g12b)3(g12b)ZdP12f(2)h(;v1;v2)(5{7) Noticethatthecoolingrateisgeneratedbyafunctionofonlytherelativevelocityofpairsatcontactandisobtainedasanaverageoverthetwobodydistributionfunctionatcontact.Inparticular,itistheaverageofthethirdmomentofthenormalcomponentoftherelativevelocityofcollidingpairs.Thisisthemicroscopicexpressionassociatedwiththeparametercharacterizingthehomogeneouscoolingdynamicsoftheunperturbedandhomogeneouslyperturbedhydrodynamicstatesofthissystem. 3{15 )and( 3{16 )usedtoobtainthehydrodynamicequationswerewrittendowntakingintoaccounttheunderlyingsymmetriesoftheuid,namelyhomogeneityandisotropy.Thesesamesymmetrypropertiescharacterizethehomogeneousreferenceensembleaswell.ThesesymmetrypropertiescanbeusedtoconcludethatthevarioustermsatEulerorderthatarezero(likethe21matrixelementwhichwouldcorrespondtoatermproportionaltothedensitygradientinthedynamicalequationofthetemperatureelds)areindeedvanishingwhencalculatedfromthestatisticalmechanicalprescriptionaswell.Asanillustrationconsiderthe21matrixelementatEulerorderarisingfromtheuxpartofthetransportmatrix.ThishasbeenidentiedinEq.( 4{74 )as where

PAGE 86

Aswasstatedearlier,itisusefultoconsiderthesecorrelationfunctionsinthelaboratoryvariables.Puttingintheformoftheuxefi2(0)andrestoringthedimensionstotheaboveequationgives @nhZd2 3esiv2h(t) 2egih(t):(5{10) Nownoticethehomogeneouscoolingstateistakentohavezeroowvelocitywhenthedensityderivativeisbeingcarriedout.Hencetheensembleaverageintheaboveexpressionisthatofavectoroverahomogeneousandisotropicdistribution.Therefore,thisintegralmustvanish.SimilarargumentsshowthateachofthezerotermsatEulerorderfollowfromthestatisticalmechanicalderivationaswell,includingthevanishingofEulerordertermsinthetransversecomponentsandthedecouplingofthetransversemodesfromthelongitudinalmodes.ThedetailsoftheseargumentsaregiveninAppendixG. 5{8 )above.Theonlynonzeroterminthisrowinthe 3esi(0)begi(0) (5{11) Asstatedinthepreviouschapter,aphysicallyinterpretableformofthisresultisaccessibleifitistransformedbacktotherealvariables.Restoringthedimensions

PAGE 87

tothevariousquantitiesgives @UjZd2 3esiv2h(t) 2egh(t):(5{12) Now,thehomogeneouscoolingstateintheaboveequationisonewithaniteowvelocityU.Itisrecognizedthattheowvelocityentersintothehomogeneouscoolingstatethroughthevelocityoftheparticles,i.e., Hence,onecanchangevariablesintheintegrationasvl!vlU,andusethetransformationpropertiesoftheheatuxandthemomentumunderGalileantransformationstoget 31 3Zdehii(0)h(Th(t))(5{14) ThedetailsofthistransformationcanbefoundinAppendixG.ComparisonwiththehydrodynamicmatrixinEq.( 5{1 )abovegivestheidentication 3Zdehii(0)h(Th(t)):(5{15) ThisgivesusthedenitionofthehydrostaticpressureastheaverageofthetraceofthevolumeintegratedmicroscopicstresstensorovertheHCSensembleatatimet.Further,aswasdoneforthecoolingrateabove,puttingintheformoftheuxehgiveninTableC-1,transformingtheexpressiontothestationaryrepresentationandcarryingouttheintegralstomapthisontoanaverageoverreduceddistributionfunctionsovertheHCSgivestheresultthatthepressureisthesumoftwotermsintheform

PAGE 88

where 3Zdv1f(1)h(v1)v1v1;(5{17) and 4(1+)Zdq1dv1dq2dv2ff(2)h(q12;v1;v2)(q12)(q12g12)(bq12g12)2g: ThersttermaboveinEq.( 5{17 )isthekineticpartofthepressurethatarisespurelyfromthetransportofmomentumassociatedwiththefreestreamingoftheparticles.ItcaneasilybeseenthatthekineticpartofthepressurepKgives andhence whichisthefullpressureinthelimitoflowdensitieswhenthecollisionsbecomeinfrequentandisindeedthefullresultforidealgases.Next,thesecondtermabove,determinedbythetwoparticledistributionfunctionatcontact,isthe\potential"partofthepressurethatarisesduetotheinteractionbetweentheparticles,whichinthecaseofhardspheresisgeneratedcollisionally.NowfocusonthecollisionalpartofthepressurepC.Asearlier,thiscanberewrittenbytransformingtorelativeandcenterofmassvariablestogive 4(1+)3Zdbdg12(bg12)(bg12)2ZdP12f(2)h(;v1;v2) (5{21) whichidentiesthecollisionalpartofthepressureasgeneratedbytheaverageofthesecondpowerofthenormalcomponentoftherelativevelocityofcollidingpairsoverthetwoparticledistributionfunction.Alsonoticethatthecenterof

PAGE 89

massvelocityofthepairP12playsnoroleinthecollisionalpartofthepressure,asshouldbeexpected. Asanillustrationforthestructureofthisquantity,letusconsideritsevaluationintheelasticlimit.When!1, whereg()isthepairdistributionfunctionatcontactanditdependsonlyonthemagnitudeof.EvaluatingEq.( 5{21 )above,inthislimitgives Hence,thecollisionalpartofthepressureisproportionaltothedensity,andathighdensities,dominatesoverthekineticpartofthepressure.Further,itisreasonabletoexpectthatpCfortheinelasticsystemhasasimilarbehavioraswell,inthesensethatthepaircorrelationfunctionatcontactwouldstillbeisotropic,beingalargelygeometricquantity.Thecoecientsofthedensitywillbemodiedbytheinelasticity,throughthecoecientofrestitution,andthevelocitycorrelationsthatwillbepresentinthetwobodydistributionfunction. Thiscompletesthecharacterizationofthemicroscopicdenitionofpressureobtainedfromthisanalysis.Also,itfollowseasilythatthetermsinthematrixelementsassociatedwiththelongitudinalowvelocityeld,namelythe31andthe32element,whicharethedensityandtemperaturederivativeofthepressure,comeoutconsistentwiththeabovedenitionofthepressure.ThedetailsofthisaregiveninAppendixG.

PAGE 90

startingfromtheformalexpressiongiveninEq.( 4{91 )earliertobe 3Zs0ds0ZdW()e( 3ZdWMiig where 3ew(0)+3 2h2 3EN+h@lnh with 4(12)XlXm6=l(qlmglm)(glmbqlm)3(qlm)(5{26) beingthesourceintheenergybalanceequationgiveninTableC-1andEisthetotalenergyofthesystemgivenby andalso Also,theadjointdensityintheaboveexpressionis aspacemomentofthevelocityderivativeoftheHCS.ThisistheGreen-KuboformoftheEulerordertransportcoecientassociatedwithgranularuids. Asarststepinunfoldingthecontentofthisexpression,itisobservedthatthephasefunctionWwasobtainedbyunfoldingtheactionof(1P)ontheenergysourceew.Andrecallthat

PAGE 91

PAGE 92

ComparethisresultwiththatobtainedforthecollisionalpartofthepressureintheuidinEq.( 5{21 )abovetoseethat Hence,itcanbesaidthatUisthecontributionofthesourcetowhatwouldphysicallyconstitutethehydrostaticpressureoftheuid.Ifasmallvolumeelementoftheuidisconsidered,thentheamountofpressurethattheuidelementcanexertonitsboundariesisdecreasedbytheenergylostlocallyduetocollisions.Partoftheeectofthistransportcoecientistodecreasetheeectivepressureinthesystem,ascanalsobeseenfromthefactthatthetransportcoeciententersthehydrodynamicequationsintheform2 3p+U.Attheleveloflinearhydrodynamics,thetwocoecientsareindistinguishableintheirphysicalconsequence. Further,thetimeintegralinEq.( 5{24 )canbecarriedoutintheaboveexpressionandhencecanberewrittenas 3ZdW()e( ThiswouldbetheHelfandformofthistransportcoecient.Aswillbeshowninthesubsequentchapter,thisformofthetransportcoecientismostconvenienttomakeconnectionwithresultsobtainedfromkinetictheory.ThiscompletestheanalysisassoiciatedwithU.

PAGE 93

4{78 )as where 4(1+)XlXm(qlm)(qlmglm)(bqlmglm)2bqilmbqjlm andMijisthesamefunctiondenedinEq.( 5{29 )above.ThisistheGreen-Kuboformoftheshearviscosity.Further,usingthefactthatbkbe1=0,Eq.( 5{35 )canbewrittenas 101 whereMtrijdenotesthetracelesspartofthetensorMijgiveninEq.( 5{29 ).Asnotedabove,intheelasticlimit andhence thebackwardmomentumuxintheelasticlimit,whichhasthesameformasthehinTableC-3,with=1.Thus,theaboveexpressionreducestoatimecorrelationfunctionovertheequilibriumensembleofthevolumeintegratedforward

PAGE 94

momentumuxehijwiththebackwardmomentumuxehijtogetherwithaninstantaneouspart[ 67 ].ThedierentuxesintheforwardandbackwardtimedirectionandtheinstantaneouspartareartifactsofthehardspherenatureoftheinteractionoftheparticlesasexplainedinAppendixC.Inthecaseofinelasticparticles,theadjointuxcontainsexplicitinformationaboutthenatureofthenon-equilibriumstateandisameasureofmomentumtransportinthisstate. Further,thetimeintegralinEq.( 5{35 )canbecarriedouttogive 101 whichistheHelfandformfortheshearviscosityofagranularuid.Lastly,theinstantaneouspartoftheGreen-Kuboformiswrittenintermsofreduceddistributionfunctionsinordertoillustratethestructureoftheformalresult.Thereductioniscarriedoutinthesamewayasforthepressureearlier,withtheresult 101 60(1+)`3nh4Zdbdg12((bg12)(bg12))ZdP12f(2)h(;v1;v2) (5{41) i.e.,itistheaverageofthenormalcomponentofthevelocityofcollidingpairsaveragedoverthetwobodyreducedHCSdistribution.Noticethattheinstantaneouspartispurelycollisional,reectingthefactthatitcomesaboutbecauseoftheboundaryconditionassociatedwithhardspheredynamicsaboutthepointofcontact.Further,usingtherstequationoftheBBGKYhierarchyassociatedwiththeHCS,theaboveexpressioncanberewrittenas 60(1+)`3nh2av(5{42) whereavistheaveragecollisionfrequencyasdeterminedbythelosspartoftherighthandsideofthehierarchy(seeforexample[ 37 ]).

PAGE 95

4{78 )inthepreviouschapter,itisshowninAppendixGthattheGreen-Kuboexpressionforthebulkviscositycanbeidentiedas 9f1 whereehijandMijarethesamefunctionsidentiedinEq.( 5{36 )andEq.( 5{29 )respectively.Thequantityeh(S)iiisthe\subtracted"momentumuxdenedas 2p@lnph 2p2 3EN: Theseadditionaltermsarenecessaryfortheexistenceofthelongtimelimitofthecorrelationfunction,asestablishedinthepreviouschapter.Withoutlossofgenerality,thesubtractedpiecescanbeaddedtotheinstantaneousterminEq.( 5{43 )aswell,astheygivezerocontributionundertheensembleaverage.Further,iftheinstantaneouspartoftheGreen-KuboexpressioninEq.( 5{43 )aboveisevaluated,itisfoundtobethesameasthatfortheshearviscosity,butforanoverallnumericalfactorreectingthedierenttensorcontractionsinvolvedineachcase. 3inst:(5{45) Finally,thetimeintegralcanbecarriedouttogive 91 ThisistheHelfandformforthebulkviscosityofagranularuid.

PAGE 96

Thusthephysicalcontentoftheexpressionsforthebulkviscosityarethesameasthoseforthatoftheshearviscosity,exceptthatthetensorcontractionsheremeasuretheisotropicpartofthemomentumtransport. 4{78 ),thedimensionlessthermalconductivityisidentiedinAppendixGas where 3ZdS(i)Ni;(5{48) and 31 Intheabove,Siistwothirdsofthevolumeintegratedheatux, 31 2XlXm(qlm)(1+)(qlmglm)(glmbqlm)2(Plmbqlm)bqilmg andS(S)iisthe\subtractedheatux"withtheinvarianttotalmomentumPsubtractedfromit, 3p+1Pi:(5{51) Lastly 2Xlrvl(vlh())ql#(5{52) isafunctiongeneratedbythenonequilibriumreferencestateassociatedwiththissystem.Eq.( 5{47 )istheGreen-Kuboexpressionforthethermalconductivityofagranularuid.Itisatimecorrelationfunctionofthesubtractedheatuxwithauxthatcharacterizesheattransportinthisnonequilibriumreferencestate.

PAGE 97

Inordertoillustratethenatureoftheadjointux,considertheelasticlimitofNiabove, 22 3bee(1)(0)ben(1)(0)eq=3 2ea(1)2eq andhence 5{47 ),whichisameasureofthispersistenceaveragedoveralltrajectorieswillbelargeandhencethesystemwillhavealargethermalconductivity.Butinthecaseathand,thereferenceensembleisanonequilibriumensemblethatisacomplicatedfunctionofitsconstrainingvariablesasthecollectionoftrajectoriesinthismacrostateismoreseverelyconstrainedthanintheequilibriumcase.Hencetheextentofheattransportinthissystemdependsstronglyonthecollectionoftrajectoriespresentinthemacrostateasreectedbythefactthattheadjointuxinthecorrelationfunctionforthethermalconductivityisnotsimplytheheatuxbutisgeneratedbythenonequilibriummacrostateitself.

PAGE 98

AsalaststepinillustratingthecontentoftheGreen-Kuboexpressionabove,theinstantaneouspartofthethermalconductivityisexpressedintermsofreduceddistributionfunctioninthehierarchyassociatedwiththeHCStogive 18`3nh4(1+)Zdbdg12dP12f2(g12b)(P12b)2+1 2(g12b)3g(bg12)f(2)h(;v1;v2): Noticethataswiththeearliertransportcoecients,theinstantaneouspartofthetransportcoecientispurelycollisionalreectingitsoriginwhichisthediscontinuityintimeforcollidingcongurationsatcontact.Also,thisisthersttransportcoecientwherethecenterofmassmomentumofthepairispartofthephasefunctionbeingaveraged,reectingthefactthatthisisthersttermsofarthatisassociatedwithheattransport,unlikealltheprevioustermsthatwereessentiallyassociatedwithmomentumtransport. Finally,asdemonstratedinEq.( 4{86 )inChapter4,thesubtractedpartoftheheatuxcanbeintroducedintheinstantaneoustermwithoutlossofgeneralityandthetimeintegralinEq.( 5{47 )canbecarriedouttogive 3ZdS(S)ie( ThisistheHelfandformforthethermalconductivityofagranularuid,withS(S)iandNiareasdenedinEq.( 5{51 )andEq.( 5{52 )respectively.Thiscompletesthecharacterizationoftheexactexpressionforthethermalconductivity.

PAGE 99

3ZdSiRiZs0ds01 3ZdS(S)ie Ls0 whereSi,S(S)i,andNiareasdenedinEqs.( 5{50 ),( 5{51 )and( 5{52 )respectively.Also, and TheexpressioninEq.( 5{56 )istheGreen-Kuboexpressionforthecoecientwhichisanewtransportcoecientassociatedwiththetransportofheatinthissystemthatisnotpresentfornormaluids.FirstnoticethatthesecondtimecorrelationfunctionisthesameasthatoccurringintheexpressionforthethermalconductivitygiveninEq.( 5{47 )above.Also,unlikethecasesofe2,...,e5,theadjointdensitye1isgivenimplicitlyintermsofthelocalHCSstate.Thisisbecause,thereferencestateashasbeencharacterizedinthisworkisahomogeneousstate.Theformalismasdevelopedheredoesnotmakeexplicitthedensitydependenceofthisstateandhenceisunknownuptothispoint.Aformalwaytoextractthedensitydependencewouldbetointroduceanexternalpotentialthatcouplesintothedensity,intheLiouvilleoperatorgoverningthedynamicsofthissystemandsolvingtheinhomogeneousproblem.Thenaformalprocedureofinversioncanbeusedtoeliminatetheexternalpotentialinfavorofthedensity.

PAGE 100

Amorephysicalwaytogetdensitydependenceofthemacrostatewouldbetogeneralizethestatetoopensystems,i.e.,denea\grandcanonical"homogeneousCoolingState.Thistreatmentwillbepartoffutureanalysisofthisproblem. Forthemoment,theresultisconsideredattheformallevelaboveandthecontentoftheexpressionelucidatedbelow.Asarststep,itisestablishedthatthistransportcoecientisuniquelyrelatedtothedissipativenatureoftheinteractionsinthesystembyshowingthatintheelasticlimit,thiscoecientvanishes.ItisapparentfromtheformofEq.( 5{56 )thatthisunfoldedexpressiondoesnothaveatransparentelasticlimit.Hence,forthemoment,themoreformalresultinEq.( 4{86 )fromwhichtheaboveexplicitexpressionwasobtained,isconsidered. 3ZdSiRi1 3Zs0ds01 First,observethat,intheelasticlimit whereec(k)istheFouriertransformoftheequilibriumdirectcorrelationfunctionoftheuid.ThedetailsofobtainingthisformaregiveninAppendixF.Itfollowsfromtheabovethat for,normalizationanduidsymmetryimplythatec(0)=1andec(1)(0)=0.Therefore, atimeindependentquantity.ThisidentiesallthequantitiesinEq.( 5{59 )intheelasticlimit.Firstnoticethattheinstantaneouspartofthecorrelationfunction

PAGE 101

PAGE 102

Restoringthedimensionstotheaboveequationandworkinginlaboratoryvariables,thiscanbewrittenas Thus,Qiisameasureofthevariationinthemacrostateathandwithrespecttothedensitywhenthecoolingrateinsteadofthetemperatureisheldxed.Theonlywaythatthecoolingratecanbeheldconstantwhenthedensitychangesisforthetemperaturetochange.Andhencethisparticularformexactlycorrespondstothechangeintemperaturethatcanbeinducedduetoavariationindensity.Sotheresultobtainedconformstothephysicalexpectationsformedfromthehydrodynamicpictureabove. Also,asafurtherinterpretationoftheresultgiveninEq.( 5{56 )above,notethatusingtheresultobtainedforthethermalconductivityintheprevioussubsection,theresultforcanberewrittenintheform 3ZdSiQiZs0ds01 3ZdS(S)ie Ls0 ItcanbeseenfromthedenitionofthehydrodynamiceigenfunctioninEq.( 3{39 )inChapter3thatthisparticularlinearcombinationcorrespondstoaresponsetoagradientinnwherethevariationthethedensityiscarriedoutinsuchawaythatthecoolingrateisheldconstant. Lastly,aswasprovedinChapter4Eq.( 4{86 )earlier,the1P>operatorcanbeintroducedintotheinstantaneouspartoftheexpressioninEq.( 5{59 ).Then,thetimeintegralcanbecarriedouttoobtain 3ZdS(S)i

PAGE 103

PAGE 104

Asnotedintheintroduction,thehydrodynamictransportcoecientsofagranularuidhavebeenextensivelystudiedatthelevelofkinetictheory,analytically,numericallyandinconjunctionwithexperiments.TheaimofthischapteristomaketheconnectionoftheresultsreportedinthisworkatthelevelofNparticleswiththeextensivebodyofworkthatexistsatthelevelofkinetictheory,thatis,theoneparticlelevel.EventhoughtheresultsinthepresentformaretimecorrelationfunctionsoveranNparticledistribution,itispossibletoformallyndanequivalentexpressionthatwouldinvolvethetwoparticlereduceddistributionfunctions.Thisreducedrepresentationservesasatractablestartingpointtondakinetictheorytodescribethedynamicsofthecorrelationfunctions. ThestructureoftheChapterisasfollows.First,startingfromtheHelfandformforthevarioustransportcoecientsidentiedinthepreviouschapter,thegeneralstructurethisclassoftimecorrelationfunctionsisexposed.ThenthisstructureisutilizedtosetupaformallyexactschemethatmapstheseNparticlecorrelationfunctionsontointegralsoverareducedoneparticledistributionfunction.Next,thisformalschemeisconsideredintheelasticlimit,andtheEnskogtheoryfornormaluidsisidentied.Inthecaseofnormaluids,aswillbeshownbelow,theEnskogtheoryissuchthatitisexactatshorttimes.AMarkovianapproximationtothedynamicsallowstheextensionofthisexactshorttimeresulttonitetimesandmakestheconnectionbetweentheresultsobtainedfromthetimecorrelationfunctionmethodtothoseobtainedfromkinetictheory.Itisshownherethatforthecaseofgranularuidsthisconnectionto 95

PAGE 105

kinetictheoryisnotapparentbecauseoftheinherentvelocitycorrelationspresentinthenon-equilibriumhomogeneouscoolingstateatalltimes.Further,theapproximationsthatneedtobemadeinordertoobtainthekinetictheoryforthisuidthathasbeenwidelystudiedusingbothanalyticandnumericaltechniquesisidentied.Thisservestoplacecontextonthedomaininwhichkinetictheorymayberelevantforgranularuids.Finally,theapproximateEnskogliketheoryforgranularuidsisusedtoevaluatethedierenttransportcoecientsandtocomparetheresultstothosealreadygivenintheliterature[ 68 ]. Ls()C1(s)(6{1) wherej()isasubtracteduxthatisoneoftheuxesorthesourcefromthedirectconservationlawswiththeinvariantsofthedynamicsprojectedoutasshownforexampleinEq.( 4{86 )inChapter4.But,fortheconsiderationshere,itsucestonotethatthej()havetheform withXldenotingthephasepointassociatedwithparticlel,i.e.,

PAGE 106

andj1andj2aresomeoneandtwoparticlefunctionsofthephasepointrespectively.Also,theadjointfunctions()aregenericallyoftheform (;s)=yh(t)(`vh(t))3NZdrbkre whereyisoneofthehydrodynamiceldsthatparameterizestheinitiallocalHCS.FurthertheC1(s)inEq.( 6{1 )istheadditionalhomogeneousdynamicstimedependencethatoccurswiththegenerator Now,thefactthatj()isasumofoneandtwoparticlefunctionsallowsEq.( 6{1 )toberewritteninaforminvolvingreduceddistributionsofoneandtwoparticlefunctionstogive withthereducedfunctions(m)'sdenedas ThesefunctionsobeyahierarchyofequationsanalogoustotheBBGKYhierarchydenedinEq.( 2{18 )inChapter2oftheform where 2{14 )earlier.NotethatEq.( 6{5 )aboveformallymapsthefullNbodycorrelationfunctiontoonethatinvolvesatwoparticledistribution.Thisisthestartingpointfromwhichtoformulateakinetictheoryfortheevaluationofthesecorrelationfunctions.Formally,akinetictheoryrepresents

PAGE 107

aclosureoftheBBGKYhierarchyattheleveloftherstequationofthehierarchythroughanidenticationofarepresentationforthetwobodydistributionfunctionintermsofafunctionaloftheonebodydistributionfunction.Inordertoformulatesuchaclosureruleforthehierarchyofthe's,supposeitispostulatedthatthereexistsaclosureruleforthetwobodyreduceddistributionfunctionassociatedwiththeinitiallocalhomogeneouscoolingstateintermsoftheonebodydistributionfunctionintheform then,fromthedenitionofinEq.( 6{4 )above,(2)cannowbeexpressedasafunctionalof(1)throughtheequation (2)(X1;X2;s)=ZdX0K(X1;X2;s:X)(1)X0;s(6{9) where ThelinearkernelKisafunctionalderivativeofthetwobodydistributionfunctionwithrespecttotheonebodydistribution,evaluatedinthehomogeneouslimit.Thepre-factorsintheaboveexpressionjustallowtheexpressiontobetransformedtothedimensionlessvariablesasinEq.( 6{4 )above.Substitutethisformforthetwobodydistributionfunctionintherstequationofthehierarchytoget Thiscanbewrittenasakineticequationgoverningthedynamicsof(1),

PAGE 108

wherethegeneratorofthedynamicsnowincludesageneralizedcollisionoperatorI(s)formallydenedas with Eq.( 6{12 )canbeformallyintegrateduptogive (1)(X1;s)=U(s)(1)(X1)(6{15) whereU(s)issuchthatitisthesolutiontotheequation togetherwiththeboundaryconditionthat Finally,theclosureequation( 6{9 )canbeusedintheexpressionforthetransportcoecienttorewritethesecondtermgiveninEq.( 6{5 )toeliminatethetwobodydistributionfunction(2)through andrelabeltheintegrationvariablessuitablytogive

PAGE 109

where Thus,Eq.( 6{19 )isaformalmappingoftheNbodycorrelationfunctiondeningthetransportcoecienttothatofaonebodycorrelationfunction.ThisformalmappingisvalidwhenevertheclosurehypothesisinEq.( 6{8 )exists.Furtherprogressforarbitraryclosuresoftheformconsideredaboveiscomplicated.But,foraspecialclassofclosuresthatareMarkovian,inthatthekernelKgiveninEq.( 6{10 )abovebecomesindependentofs,Eq.( 6{19 )canberecastintoareducedGreen-Kuboformbyintegratingbypartswithrespecttotimetogive (6{21) NoticethatthereducedGreen-KuboexpressionhasthesameformastheoneattheNbodylevel,inthatithasaninstantaneouspartandatimecorrelationpart.Thetimecorrelationpartisstilloftheformofadirectuxadjointuxcorrelationfunction,butnowreducedtotheoneparticlelevel.Thedirectuxisnowg(X)denedinEq.( 6{20 )above,andtheadjointuxis whereKhydTisunderstoodtobetheappropriateelementsofthehydrodynamicmatrix,dependingontheparticulartransportcoecientunderconsideration. Asanillustrationofthecontentoftheaboveformaltreatment,considerthesimplestpossiblecircumstancewhereanadhocclosurecriterionisproposedintheform

PAGE 110

i.e.,thetwobodydistributionfunctionassociatedwiththeinitiallocalhomogeneouscoolingstateissimplyaproductoftheonebodydistributions.Thisassumptionentailstheneglectofallcorrelationsinthesystemandmaybethoughtofasrepresentingthedynamicsatverylowdensitieswheretheinteractionbetweenparticlesisverysmall.Inthiscase,thevariousstepsintheformalanalysisabovecanbecarriedout.First,Eq.( 6{10 )becomes wheref(1)histhereducedoneparticledistributionfunctionassociatedwiththetruehomogeneouscoolingstateh.ItthenfollowsthatthekineticoperatorI(s)inEq.( 6{12 )becomes, whichispreciselytheBoltzmann-Bogoliubovcollisionoperatorassociatedwithinelastichardspheres,linearizedaboutthehomogeneouscoolingstate.Theaboveexampleillustratesthat,forthesimpleclosureimposedthroughEq.( 6{24 ),theformalprocedureabovereproducesafamiliarresult.Inwhatfollows,moresystematicapproachestotheclosurecriterionarediscussed.

PAGE 111

Intheelasticlimit,theensemblewithrespecttowhichtheclosurecriterionmustbeformulatedisthelocalequilibriumensembledenedinAppendixH.Thelocalequilibriumensemblehastheform whereW()istheexcludedvolumefactordenedas thatcharacterizesthehardspherenatureoftheparticlesinthesystem,and 2mU2;=1 Inthisformitisapparentthatthemomentumpartofthisdistributionfunctionisaproductofoneparticledistributionfunctions.Also,thetwobodyreduceddistributionfunctionisdenedas Notethatattimet=0,thisrelationshipgivesanidenticationofthetwobodydistributionas

PAGE 112

wherethelocalpaircorrelationfunctionisdenedas withexp(Q1)beingthenormalizationconstantfortheoneparticlelocalequilibriumdistribution.Thus,attheinitialtime,Eq.( 6{31 )givesanexactidenticationofthetwobodylocalequilibriumdistributionasafunctionaloftheoneparticledistribution. Now,supposeitisassumedthatthisfunctionalrelationshippersiststonitetimes,i.e., whichamountstoaMarkovianapproximationtothedynamicsofthesystem.ThenusingthisforminEq.( 6{10 ),thekernelthatrelates(2)totheoneparticlefunction(1)canbeidentiedas Inorderthatthiskernelbedeterminedentirelybyoneandtwoparticlefunctionsgeneratedfromthelocalequilibriumensemble,thefunctionalderivativeofthepaircorrelationfunctionhastobesimplied.ItturnsoutthattheformofthelocalequilibriumdistributioninthedenitionofglgiveninEq.( 6{32 )canbeusedto

PAGE 113

obtaintherelationship[ 70 ](seeAppendixIforthedetails) (6{35) wherec(r)isthedirectcorrelationfunctionwhichisrelatedtotheequilibriumpairdistributionfunctiongthroughtheOrnstein-Zernickeequation andg(3)isthethreeparticledistributionfunctionrelatedtothethreebodyreduceddistributionintheequilibriumhierarchythrough SubstitutingtheformofthefunctionalderivativeinEq.( 6{35 )intheexpressionforKandusingthedenitionEq.( 6{13 )forthecollisionoperatorgivestheresult Inwhatfollowsconsiderthelasttermintheaboveequation.SubstitutetheformofthefunctionalderivativegiveninEq.( 6{35 )aboveandnotethatusingthesecondequationofthehierarchyassociatedwiththeequilibriumstate,itiseasyto

PAGE 114

derivethefollowingidentitynZdX2 withthenotation ThisidentityexpressesthethreeparticlefunctioninEq.( 6{38 )aboveintermsofthepaircorrelationfunction.PuttinginthisrelationshipintothelastterminEq.( 6{38 )andusingtheresultingexpressionthecollisionoperatorbecomes wherethe'sonthevariousdimensionlessquantitieshavebeensuppressedforcompactnessofnotation.Observethatthepartofthecollisionoperatorthatcomesfromthefunctionalderivativeofthepaircorrelationfunctionisameaneldoperatorgeneratedbythedirectcorrelationfunctionofthehardsphereuid.TheIgivenaboveisthegeneralizedEnskogcollisionoperator,linearizedaboutthehomogeneousequilibriumstate,extensivelystudiedinthecontextofnormaluids[ 48 69 ]. TheEnskogkinetictheoryisknowntoworkwellfornormaluidsuptomoderatelyhighdensities.TheBoltzmannkineticequationcanbethoughtofasthezerodensityanalogoftheEnskogequation.TheaboveanalysisshowsthattheEnskogtheoryisexactintheshorttimelimitandhenceprovidesthe

PAGE 115

PAGE 116

notasdirectasthecaseofanormalelasticuidthathastheequilibriumstateasitshomogeneousreferencestate. Beforeconsideringanyclosurecriterionforthedistributionfunction,letuslookattherstequationofthehierarchyassociatedwiththelocalHCSdistributionrewritteninthefollowingform, 2hrv1v1+v1rq1f(1)(X1;s)=ZdX2(q12)(g12^q12)1 (6{42) wherethe NoticethatthisboundaryconditionisthetwobodyanalogofthecollisionalboundaryconditionsforhardspheredistributionsgiveninAppendixAinEq.( A{24 ).Aswillbeseeninthefollowingthisistruefortheconsiderationsassociatedwithtimecorrelationfunctionsalso.Thereforeinallthatfollows,theclosurecriterionisappliedonlyontheprecollisionhemisphere. Inwhatfollows,akinetictheoryofgranularuidsthatistheanalogoftheEnskogtheoryfornormaluidsisderivedbymakingtheadhocassumptionthat,

PAGE 117

PAGE 118

derivationinvolvestakingintoaccountthenitesizeoftheparticlesindeterminingthecollisionfrequencyaswasdoneinthecaseofthepressureofaVanderWaal'sgasforinstance.Thepresenceofathirdparticleinthevicinityofthecollidingpairistakenintoaccountthroughaneectivescreeningofthecollisioncenterfromtheincomingparticle.Thus,theEnskogkineticequationcapturesthegeometriceectsduetootherhardparticlesintheneighborhoodofthecollidingpairtovarygoodaccuracyandhenceworksverywellforelastichardspheresinawiderangeofdensitiesoftheuid.Thisgeometricscreeningshouldbethesameforinelastichardspheresaswell.Hence,anotherapproximationcanbeformulatedwherethegistakentobetheequilibriumpaircorrelationfunctiontogetanEnskogliketheoryforinelastichardspherecorrelationfunctions.So,thefailureofEnskogtheorycanbetakentoindicatethatsomeotherdynamicaleectoverwhelmsthispurelygeometricapproximation. ProceedingfurtherwiththechoiceofglinEq.( 6{45 ),thekernelthatrelates(2)to(1)isfoundtobe (6{46) whichhasthesamestructureasthecorrespondingresultfornormaluidsgiveninEq.( 6{34 )earlier.Inthecaseofnormaluids,theanalogofthethreeparticlefunctiongl(q1;q2)

PAGE 119

sucienttoknowthepaircorrelationfunctionatcontactinthetruehomogeneousstate.Sincethesystemathandisoneofhardparticles,thepaircorrelationfunctionatcontactisalargelygeometricfactorandhencetheanalogousquantityforelastichardspherescanbeusedtogoodapproximation.Sincetheaimhereistoidentifythekinetictheoryataformallevel,nofurtherdiscussionofthenatureofthepaircorrelationfunctionisgiven. NowmaketheapproximationEq.( 6{44 ),anduseitinEq.( 6{10 )andEq.( 6{13 )intheformalanalysisabovetogetthecollisionoperatorIintheform wherethedenition hasbeenusedand whichistheanalogofthemeaneldtermunfoldedexplicitlyinthecaseofelastichardspheresearlier.Noticethatforthisclosurerule,thedynamicsisMarkovianinthesensethatthecollisionoperatordoesnotdependonthescaledtimes,aswasthecaseintheprevioussection.ThisisthelinearizedformoftheEnskogoperatorstudiedin[ 68 ]. Summarizing,thekernelKthatgivestheformalmappingofthetwoparticlefunction(2)ontotheoneparticlefunction(1)fortheclosurecriterionunderconsiderationhasbeenidentiedinEq.( 6{46 )above.Usingthiskernel,thecollision

PAGE 120

operatorthatgeneratesthelinearkinetictheoryfortimecorrelationfunctionsunderconsiderationhasbeenidentiedinEq.( 6{47 ).Thus,theseresultstogetherwiththeobservationthatthegeneratorIisindependentofs,leadstothereducedrepresentationfortheHelfandformofatransportcoecientas 2hrvvvrq+IE+KhydTs(1)X0(6{50) whereg(X0)isasdenedinEq.( 6{20 )aboveandtheinitialcondition(1)(X0)isobtainedfromEq.( 6{6 )bytakingthes=0limitatxed,explicitformsofwhicharegiveninthefollowingsection.Further,theaboveexpressioncanbeintegratedbypartswithrespecttotimetoobtainareducedGreen-Kuboformforthetransportcoecientas 2hrvvvrq+IE+KhydTs0(1)X0 where 2hrv0v0+v0rq0IEKhydT(1)X0:(6{52) ThiscompletesthederivationoftheEnskoglikekinetictheoryofthetimecorrelationfunctionsthatdeterminethehydrodynamictransportcoecients.Inthefollowingsections,thedetailsoftheuxesinvolvedintheseexpressionsaregivenandtheexistenceofthelongtimelimitdemonstrated.Then,asanillustration,thedetailedformoftheaboveexpressionforthecaseoftheshearviscosityderived.TheformsoftheothertransportcoecientsaregiveninAppendixI.

PAGE 121

6{50 )andEq.( 6{51 )areconsideredanditisshownthatthelongtimelimitofthesetimecorrelationfunctionsthatneedstobetakeninordertogetthehydrodynamictransportcoecientsiswelldened.Asarststep,notethattheinitialconditionforthekineticequation,namely(1)X0isdenedthroughtheequation wherethehydrodynamiceldsareasearlier AsobservedattheNparticlelevel,thelocalhomogeneouscoolingstatedependsontheoweldUandthetemperatureTthroughthevelocitiesoftheparticles.Andhence,itreadilyfollowsthat(1)2(X)=bkq1 2rv(vh);(1)3=1 2bkqbkrvh(1)4;5=1 2bkqbe1;2rvh: Further,usingtheform onecanobtain (1)1(X)=bkqh(v):(6{57) Thisidentiestheinitialconditionsinthereducedrepresentationforthetimecorrelationfunctionsdeterminingthetransportcoecientsintheprevioussection.

PAGE 122

Next,notethattheNparticleuxj()fromwhichthedirectreduceduxg(X)isgeneratedisoftheform where 3s^g;hij;(6{59) andPistheprojectionoperatordenedinEq.( 4{45 )inChapter4earlierthatprojectsontotheinvariantsoftheNparticleLiouvilleoperator.Thisstructureisinheritedbythereducedrepresentationalso,inthatEq.( 6{51 )indetailtakestheform(seeAppendixIforthederivation) LE+KhydTis0(1)(X); withthedenitions 2hrvv+vrqIE;(6{61) and wherej1isthekineticpartoftheuxfandj2isitscollisionalpart.Further,thereducedprojectionoperatorintheaboveexpressionis

PAGE 123

with 3v213 2;v1;'(v)!h;1 2rv(vh);rvh:(6{64) Also,itisshowninAppendixIthatthefunctions'(v)satisfythepropertythat 2hrvv+vrqIE'(v)=KhydT'(v);(6{65) i.e.,'(v)arepreciselytheinvariantsofthegeneratorofthedynamicsinthereducedtimecorrelationfunctions.ThustheprojectionoperatorP(1)projectsorthogonaltotheinvariantsofthisgenerator.Further,aswasdoneintheNparticlelevel,anadjointoperatorP(1)>as andrewritetheexpressioninEq.( 6{60 )as LE+KhydTis0(1)(X): ItisshowninAppendixIthat,aswasfoundinthecaseoftheNparticledynamics,thefollowingidentityholds Inthisformitisevidentthatthegeneratorofthedynamicsactsonaquantitythatisorthogonaltoitsinvariants.UsingthispropertythereducedGreen-Kubo

PAGE 124

expressionforanytransportcoecienttakestheform LE+KhydTis01P(1)>(1)(X): Thus,thereducedGreen-KuboexpressionhasthesameformalstructureasattheNparticlelevel,inthatitconsistsofadirectuxadjointuxtimecorrelationfunctionwiththeprojectionoperatorsmakingthelongtimelimitofthisexpressionwellbehaved,andaninstantaneoustermthatgivespartofthecontributionfromcollisionaltransfertothevaluesofthetransportcoecients.ThisGreen-KuboformturnsouttobetheonemostconvenientforformalcomparisonwiththeresultsobtainedusingtheChapman-EnskogprocedurefromtheEnskogkineticequationin[ 68 ]. 6{69 )aboveareunfoldedfortheparticularcasewhenthetransportcoecientunderconsiderationistheshearviscosityforthepurposeofcomparisonwiththeanalogousresultobtainedusingtheChapman-EnskogmethodfromtheRevisedEnskogTheoryforinelastichardspheresgivenin[ 68 ].TheunfoldingofothertransportcoecientsisdoneintheAppendix. Beforegivingthedetailedformofexpression( 6{69 )fortheshearviscosity,rsttheresultsobtainedfromtheChapman-Enskogmethodaresummarized.Inthismethod,anormalsolutionwhosetimeandspacedependenceoccursonlythroughthehydrodynamiceldsisassumedtoexistforthekineticequationathand.Then,thisnormalsolutionisconstructedselfconsistentlyinpowersofthegradientsofthehydrodynamiceldsusingthekineticequationandthemacroscopicbalanceequationsassociatedwiththehydrodynamicelds.Carrying

PAGE 125

outsuchaproceduregivestheshearviscosityintheform 45I(): Thersttermiscalledthekineticpartoftheshearviscosityandisgivenby 10ZdvDij(v)Cij(v)(6{71) where 3ijv2(6{72) andCijisdeterminedasasolutiontoanintegralequationoftheform 2hrvvg()IB+h withIBbeingthelinearizedBoltzmanncollisionoperatoridentiedinEq.( 6{25 )aboveand Lastly,thesecondtermintheexpressionfortheshearviscosityinEq.( 6{70 )iscalledthecollisionaltransferpartoftheviscosityandthecollisionintegralIinthisexpressionisfoundtobe Thisistheresultgivenin[ 68 ]. Next,thevariousphasefunctionsinEq.( 6{69 )areidentiedtogettheexplicitformoftheshearviscosityinthisclosureapproximationfromtheexactexpression

PAGE 126

obtainedfromlinearresponse.Asarststep,notethattheshearviscositycorrespondstothechoice=4and=4inEq.( 6{69 ).AsstatedinEq.( 5{35 ),thedirectuxofinteresthereis 4(1+)XlXm(qlm)(qlmglm)(bqlmglm)2bqilmbqjlm: Thereforeinthiscase, 4(1+)(q12)(q12g12)(bq12g12)2bqi12bqj12:(6{79) UsingthekernelKidentiedinEq.( 6{46 )earlier,thereduceduxdenedinEq.( 6{62 )takestheform 4(1+)@g() Carryingouttheangularintegrationsovertherelativecoordinateintheaboveexpressionsyieldstheresult Now,usingthefactthattheoneparticlehomogeneouscoolingstatemustbesphericallysymmetric,theaboveexpressionreducesto 2(1+)3n`3g()4

PAGE 127

Thisidentiesthereducedrepresentationofthedirectux. Similarly,thereducedrepresentationfortheadjointuxisobtainedfromitsdenitioninEq.( I{34 )abovewith (1)4(X)=qkrvlh(v):(6{83) ThedetailsofthisaregiveninAppendixIwiththeresult where Observethattheexpressionfortheuxin( 6{84 )isthesameasthequantityCijdenedinEq.( 6{74 )above.Alsonoticethat,asinthecaseofthefullGreen-Kuboexpression,theactionoftheprojectionoperatoronboththedirectandadjointuxesvanishesbecauseoftheorthogonalityofbkandbe.Lastly,thereducedexpressionfortheinstantaneouspartoftheviscositywasobtainedinEq.( 5{41 )inChapter5.Inthisexpression,iftheEnskogclosureissubstitutedandtheangularintegrationsarecarriedout,itresultsin 45Zdv1dv2g12h(v1)h(v2):(6{86) Puttingalltheseresultstogether,theshearviscosityinthisEnskogapproximationisfoundtobe 2(1+)n`3g()4 LEs0h

PAGE 128

Nowconsiderthetimecorrelationfunctionpartofthiscoecientinthefollowing.Thetimedependentpartofthiscorrelationfunctionis LEs0h InordertoshowtheequivalenceofthisexpressionwiththatobtainedfromtheChapman-EnskogproceduregiveninEq.( 6{70 )above,consider LEs0h 2hrvv+vrqIE+h NotethatfromitsdenitioninEq.( 6{74 )above,Cklisahomogeneousfunction.Therefore,actionofthefreestreamingtermintheaboveexpressionvanishes.Further,theformofthelinearizedEnskogoperatorsimpliestotheformgiveninAppendixIinEq.( I{35 ),togive LEs0h 2hrvvg()IB@g() 2hrv(vh)Zdv0b(v;v0)+h (6{90) whereb(v;v0)isasubstitutionoperatorsuchthat ItfollowsfromthesphericalsymmetryofthehomogeneouscoolingstatethattheaverageoverthevelocitiesofCklmustbeproportionaltokl.Andhencethetermproportionalto@g() 6{90 )reducesto LEs0h 2hrvvg()IB+h

PAGE 129

Itiseasytoseethat,bymakingtheaboveargumentstoeachorderinwthat LEs0h 2hrvvg()IB+h ThereforeEq.( 6{88 )abovebecomes 2hs0rvv+g()IBs0h i.e.,theadjointpartofthecorrelationfunctioninEq.( 6{87 )aboveisasolutiontoanintegralequationoftheform 2hrvvg()IB+h ComparingthiswithEq.( 6{73 )aboveleadstotheconclusionthat Lastly,carryingouttheexternaltensorcontractionswithrespecttobkandbegives, 2(1+)n`3g()4 10ZdXDij(v)Cij(v): Thus,theinstantaneouspartofthisreducedGreen-KuboexpressiongivesthecollisionaltransferpartofthetransportcoecientinEq.( 6{70 ),whilethetimecorrelationfunctionpartcanbewritteninaformequivalenttotheintegralequationthatneedstobesolvedtoidentifythekineticpartofthetransportcoecientinEq.( 6{70 ).Theseobservationsaregenericinthattheyapplytotheothertransportcoecientsaswell(seeAppendixI).Thiscompletesthetaskathand,namelytoshowthattheresultobtainedfromtheChapman-Enskogmethodin[ 68 ]andthatobtainedhereasresultofanapproximateevaluationoftheexactresultobtainedfromthelinearresponsemethod.

PAGE 130

Inclosing,thekeypointsintheChapterabovearesummarized.IntherstpartoftheChapter,thegeneralproceduretogofromtherepresentationofthetransportcoecientintermsofanNparticletimecorrelationfunctiontoareducedrepresentationinvolvingoneparticlefunctions,forarbitraryclosuresoftherstequationofthelocalHCSensemblehierarchyisdetailed.Next,thecaseofelastichardspheresisconsideredandtheEnskogkinetictheoryforthesetimecorrelationfunctionsisderived,withspecialnotetakenoftheroleplayedbytheequilibriumreferenceensembleintheprocess.Then,thecorrespondingproblemforgranularuidsisconsideredanditispointedoutthattheanalysisiscomplicatedbecauseofthenon-equilibriumreferencestate.AclosurecriterionisproposedthatentailstheneglectofvelocitycorrelationsinthetwoparticledistributionfunctionintheprecollisionhemisphereandanEnskoglikekinetictheoryforasystemofinelastichardspheres.ThisisusedtoderivereducedGreen-Kuboexpressionsforallthetransportcoecientsconsideredinthiswork.Also,theexistenceofthelongtimelimitisestablishedbystudyingtheinvariantsofthereduceddynamics.Finally,theshearviscosityisconsideredasanillustrativeexampletocomparetheresultsobtainedfromthelinearresponseproceduretothoseobtainedbyusingtheChapman-EnskogmethodtosolvetheRevisedEnskogTheoryforinelastichardspheres. TheworkinthischapterservesasanillustrationinthatitisanexampleofanapproximateevaluationoftheHelfandandGreen-Kuboformsofthetransportcoecients.Also,theresultsinthischapterprovidetheconnectionbetweenthisworkandthevastbodyofliteraturethatexistsassociatedwiththestudyofhydrodynamicsusingthekinetictheoryofinelastichardparticles.

PAGE 131

Granularuidsbelongtothecategoryofphysicalsystemsthatareofpracticalrelevancetoindustriesandatthesametimearereasonablyrepresentedbymodelsthatareamenabletotheoreticalinvestigations.AwidelystudiedmodelsystemisoneconsistingofNhardparticlesthatcollideinelasticallysoastoloseafractionoftheirkineticenergyduringeachcollision.Theoreticalstudiesuptothispointintimehavefocussedonthekinetictheoryofthismodelsystem.Thisworkisarststepinthedirectionofusingthetoolsofnon-equilibriumstatisticalmechanicstostudythedynamicsofthissystem.InthisChapter,theprimaryresultsinthispresentationaresummarized.Then,theimmediateconsequencesofthisworkinthecontextofunderstandingthehydrodynamicdescriptionofagranularuidaregiven.Finally,theavenuesavailableforfuturetheoreticalexplorationsarediscussed. 122

PAGE 132

PAGE 133

PAGE 134

quantitativelycapturesthephysicsofthesystem.Inthecaseofnormaluids,theonlylimitsontheapplicabilityofhydrodynamicsaretheparametersandtransportcoecientsusedintheseequations.Ifthetransportcoecientsusedarethoseobtainedfromkinetictheory,thentheresultinghydrodynamicdescriptionisapplicableonlyinthelowtomoderatedensityregimes.Amoresophisticatedevaluationofthesetransportcoecients,forexamplethroughthenumericalevaluationoftheexactformsobtainedfromthetimecorrelationfunctionmethod,extendsthedomainofapplicabilityintodenseregimesaswell.Inthecaseofgranularuids,thedomainofapplicabilityofahydrodynamicdescriptionisnotonlysetbythequalityofthetransportcoecientsbutalsobytheneedtoidentifyclearlytheconditionsunderwhichthegranularmaterialissucientlyuidizedforthedescriptiontoberelevant.Thisworkpotentiallygivesaccesstohydrodynamicequationswhoseparametersarenotlimitedtolowdensitiesorweakinelasiticitiesfortheirapplicability.Hence,oncetheseparametersareavailable,itispossibletoestablishthedomainsofapplicabilityofthehydrodynamicdescriptiontoagranularuidwithouttheproblemofdistinguishingwhenthetransportcoecientsbecomeinvalidratherthanthehydrodynamicdescriptionitself. Insummary,notethattheusefulnessoftheresultsobtainedinthisworkrestsontheirnumericalevaluationandontheirservingastractablestartingpointsforanalyticalapproximationschemes.First,letusconsiderthequestionofnumericalevaluation.Thestartingpointforthenumericalanalysisoftheseexpressionsistheidenticationofthereferencestateoverwhichthecomputationsarecarriedout.Theexplicitformofthisreferencestateisunknownatthistimeexceptforitsspecicationasthe\normal"homogeneoussolutiontotheLiouvilleequation.ButnoticethattheonlycharacteristicpropertyofthisstateusedinthisanalysisistheHa'slawcoolingexhibitedbythetemperatureinthisstate.Hence,in

PAGE 135

PAGE 136

PAGE 137

Inthisappendix,thegeneratorsofdynamicsfortrajectoriesanddistributionfunctionsforasystemofinelastichardspheresarederived.Notethathardspheredynamicsischaracterizedbypiecewiseconstantvelocitiesthatchangeinstantaneously(anddiscontinuously)atthetimeofcollision.Thisfactallowsthegeneratoroftrajectoriestobederivedusinggeometricarguments[ 71 ].Thisderivationiscarriedoutbelow. Forthispurpose,letusbeginbyconsideringasystemoftwoinelastichardspheresthatcollidesothatthecenterofmassvelocityP12=(v1+v2)=2isunchangedandtherelativevelocityg12=v1v2changesinstantaneouslyaccordingtotherule whereisthehardspherediameter,bistheunitvectorgoingfromthecenterofparticle2tothecenterofparticle1,isthecoecientofrestitution,andthe0onanyquantitydenotesitspost-collisionvalue.Then,thetrajectoryofthissystemisgoingtobeoftheform(t)=fqi+vit;vigi=1;2,i.e.,freestreaminguntilacollisionoccurs.Supposetheparticlescollideatatime,thenthepostcollisiontrajectoryisoftheform0(t)=fqi+vi+v0i(t);v0igi=1;2,whereacollisionhasoccurredatatime()2[0;t]obtainedasasolutiontotheequation (q12+g12)22=0,(A{2) andv0iisgivenbythecollisionruleinEq.( A{1 )above.ThereforethetimeevolutionofanyphasefunctionA()canbegivencompactlybyanequation 128

PAGE 138

oftheform with(t)and0(t)ascharacterizedabove. Dierentiationoftheaboveequationwithtimegives (A{4) whereusehasbeenmadeofEq.( A{3 ).Since, theEq.( A{4 )abovebecomes whereb12isasubstitutionoperatoroftheform Next,toeliminatetheinitialconditionoccurringthrough(),Eq.( A{2 )isused.Therearetworootstothisequation,thatcorrespondtothetimeatwhichthetwoparticlesarerstincontact,andthetimeatwhichtheywerelastincontactiftheywereallowedtostreamthrougheachother.Clearlytherstcorrespondstothephysicalcollisiontime.Thisispickedoutbyrecognizingthatatthephysical,

PAGE 139

AndhenceEq.( A{6 )takestheform (A{9) Finally,usingthefactthatthegeneratoroftrajectoriesisdenedas thegeneratorLcanbeidentiedas @qi+(q12)(q12g12)jbq12g12j[b121] (A{11) =2Xi=1vi@ @qi+T(12) (A{12) Thisgivestheidenticationofthegeneratorforthetwobodyproblem. TogofromheretothegeneratoroftrajectoriesforasystemofNparticlesrequirestheadditionalassumptionthatonlybinarycollisionsoccurbetweenparticles(thebasisofwhichisdiscussedinthemaintext).Thisgivesthegeneratoras 2Xi;jT(i;j);(A{13) whichistheresultquotedinthetextinEq.( 2{9 ). Next,considertheLiouvillegeneratorassociatedwiththedistributionfunctionsoverphasespace.RecallthattheadjointLiouvilleoperatorisdened

PAGE 140

throughtheequation Inparticular, Itturnsoutthat,forhardparticlesingeneral,theLiouvilleoperatorsassociatedwithphasefunctionsandwithdistributionfunctionsaredierentbecauseoftheexcludedvolumeassociatedwithanydistributionofsuchparticles.Hardparticlescannotoverlap.Hence,anyvaliddistributionofhardparticlescanbewrittenas where, istheexcludedvolumefactorthatguaranteesthenonoverlapoftheparticles.IntherepresentationgivenontheleftsideofEq.( A{15 ),theexcludedvolumefactorensuresthattheintegrationoverspaceisonlyoveracceptablecongurations.Butontherightside,theintegrationisunconstrainedandhencethegeneratorissuitablymodiedtoensurethecorrectresult.Inwhatfollows,theformoftheadjointgeneratorisidentiedbychangingvariablesintheintegrationasdescribedbelow.

PAGE 141

StartwiththelefthandsideoftheEq.( A{15 ).PuttingintheformofLobtainedabove,thisbecomes 2Xi;jZd()(qij)(qijgij)jbqijgijj[bij1]A()=NXi=1Zd(virqi())A()+NXi=1ZSdbqiZdvidj6=i(vi()A())+1 2Xi;jZd1 2Xi;jZd()(qij)(qijgij)jbqijgijjA() (A{18) where,inthecollisionalterm,achangeofvariablestothepostcollisionvelocitiesv0iandv0jhascarriedoutandusehasbeenmadeofthefactthattheJacobianassociatedwiththistransformationis Theoperatorb1ijisdenedthroughtheidentity andcanbedirectlysolvedfor,whichyields (bgij)b:(A{21)

PAGE 142

Then,fortheclassoffunctionsforwhichthesurfaceintegralinEq.( A{18 )vanishes,theadjointLiouvilleoperatorcanbeidentiedtobe 2NXi=1NXj6=i where,thenewbinarycollisionoperatoris T(i;j)=(qij)jgij^qijj(^gij^qij)2b1ij(^gij^qij):(A{23) Thiscompletesthederivationofthegeneratoroftheadjointdynamics. Lastly,theoperatorLdenedinthetextinEq.( 2{54 )isderived.Inordertoderivesuchanoperator,useismadeofaspecialcollisionalboundaryconditionthatischaracteristicofhardspheredistributions.Theboundaryconditionforanypairatcontactis[ 53 ] Thisboundaryconditioncanbederivedasfollows.Sincethecollisiontimeforhardparticlesiszero,itiscorrecttosaythatatagiventime,theuxofparticleswithrelativevelocitygwithrespecttoanychosenparticleinitspre-collisionhemisphereshouldbeequaltotheuxofparticlesinthepost-collisionhemispherewithrelativevelocityg0generatedfromgusingthecollisionrulegiveninEq.( A{1 ).Henceforagivenparticlej, where0ijdenotesthephasespacemeasureafterparticlesiandjhavecollided,allotherphasepointsstayingthesame.SinceEq.( A{25 )isasumoverfunctions,theequalitymustholdpairwise.Moreover,notethatfromEq.( A{19 )above,itfollows

PAGE 143

that and PuttingallofthesetogetherinEq.( A{25 )aboveandrenaming0asgives whichistheboundaryconditionstatedearlier. Now,inordertoidentifythecollisionoperatorassociatedwiththebackwardgenerator,usetheidentityabovetorewritethe T(i;j)(A)=(qij)jgij^qijj(gij^qij)2b1ij(gij^qij)(A)=(qij)(gij^qij)jgij^qijjb1ij1A+A(qij)jgij^qijj(gij^qij)2b1ij(gij^qij)T(i;j)A+AT(i;j) Hence,itfollowsthat L(A)=AL+LA(A{30) with 2NXi=1NXj6=iT(i;j)(A{31) and whichistheresultquotedinthetext. Thatconcludesthederivationofthegeneratorsassociatedwiththeinelastichardspheredynamics.Theprimaryresultsinthisappendixaresummarizedinatablebelow.

PAGE 144

TableA{1. HardSphereCollisionOperators TrajectoriesT(i;j)(qij)jgijbqijj(bgijbqij)(bij1)DistributionfunctionsT(i;j)(qij)jgij^qijj(bgij^qij)2b1ij(bgij^qij)BackwarddynamicsT(i;j)(qij)jgij^qijj(bgij^qij)b1ij1

PAGE 145

Inthisappendix,thegeneratorsofthedynamicsinthestationaryrepresentationoftheHomogeneousCoolingStateensemblearederived.FirstconsidertheLiouvilleequation TherepresentationforthedynamicsthatisderivedhereaccordsaspecialstatustothatpartofthedynamicsassociatedwiththecoolingofthehomogeneoustemperatureTh(t).Hence,thetimederivativeintheaboveequationispartitionedas @t=@ @tjTh(t)+@Th(t) @Th(t):(B{2) Also,recallthat ThistransformstheLiouvilleequationabovetotheform @tjTh(t)h(Th(t))Th(t)@ @Th(t)+ Next,deneadimensionlessdistributionfunctionthroughtheidentication where 136

PAGE 146

ands(t)isanasyetunspeciedfunctionoft.Then,itfollowsthat 2h(Th(t))(`vh(t))3N(;s)1 2h(Th(t))(`vh(t))3NXivirvi(;s); whereusehasbeenmadeofthefactthatforhardspheres,thereisnointernalenergyscaleandhencetheonlywaythetemperaturecanenterthedistributionisthroughthescalingofthevelocitiesrepresentedthroughthevariables.SubstitutingthisforminEq.( B{4 )abovegives vh(t)@s @t@ @sj+1 2hXirvivi+ where vh(t)h(t); vh(t) with vh(t)@s @t1(B{10) Thisidentiessas orequivalentlyusingEq.( 2{29 )andintegratinguptheaboveform, 2`ht(B{12) Summarizing,thedynamicsofanydistributionfunction(;t)takestheform

PAGE 147

withthenewgeneratorforLiouvilledynamics 2hNXi=1rVi(Vi); L=` vh(t) whichistheresultstatedinEq.( 2{37 )inChapter2. Next,theadjointdynamicsofphasefunctionsinthisstationaryrepresentationisconsidered.RecallthattheensembleaverageofaphasefunctionAhasthetwoequivalentrepresentations (B{15) Thesecondrepresentationabovecanberewrittenas Ls()A(f`qi;vh(t)vig)(B{16) andhence L()A(f`qi;vh(t)vig):(B{17) BycarryingoutanintegrationbypartsinthespaceandgoingtoprecollisionvariablesasinthepreviousappendixEqs.( A{18 )-( A{21 ),theaboveexpressioncanberewrittenas L()A(f`qi;vh(t)vig)=Zd()(LA(f`qi;vh(t)vig))(B{18) where 2hNXi=1virvi;L=` vh(t)L:(B{19) ThisidentiesthegeneratorassociatedwithphasefunctionsinthestationaryrepresentationoftheHCS.

PAGE 148

NextthestationaryrepresentationoftimecorrelationfunctionsovertheHCSensembleisconsidered.Forthispurpose,notethatthemostgeneraltimecorrelationfunctionisathreetimecorrelationfunctionoftheform Changingvariablesintheintegrationoverphasespacesuchthat!t0theaboveequationcanbewrittenas (B{21) =Zdh(;t0)heL(tt0)A()iB() (B{22) Next,achangeofvariablesfromto=nqi AlsorecognizethatforanyfunctionF(fvig)thefollowingidentityholds. 2hs(t;t0)PivirviF(fvig)=Fne1 2hs(t;t0)vio=Fvh(t) Thisallowsthecorrelationfunctionabovetoberewrittenas where 2hs(t;t0)Pivirvi:(B{26) Noticethat @s=S(t;t0)"vh(t0) 2hs(t;t0)L+1 2hXivirvi# =S(t;t0)L

PAGE 149

SinceLisindependentofs,thiscanbeintegrateduptoget Allthephasefunctionsofinterestinthispresentationarehomogeneousfunctionsofthevelocityandhenceonecanwrite (B{30) Usingthis,thetimecorrelationfunctionabovecanbewrittenas where Thisistheclassofdimensionlesscorrelationfunctionsthatisconsideredinthetextinthecontextoflinearresponse. Further,usingthepropertyEq.( A{30 )inthepreviousappendix,anoperatorLcanbeidentiedsuchthat with 2hNXi=1virviandL=` vh(t)L:(B{35)

PAGE 150

andLbeingtheoperatoridentiedinEq.( A{31 )earlier.Hence,thetimecorrelationfunctionabovecanbewritteninthreeequivalentformsas Lsh()B()iA()=Zdh()eLsB()A() (B{36) whichistheresultstatedinthetext. 2{34 )isbasedonscalingrelativetovh(t).Thiswasdonesoastobeabletoposetheoreticalquestionsofinterestinanelegantselfconsistentform.However,thisisinconvenientinpracticesincethecoolingrateisgivenimplicitlyintermsofthestationaryHCS.Instead,thesameanalysiscanbeperformedbyscalingwithaknownfunction!(t)insteadofvh(t);togetaLiouvilleequationintheform whereasearlier`isaconstantcharacteristiclengthinthesystemand ~=(`!(t))3N(;t):(B{38) Deneanewtimevariableby andchoose!(t)tomakethecoecientsofthisequationindependentofs!(t) dt!1(t)1 2~! `

PAGE 151

where~!isanarbitraryconstantthatcanbepickedforconvenience.Thisgives 2~! `t;d~s=2` 2~!NXi=1r~Vi(~Vi~)+e Thisisformallythesameas( B{13 )exceptherethecoolingratehasbeenreplacedbythearbitraryconstant~!,whichmaybechosenforconvenience(e.g.,~!=1). Thereisastationarysolutionto( B{41 )givenby 2~!NXi=1r~Vi(~Vi~)+e Clearly~~!isthesameashwithonlytheunknownvaluehreplacedby~!.However,itispossibletodeterminehfromthechosenvalueof~!andthemeasuredvalueofthesteadystatetemperaturefrom( B{42 ) Thisrelationshipmaybederivedasfollows.Dene ~T(~s)=T(t) i.e.,thetemperature~Tistherealtemperatureexpressedinthearbitraryscalingvariables.Nowconsiderthedynamicalequationassociatedwiththisscaledtemperature, @~s~T(~s)=T(t)@ @~s1 @~sT(t)

PAGE 152

UsingEq.( 2{23 )inthetexttosimplifythesecondtermandEq.( B{40 )abovetosimplifytherstterm,thiscanbewrittenas @~s~!~T(~s)=` !(t)(t)~T(~s):(B{45) Inparticular,usingthescalingformoftheHCSthisbecomes @~s~!~T(~s)=p Integratingtheaboveequationdirectlygives ~T(s)=~!2 hq whichinthelongtimelimitgoestoEq.( B{43 )above.Therefore,inpractice,oneimaginesmeasuring~T(1)ratherthansolvingforhself-consistentlyintheHCSstate.Also,thedierentgeneratorsdenedearlierandthestationaryrepresentationoftwo-timecorrelationfunctionsovertheHCSensemblecanbetranslatedintothislanguageofarbitraryscaling.

PAGE 153

Inthisappendixthemicroscopicbalanceequationsassociatedwiththephasefunctionsfea(k;t)garederivedandtheexplicitformsofthevariousuxesinvolvedareidentied.Therearetwodistinctcasestobeconsidered.Oneisthebalanceequationsassociatedwithdynamicsatt>0,i.e.,forwarddynamics,wherethegeneratoristheLoperatordenedinEq.( A{13 )inAppendixA.Theotheristhebackwarddynamicsfortimet<0thatisgeneratedbytheLoperatordenedinEq.( A{31 )inAppendixA.Bothformsoftheuxeswillprovenecessaryforcomparisonofresultstothoseobtainedintheelasticlimit,wherethetransportcoecientsturnouttobeforwardux-backwarduxtimecorrelationfunctions(seeChapter5inthemaintext).Tobeginwith,thedynamicsfort>0isconsidered. Inthiscasethedynamicsofthephasefunctionsisgivenbyanequationoftheform with 2Xi;jT(i;j)(C{2) where Inwhatfollows,theactionoftheLoperatoronthephasefunctionsofinterestisevaluatedinordertoextractabalanceequationfromtheaboveLiouvilleequation. 144

PAGE 154

Firstconsiderthenumberdensity, TheassociatedLiouvilleequationgoverningitsforwardintimedynamicsis ClearlytheactionoftheToperatoronthisdensityvanishes.Hence, withtheidenticationthat Next,considerthemomentumdensitybg(r)andrecognizethat

PAGE 155

and Nowusethefactthat (bij1)vj=(bij1)vi(C{11) torewritethisas Also, @(rqi+qij)=Z10drr(rqi+qij)qij; andhence Further, (bij1)vj=1 2(1+)(bgij)^:(C{15)

PAGE 156

SubstitutingtheseresultsintoEq.( C{12 )aboveyields 2(1+)(bqijgij)bqijbqijZ10d(rqi+qij)g Puttingallofthistogether,theconservationlawforthemomentumtakestheform with 4(1+)XiXj(qij)(qijgij)jbqijgijj(bqijgij)bqijbqijZ10d(rqi+qij)g wherethersttermisthekineticpartofthemomentumuxandthesecondtermisthecollisionaltransferpart. Finally,considertheenergydensitybe(r)=Pi1 2mv2i(rqi).Inthiscase,theactionofLconsistsof 2mv2j(rqj)=rrXi1 2mv2ivi(rqi)(C{19) andT(i;j)NXk=11 2mv2k(rqk)=(qij)(qijgij)jbqijgijj[bij1]NXk=11 2mv2k(rqk)=(qij)(qijgij)jbqijgijj[bij1]1 2mv2i(rqi)+1 2mv2j(rqj):

PAGE 157

Further, (bij1)v2j=(1+)2 4(gijbqij)2+(gijbqij)(vjbqij) (C{20) and (bij1)v2i= 4(gijbqij)2(gijbqij)(vibqij): ReorderingthedummyindicesinthesumovertheToperatorsandusingtheidentityEq.( C{11 )above,gives 1 2Xi;jT(i;j)Xk1 2mv2k(rqk)=rrm (C{22) wheretheresultingexpressionhasbeensuggestivelyregroupedtodisplaythepartofthecollisionaltermthatcanbewrittenasagradientandanotherpartthatisinherently\local".Thersttermisthepartthatgivestheconventionalheatuxintheelasticlimit.Thesecondtermisthesourcethatgeneratesthelocalcoolingrate.Hence,thebalanceequationassociatedwiththeenergydensitycanbewrittenintheform

PAGE 158

withtheidentication 2mv2ivi(rqi)m and Thatcompletestheidenticationoftheuxesandsourceinthemicroscopicbalanceequationsfortheforwarddynamics. Asimilaranalysiscanbedoneforthebalanceequationsassociatedwiththebackwarddynamicsofthephasefunctions.Thebackwarddynamicsofaphasefunctiona(r)isdenedthroughtheidentity Then,aswasdoneearlier,theformoftheuxeshavetobeidentiedfromthedynamicalequation Theformoftheconservationlawitselfisthesameasearlier,theonlydierencebeingthatthecollisionaltransferpartoftheuxeswillbegeneratedbytheT(i;j)operatorgiveninTable1earlier.Suchananalysisyieldstheformoftheuxesas

PAGE 159

withthesuperscriptonthesequantitiesisusedtodenotethefactthattheyareassociatedwiththebackwarddynamicsequations.Noticethatthesourcetermisnowpositiveaccountingforthefactthatthebackwardtrajectoryofasystemofinelastichardsphereswillbesuchthattheenergyofthesystemincreaseswithtime. Thiscompletestheidenticationofthemicroscopicuxesassociatedwiththephasefunctionswhoseensembleaveragegivesthehydrodynamicvariablesofinterest.Inwhatfollowstheprimaryresultsofthisappendixaretabulated. TableC{1. MicroscopicBalanceEquations DensityBalanceEquation numberdensity@bn(r;t)

PAGE 160

TableC{2. FormsoftheForwardFluxesandSource FluxForwardintimeform momentumuxh(r)Pimvivi(rqi)m1 4(1+)PiPj(qij)(qijgij)(bqijgij)2bqijbqijR10d(rqi+qij) energyuxs(r)Pi1 2mv2ivi(rqi)m energysourcew(r)m TableC{3. FormsoftheBackwardFluxesandSource FluxBackwardintimeform momentumuxh(r)mPivivi(rqi)+m energyuxs(r)m energysourcew(r)m

PAGE 161

Inthisappendix,thespecialfunctions'sthatarethe\microscopicprecursors"tohydrodynamics,i.e.,thosefunctionswhichhavethesamedynamicsasthelinearizedhydrodynamicequationsgiveninthetextinEq.( 4{28 )areidentied.Thestartingpointforthisanalysisisthedeningequationofthehomogeneousreferencestate,namely 2h(t)NXi=1rvi((viu))+ Usingthesamemethodusedin[ 38 ]toderivetheeigenfunctionsassociatedwiththehydrodynamicmodesattheleveloftheBoltzmannequation,thederivativeoftheaboveequationwithrespecttotheparametersofthisensemble,namelythedensity,temperatureandowvelocityareconsidered.First,thederivativewithrespecttothetemperatureTatxeddensity,owvelocity,phasepointandtimetisconsidered. @T Usingtheformof 2@h(t) Next,itisnotedthatthescalingpropertyoftheHCSgivestheproperties 2NXi=1rvi((viu)h)=T@h 152

PAGE 162

Useofthesein( D{3 )givestheidentity 2h(t)@h Inananalogousway,dierentiatingEq.( D{1 )withrespecttothedensitygives 2@h(t) where( D{4 )hasbeenusedinthelastequality.Finally,threemorerelationshipsfollowfromdierentiatingEq.( D{1 )withrespecttothethreecomponentsoftheowvelocity @uh=1 2h@ @uh:(D{7) Itcanbereadilyinferredfromtheabovethat whereKhydT(0)isthetransposeofthegeneratorofthehydrodynamicequationsinthehomogeneouslimitidentiedinEq.( 3{30 )and'sarethefunctions (;s)=(`vh(t))3Nyh(t)@h(;t) wherethedimensionlessformsoftheabovefunctionsaredenedinananalogouswaytothatusedfordistributionfunctionsinEq.( B{5 )inAppendixB.Thisistheresultquotedinthetext. Further,theexpressioninEq.( D{6 )canberearrangedas

PAGE 163

Thus,anequivalentexpressiontoEq.( D{8 )canbeobtainedas where=()(k=0)arethehydrodynamicmodesatk=0giveninEqs.( 3{34 )-( 3{37 )inthetext, 2h;1 2h;1 2h;1 2h;(D{12) and 1=@h 2=@h 3;4;5=vhcs@h withtheirdimensionlessformsdenedthrough (;s)=(`vh(t))3N(;t):(D{16) Therefore,anequivalentcharacterizationofthesespecialfunctionsisthattheyareeigenfunctionsofthescaledLiouvilleoperatorwiththesameeigenfunctionsasthehydrodynamicmodes.

PAGE 164

Inthisappendix,themicroscopicbalanceequationsassociatedwiththenumberdensity,momentumdensityandowvelocityarerewritteninthedimensionlessvariablesgiveninEq.( 2{34 ).Also,thespecialhomogeneousfunctionscharacterizedareusedtogetasetof\adjoint"conservationlawsassociatedwiththefunctions'sdenedinEq.( 4{36 ),thatconstitutethespecialinitialpreparationconsideredinthetext. First,recallthatthemicroscopicbalanceequationsassociatedwiththedensitiesfbn(r);be(r);bg(r)gwerederivedinAppendixCabove.InorderthattheensembleaverageoftheseequationsinthedimensionlessformgoovertothedimensionlesshydrodynamicequationsgiveninEq.( 3{25 )inChapter3earlier,theyarenon-dimensionalizedwithrespecttothehydrodynamiceldsinthehomogeneousstate,i.e., ^n(r)^n(r) (E{1) (E{2) (E{3) Withthischoice,thebalanceequationsinthedimensionlessformbecome (E{4) @s^g1 2h^g+rrh=0 (E{5) 155

PAGE 165

with (E{7) (E{8) Further,thebalanceequationsofinterestarethosethatcorrespondtothelinearcombination 3be3 2bn;bg;(E{10) Considertheparticularlinearcombinationthatcorrespondstoa2.Inthiscase,oneobtains@ @s2 3bebnh2 3be+rr2 3s^g=2 3w E{10 )abovetowrite@a2(r;s) 3s(r;s)^g(r;s)=l(r;s); 3w(r;s)+3 2h2 3be(r;s)bn(r;s)+h@lnh hasbeenintroduced.Or,equivalentlyintheFourierrepresentation, 3esbeg=el(k;s)(E{12) Next,thehomogeneouslimitofthesourcetermel(k;s)ischaracterized.Forthispurpose,rstnotethatfromthedenitionofthecoolingrate,itmustbetruethat 2n(r;t)T(r;t)(r;t):(E{13)

PAGE 166

ConsiderthesecondterminEq.( E{11 )inthehomogeneouslimit.ItfollowsfromthedenitioninEq.( E{13 )that 3 2h2 3bee(0;s)ben(0;s)+h@lnh 2hea2(0)+h@lnh 3w(0)e(0) whereusehasbeenmadeofthefactthath= 3ew(0)(E{15) withthenotation and Hence,ingeneralel(k;s)hasthepropertythat 3ew(0;s);(E{18) i.e.,itisorthogonaltotheinitialpreparationinthelongwavelengthlimit.Thisisjustarestatementofthefactthatinthelongwavelengthlimit,thisinitialpreparationexcitesonlythehydrodynamicmodes,ascanbeinferredbyconsideringtheensembleaverageofEq.( E{12 )above. Insummary,theconservationlawsassociatedwiththechosenphasefunctionsfagindimensionlessvariablestaketheform

PAGE 167

with 3s^g;h(E{20) Next,considertheadjointfunctions(r)denedinEq.( 4{36 )inChapter4as Thedynamicalequationassociatedwiththesefunctionsis Notethatinthehomogeneouslimit, UsingthepropertyinEq.( D{8 )inthepreviousappendix,theabovedynamicalequationcanbewrittenintheform withthedenition for,asestablishedbyEq.( E{23 )above,theabovetermiszeroatk=0andhenceisofleadingorderink. Thiscompletestheidenticationofthedirectandadjointconservationlawsinthestationaryrepresentation.

PAGE 168

Inthisappendix,thedetailsofobtainingtheresultsgiveninChapter4associatedwiththeperturbativeexpansionofthetransportmatrixK(k;s)aregiven. 1.HomogeneousLimit UsingthepropertiesofthefunctionsinAppendixDearlier,itfollowsthat Further,when=1followsfromtheeigenvalueequationthat andhence 159

PAGE 169

Thisequationcanbeintegrateduptogive Also and Hence,thetransportmatrixinthehomogeneouslimitbecomes

PAGE 170

whichispreciselythehydrodynamictransportmatrixidentiedinChapter3.Furthernotethatbyexplicitcalculation,itcanbeconcludedthat ThesepropertiesturnouttobeusefulintheexplicitevaluationofthetransportcoecientsinChapter4andChapter5. 4{72 )inChapter4afteritsintrinsickorderingwasexposedusingthedirectandadjointconservationlawswas (F{10) + (F{11) 3ew(0)

PAGE 171

Theabovefollowsfromthefactthat(1P)projectsorthogonaltothe F{9 )earlierandthedenitionof 4{62 )as that theidentitymatrix.So,asdenedinEq.( 4{53 )inChapter4,calling andidentifyingthecorrespondingtermsfromEq.( F{11 )abovegives (F{18) + 4{73 ) onecanrecognizethat

PAGE 172

(F{23) 4{78 ),Eq.( 4{87 )andEq.( 4{93 )respectively. Next,theexplicitformsofthevariouskexpandedcorrelationfunctionsaboveareconsideredandthenonzerotermsidentied.Firstconsider 4{63 ),itfollowsthat Itisprovedinthefollowingthatthersttermintheaboveequationiszero.Forthispurpose,noticefromtheexpressionsoff(r;s)giveninTable2inAppendixCthattheyaregenericallyoftheform withhbeinganevenparityfunctionwithrespecttointerchangeofindicesiandj.Thus,Fouriertransformingthisequationgives (F{26) =Xij(vi)eikqi+Xi;jhgij;Pij;qijeik(qi+qj) kqij=2:

PAGE 173

NotethatintherstterminEq.( F{24 ),ef(1)isaveragedoverthehomogeneousfunction 1 Hence, asclaimedinthetext. Nowconsiderthecorrelationfunction 4{64 ),itfollowsthat Againasearlierthegenericformoflcanberecognizedtobeoftheform ThereforeduetothesameargumentsasweremadeinthecaseofEq.( F{25 )above,itcanbeconcludedthat 1 andhence asclaimedinthetext.

PAGE 174

Next,considerthecorrelationmatrix 4{64 )thisisrecognizedas (F{33) Thersttermintheaboveequationiszero.Inordertoprovethisconsidertheformofel(k). 3ew(k)+3 2h2 3bee(k)ben(k)+h@lnh Itcanbeseeneasilythattheaboveexpressioncanberewrittenintheequivalentform 3ew(k)ea(k)Zd2 3ew(0) ConsiderZdel(k) 3ew(k)Zd 3ew(0) 3ew(0)k;0Zd2 3ew(0) Hencethisfunctioniszerotoaallordersink.Inparticular 1 Finally,considertheform 4{70 ),itfollowsthat whichistheformgiveninthetext.

PAGE 175

and Theseareprovedbelow. InordertoseethattherelationshipinEq.( F{38 )istrue,notethat andthefunctionseaand ExpandingthesecondequationorderbyorderinkandusingthesphericalsymmetryoftheHCSgivesthedesiredresult. Next,toprovethesecondidentity,expandtherighthandsideoftheaboveequationtogete

PAGE 176

Therefore1P>e ThesearetheresultsusedinthetextinChapter4.

PAGE 177

Inthissection,thedetailsofthederivationofvariousresultsquotedinthetextinChapter5arederived.EachmatrixelementinthetransportmatrixKisconsideredateachorderinktoidentifythetermsidentiedonphenomenologicalgroundsinthehydrodynamictransportmatrixKhydgiveninEq.( 3{25 )inChapter3.BeforetheperturbativeexpansionofKisconsidered,someresultscanbeprovedtobetrueatallorders.First,itisshownthatthecontinuityequationassociatedwiththenumberdensityisshowntocomeoutofthefullKmatrix,toallordersink.Then,itisshownthatthetransversedegreesoffreedomdecouplefromthelongitudinaldegreesoffreedomforallk. Inordertoshowtherstresult,namelythecontinuityequation,considerthepartofthetransportmatrixwhentheobservableisthenumberdensity,namelythe1matrixelements where Carryingoutthetimederivativeandusingtheconservationlawassociatedwitha1resultsin (G{3) 168

PAGE 178

whereusehasbeenmadeofthefactthattheuxassociatedwiththenumberdensityisthemomentumdensity,whichisoneofthehydrodynamicvariablesunderconsideration.Hence whichgivesthecontinuityequationtoallordersink. Next,considerthetransportmatrixelementsassociatedwiththetransversedegreesoffreedom.Forthiscasethecorrelationfunctionofinterestis Whenisoneofthelongitudinaldegreesoffreedomtheonlyothervectorsintheproblemwillbebk's.Therefore,theorthogonalityconditionsbe1bk=0togetherwiththesphericalsymmetryoftheunderlyinghomogeneousstateguaranteesthatallsuchmatrixelementsarezero.Further,sincebe1be2=0also,theaboveequationreducesto Asimilarresultfollowsfortheothertransversedegreeoffreedomaswell.Thus,thetransversedegreesoffreedomdecouplefromthelongitudinalonesandarediagonaltoallordersink.Thiscompletesourgeneralconsiderations.IntherestoftheappendixattentionisfocussedonthekexpandedformsofthetransportmatrixgiveninEqs.( F{20 )-( F{23 )inthepreviousappendix. F{20 )inthepreviousappendixisconsideredtermbytermandthevarioustermsrelatedtothepressureoftheuidareidentied.

PAGE 179

Firstconsiderthe21matrixelement,thisisgivenby Noticethat (G{8) Noticethat Next,considerthe23matrixelementgivenby Asarststeprewritetheabovematrixelementinthelaboratoryvariablestogive 3esi(0)1 @UjZdf2 3esiv2h(t) 2begigh(fqi;viUgi;t) (G{10)

PAGE 180

Next,makeachangeofvariablesintheintegrationsuchthatvi!viUandnotethattheheatuxsandthemomentumdensitygtransformunderaGalileantransformas 3esi+1 2mU2begi+Ujhij+beeUi toobtain 3(hij+^eij)v2hcs andhence 31 3Zdhiih(G{14) ThisistheresultquotedinthetextinEq.( 5{14 ). Nowlookatthe31matixelement, Again,restoringthedimensionstotheaboveequation,thiscanbewrittenas FollowingtheidenticationofthepressureasthetraceofthemicroscopicstresstensoraveragedovertheHCS,itiseasytothatthiscorrespondstotheresultgiveninthephenomenologicallinearhydrodynamicequationsgiveninEq.( 5{1 ),namelythedensityderivativeofthehydrostaticpressure.Similarly,the32matixelementcanbesimpliedtoidentifythatitisthetemperaturederivativeofthepressure.

PAGE 181

G{7 )and( G{8 )thattheonlynonvanishingentryinthispartofthetransportmatixisthe23element.TheformalexpressionofthismatrixelementwasderivedinEq.( 4{91 )earliertobe 3ew(0)1P> 3ew(0)1P> Therststepistoidentifyexplicitlythefunction Further,asobservedinEq.( 4{35 )earlier,thelocalhomogeneouscoolingstateisdenedthrough Hence, Now,inordertoidentifytheinitialconditionthatoccursinEq.( G{17 )intermsofthescaledvariables,notethat and (G{22) =Xlbkrvih()(rqi) (G{23)

PAGE 182

Therefore astheintegratingfactorin ThisistheresultquotedinthetextinEq.( 5{31 ).Further,fromthedenitionoftheadjointuxinEq.( 4{60 ),itfollowsthat ThiscompletestheidenticationoftheadjointdensitiesanduxesassociatedwiththisEulertransportcoecient. Next,notethatitwasshowninAppendixDEq.( E{11 )that (1P)2 3ew(0)=2 3ew(0)+3 2h2 3bee(0)ben(0)+h@lnh where ThusfornotationalsimplicitydeneaquantityWas 3ew(0)=2 3ew(0)+3 2h2 3EN+h@lnh SubstitutingtheseidenticationsinEq.( G{17 )abovegivestheresultinthetextinEq.( 5{24 ).

PAGE 183

4{78 ). Firstnoticethat,asshowninEq.( G{6 )above,therearenoEulertermsinthetransversemodes.Hencethesecondtermunderthetimecorrelationfunctiondropsout.Substitutingtheexplicitformsofthesecorrelationfunctionsgives Toidentifyallthedierentquantitiesintheaboveexpression,rstnotethatthebalanceequation( E{19 )givestheidentication 4(1+)XlXm(qlm) (G{32) G{18 )-( G{24 ),itcanbeshownthat HenceitfollowsfromthedenitionoftheadjointuxEq.( 4{60 )that SubstitutingalltheseidenticationsinEq.( G{31 )abovegives

PAGE 184

Theorthogonalityofbkandbe'sgives=1 10VZdehijMij+Zs0ds01 10VZdehije( 301 30VZdehiie( 4{78 ). 4 3+=bkibkjf Notingthat 4 3+=bkibkjf1 2p@lnph 2p1 Thephasefunction G{25 )aboveinthecontextoftheEulertransportcoecient.SimilarlytheuxisgivenbyEq.( G{26 )earlier.Puttingintheformofthemomentumuxgives 4 3+=bkibkjbki0bkj0f1 with 2p@lnph 2pea2(0)(G{39)

PAGE 185

AppropriatetensorcontractionsoftheaboveexpressionsgivetheresultinthetextinEq.( 5{43 ). 4{78 )andsubstitutingtheformsofthevariouscorrelationfunctionsgives 3p1 (G{40) First,thevarioustermsintheaboveexpressionarerecognizedas 3esibegi=2 31 2XlXm(qlm)(1+)(qlmglm)(glmbqlm)2(Plmbqlm)bqilmgPi:Si Also,recallthat2isdenedinEq.( 4{36 )as This,togetherwiththeformofthelocalhomogeneouscoolingstategiveninEq.( 4{35 )itiseasytoseethat 2Xlrvi(vih(;t))(rqi):(G{43) TransformingthistothedimensionlessvariablesandFouriertransformingwithrespecttorgives 2Xlrvl(vlh())eikql:(G{44)

PAGE 186

Therefore 2Xlrvl(vlh())qil(G{45) and 2Xlrvl(vlh())ql# Forcompactnessofnotation,dene 2Xlrvl(vlh())ql(G{47) SubstitutingtheseresultsintheexpressioninEq.( G{40 )gives 31 3VZdS(S)ie( with 3pPi(G{49) ThisistheresultquotedinEq.( 5{47 )inthetext. =bkibkjf1 3p1

PAGE 187

Thedirectuxefi2isthesameasthatidentiedforthethermalconductivityinEq.( G{41 )earlier.Now,fortheadjointfunctions,rstrecallthat,bydenition Unlikeinthecaseoftemperatureandtheowvelocity,thedensitydependenceofthelocalHCSisnotmadeexplicit.Hence,atthisstagetheidenticationofe(1)j1(0)canonlybeformal.Proceedingasintheearliercases,itiseasytoseethat wherethenewnotationhasbeenintroducedforcompactness.Next,tocalculatethequantities and Finally, Le2(k;s)h@lnh Observingthat2=h G{50 )givesEq.( 5{56 )inthetext.

PAGE 188

Inthisappendix,thederivationoftheGreen-Kuboexpressionforelastichardspheresisoutlined.However,whatisdonehereisnotstrictlyspeakingthe!1limitofthederivedGreen-KuboexpressionsforthetransportcoecientsgiveninChapter5ofthiswork.ThisisbecausetheHCSensembleintheelasticlimitgoesovertoaxedmomentummicrocanonicalensemble.Thelinearresponseprocedureasoutlinedinthisworkistechnicallycomplicatedforaclosedensembleofthisform.SincetheelasticlimitexpressionsforthevarioustransportcoecientsareusedasaguidetothephysicalcontentoftheGreen-Kuborelationsfor6=1,advantageistakenofthefactthatphysicalobservablescalculatedusinganyoftheequilibriumensemblesgivesthesameresultinthethermodynamiclimitandtheGrandCanonicalEnsembleisusedtoderivethedesiredresults. Recallthatthelocalgrandcanonicalensembleisoftheform[ 37 ], wheretheparametersoftheensemblearedenedas 2mU2;=1 179

PAGE 189

Also,W()istheexcludedvolumefactordenedinEq.( A{17 )inAppendixA.Further,theensemblecanbewritteninanequivalentformas wherethedensitiesbn(r),be(r)andbg(r)areasdenedinEq.( 4{6 )andthenormalizationconstantQLisdeterminedasearlier. Startingfromthislocalstate,thevariousquantitiesthatdeterminetheadjointfunctionsandtheirassociateduxesarerstcalculated.Recallthattheadjointfunctionsaredenedthrough where Butthenaturalindependentvariablesforthegrandcanonicalensembleare Firstconsiderthecase=1.Thehydrodynamiceldhereisthedensity.Inthiscase (r0)jeq;T (r0) (r0) whereusehasbeenmadeofthefactthat (r0)jeq;T=hbn(r)bn(r0)ieqgnn(rr0):(H{8)

PAGE 190

Next,consider=2.Inthiscase, whereusehasbeenmadeoftheidentitythat togetherwiththerecognitionthat andtheanalogousrelationinEq.( H{8 )above.But,usingtheexplicitformofthegrandcanonicalensembleinEq.( H{1 )above,itiseasytoseethat 2n(r)T(r)(H{12) Andhencethecorrelationfunctionsabovecanbecalculateddirectlytogive 2Tbn(r)G:C(H{13) Finallyconsiderthederivativeswithrespecttotheowvelocity.Inthiscasetheresultis

PAGE 191

Insummarytheadjointfunctionsare 2Tbn(r)G:C wherevh=q m,asinthecaseoftheinelasticparticles. Itiseasytoseethatthesefunctionsarebi-orthogonaltothesetofphasefunctionsa(r)denedearlierinEq.( 4{6 ).Further,forthepurposeofevaluatingthetransportcoecientstheFouriertransformedformsofthese'sarerequired.Forthecaseof1itfollowsfromEq.( H{15 )that Inordertorecognizethekdependentprefactor,notethat wheregontherighthandsidedenotestheequilibriumpaircorrelationfunction.Dene

PAGE 192

Then,itfollowsthat Therefore TheotherfunctionsintheFourierrepresentationbecome 2Tben(k)G:C Further,forthepurposesofevaluationofthetransportcoecients,itissucienttoknowthesefunctionsupthroughlinearorderinkintheirFourierrepresentation.Theseturnouttobe =S11 whereithasbeenrecognizedthatthepre-factorintheaboveequationispreciselytheinverseofthestaticstructurefactorfortheuidinthehomogeneouslimit[ 72 ].Also, 2Tben(0)G:C=Xip2i 2 wherePisthetotalmomentuminthesystem.Further,makinguseofthefactthattheuidissphericallysymmetric,theabovefunctionstolinearorderinkare 2(H{30)

PAGE 193

and ThesearetheresultsusedinthetextinChapter5.

PAGE 194

Inthisappendix,someofthedetailsassociatedwiththekinetictheoryoftimecorrelationfunctionsdescribedinChapter6aregiven. 6{35 )inChapter6isproved.FromthedenitionofthepaircorrelationfunctiongiveninEq.( 6{32 ),itcanbeseenthatthefunctionalderivativeofthisfunctionwithrespecttothedensitygives (I{1) wherethethreeparticlecorrelationfunctionisdenedas (I{2) Allthefunctionalderivativesintheaboveexpressionsareunderstoodtobeevaluatedatthetruehomogeneousequilibriumstate.Therefore,oneneedstoidentify(q1) 185

PAGE 195

this,considerrstthedenitionofdensityinthelocalgrandcanonicalensemble, (N1)!ZdX2:::dXNW()expQL+Xi>1"(qi)(qi)(qi)(pimU(qi))2 CarryingoutthefunctionalderivativewithrespecttothechemicalpotentialandcomparingtheresultingexpressionwiththedenitionofthepaircorrelationfunctioninEq.( 6{32 )givestheresult (I{4) wherethepaircorrelationfunctionisnowthatofthetrueequilibriumstateandhenceistranslationallyinvariant.Next,notethattheOrnstein-Zernickeequationrelatesthepaircorrelationfunctiontothedirectcorrelationfunctionc(r)as whereh(r)=g(r)1,thepaircorrelationfunctionwithitsasymptoticvaluesubtractedout.Usingthis,itcanbeinferredfromEq.( I{4 )that Now,substitutethisidenticationinEq.( I{1 )abovetoconcludethat (I{7) whichistheresultquotedinthetextinEq.( 6{35 ).

PAGE 196

6{61 ),thegeneratorofdynamics 2hrvv+vrqIE(I{8) withIEbeingtheoperatorgiveninEq.( 6{47 )inthetext.Forthepurposeofidentifyingtheinvariantsofthisoperator,notethattheoneparticlehomogeneouscoolingstatef(1)hisdeterminedbytheEnskogkineticequationoftheform 1 2hrv1v1f(1)h(v1)=JEhf(1)h;f(1)hi(I{9) where Theoneparticlestateisparameterizedbyhomogeneoushydrodynamiceldsn;TandUthrough 2h;f(1)h=nv3hhvU AswasdoneinthecaseoftheeigenvalueproblemfortheNparticleLiouvilleoperator,dierentiatingEq.( I{9 )withrespecttotemperature,owvelocityanddensity,andusingEq.( I{11 )togetherwiththerecognitionthat 2rv1v1f(1)h;(I{12)

PAGE 197

andthattheactionofthelinearizedEnskogoperatorinEq.( I{8 )onspatiallyhomogeneousstatesis @n1 2hrv1(v1h(v1))Zdv0'(v0): givestheresult 2hrvvIE1 2rv1v1h=1 2h1 2rv1v1h;(I{15) 2hrvvIE(rvh)=1 2h(rvh)(I{16) and 2hrvvIEh=1+@lng() 2hrv(vh)=h@lnh 2rv1v1h Insummary,theresultis 2hrvv+vrqIE'(v)=KhydT'(v);(I{18) where 2rv(vh);rvh:(I{19) ThisistheresultquotesinthetextinEq.( 6{65 ). 6{60 )inthetextisgiven.AsstatedinChapter4,theHelfandexpressionsofallthetransport

PAGE 198

coecientsareoftheform Ls()C(s)(I{20) where withthephasefunctionsea(0)beingsumsofsingleparticlefunctionsoftheformea(0)=Xia1(v1);a1=1;2 3v213 2;v1 Hence,thetransportcoecientconsistsoftwotimecorrelationfunctionoftheform with Ls()C(s)(I{24) and Ls()C(s)1 ThereducedformofFisasobtainedinEq.( 6{50 )inthetext.Also,usingthepropertythatea(0)isasumofsingleparticlefunctions,itfollowsthat Ls()C(s)=limn`3ZdX0fa1(v) (I{26) LE+KhydTis(1)X0g

PAGE 199

Nowconsider wheretheformoftheuxfhasbeenusedtoexpressthecorrelationfunctionintermsofaveragesovertheoneandtwoparticlefunctionsintheBBGKYhierarchyassociatedwiththefunctiongiveninEq.( I{22 ).Inparticular (I{28) and Observethatthecollisionalpartoftheuxf2isgeneratedbytheactionofaToperatoronafunctionofthephasepointassociatedwithoneparticle.Therefore,itissucienttospecifyf(2)lhontheprecollisionhemisphere.ThenthekernelKaboveisthesameastheoneidentiedinEq.( 6{46 )fortheadjointfunctions(2):Usingtheidenticationof(1)'sinthecontextoftheeigenvalueprobleminthe

PAGE 200

previoussection,Eq.( I{27 )becomes whereg(X1)isthefunctionidentiedinEq.( 6{20 )inthetext.Therefore, LE+KhydTis(1)X0=ZdX1P(1)g(X1)exph LE+KhydTis(1)X0 Andnally LE+KhydTis(1)X0 whichistheresultquotedinthetext. (1)(X)=bkkbe1lqkrvlh(v)(I{33) Theaimhereistoevaluatetheoneparticleuxoccurringinthistransportcoecient,namely 2hrvv+vrqIEKhydT(1)(X):(I{34) Notethatthistransportcoecientisdeterminedbythe44matrixelementassociatedwiththetransportmatrixK,andhenceKhydT=h=2.First,evaluate

PAGE 201

theactionofthelinearizedEnskogoperatoronthefunction(1): Allthevariablesontherighthandsideoftheaboveequationareunderstoodtobeindimensionlessform.The*'shavebeensuppressedforcompactnessofnotation.Therstsimplicationistheobservationthatthemeaneldterminthelastlineoftheaboveequationdropsout.Further,thecollisionoperatortermscanberegroupedandwrittenas where Therefore,theadjointuxbecomes 2hrv0v0IE+h=2'4(v0)+bkkbe1lv0krv0lh(v0)+Qkrv0lh(v0) TheeigenvaluepropertyestablishedinEq.( I{18 )aboveshowsthatthersttermintheaboveexpressionvanishes.Hence,theresultgiveninthetextinEq.( 6{84 )isobtained.

PAGE 202

Next,toidentifythereducesformofthetimecorrelationpartofthetransportcoecient.Notethatthedirectuxjinvolvedinthistransportcoecientis 31 4(12)XlXm6=l(qlmglm)(glmbqlm)3(qlm);(I{40) thesourceinthemicroscopicenergybalanceequationgiveninTableC-2.Therefore,usingtheformofthekernelKinEq.( 6{46 )thecorrespondingreduceduxbecomes 3n`3ZdX1dX21 4(12)(q12)(q12g12)(g12bq12)3fh(X1X0)g(q12)f(1)h(X2)+(X2X0)g(q12)f(1)h(X1)i+g(q1;q2) 31 4 (I{41) Alsoobservethattheadjointuxisthesameasthatidentiedinthecontextofshearviscosityearlierandinthiscase,iscontractedto 3vkrvkh(v)+Qkrvkh(v)(I{42)

PAGE 203

whereQkhasbeendenedearlierinEq.( I{37 ).Inwhatfollows,the'sonthedimensionlessphasepointsissuppressedfornotationalcompactness. 31 4 LE+h 3v0krv0kh(v0)+Qkrv0kh(v0) ThisidentiesthespecicfunctionsthatgointothegeneralreducedGreen-KuboexpressiongiveninEq.( 6{69 )forthisparticulartransportcoecient. Firstnoticethatthepresenceoftheprojectionoperatorensuresthattheconstantpartofthedirectuxdoesnotcontribute.Nexttheactionoftheprojectionoperatorontheadjointuxisevaluated.Inthiscase,itiseasytoverifythat Therefore,onlythecollisionalpartcontributes.Theactionoftheprojectionoperatoronthispartcanbeevaluatedasfollows.Firstnoticethat (I{45) Henceonlythetermintheprojectionoperatorthatisalongtheenergycontributes.Furtherusingthefactthat 4(1+)2(bg)2(1+)(bg)bv1;(I{46)

PAGE 204

thiscanbeevaluatedas 3Qkrv0kh(v0)=1 2rv0(v0h)n`3g()22 31 12Zdv2Zdv1Zd^h(v1)h(v2)(1+)[13](bg)2(g12b) (I{47) ComparingthiswiththedenitionofthecollisionalpartofthepressuregiveninChapter5,itfollowsthat 3Qkrv0kh(v0)=pC13 SubstitutingtheaboveresultsinEq.( I{43 )aboveyields 31 4 (I{49) with LE+h 3v0krv0kh(v0)+Qkrv0kh(v0) Puttingintheevaluatedformsofthesubtractedadjointuxandusingargumentsalongthelinesdetailedinthecaseoftheshearviscositytorecognizethatthefreestreamingandthemeaneldtermsinthecollisionoperatorgeneratingthedynamicsdropout,thefollowingresultcanbeobtained, 2hrvv+g()IB+h where 3Qkrv0kh(v0)pC13

PAGE 205

Further,Eq.( I{51 )canberecastasanintegralequationsuchthatthefunctionDisasolutiontotheequation 2hrvvg()IBh ThiscompletestheformalanalysisofthistransportcoecientatthelevelofEnskogkinetictheory.Inordertocomparethisexpressiontotheoneobtainedin[ 68 ],theactionoftheprojectionoperatorneedstobeunfolded.Forthispurposerstnotethatitcanbeveriedbydirectevaluationthat 2hrvvh Next,observethat 3v23 2IBD(v)=g()2n`31 2Zdv12 3v213 2Zdv2Zd^(g12b)(g12b)f2b112[(1)h(v1)D(v2)+(1)h(v2)D(v1)]h(1)h(v1)D(v2)+(1)h(v2)D(v1)ig=g()21 612n`3Zdv1Zdv2Zd^(g12b)(g12b)3(1)h(v1)D(v2)=n2g()2 (I{55) Usingthisfact,theprojectionoperatorcanbeevaluatedandEq.( I{53 )canberewrittenas 2hrvvg()IBh 2rv(vh)=D(v)(I{56)

PAGE 206

wherethenotationTCU(D)denotesthetimecorrelationpartofthetransportcoecientinEq.( I{51 )above.ThiscompletesthecharacterizationofthetransportcoecientU: 2(1+)3n`3g()4 3Trvivj=1+1 2(1+)3n`3g()4 Clearlytheactionoftheprojectionoperator(1P)onthistermvanishes.Hencetherearenokineticcontributionstothebulkviscosity.Asfortheinstantaneouspart,thatisthecollisionaltransfercontribution,ithasbeenshowninChapter5that 3inst:(I{58) ThereforethecollisionaltransferpartcanbeidentiedusingEq.( 6{86 )earlier.ThiscompletestheanalysisoftheEnskogevaluationofthebulkviscosity. 5{54 )as 18`3nh4(1+)Zdbdg12dP12f2(g12b)(P12b)2+1 2(g12b)3g(bg12)f(2)h(;v1;v2):

PAGE 207

SubstitutingtheEnskogclosureforthetwoparticledistributionfunctionandevaluatingtheangularintegralsgives 2bgbP+1 4g3 Thisisthecollisionaltransferpartofthethermalconductivity. Next,letusconsiderthetimecorrelationpartofthistransportcoecient.ThedirectuxinthereducedrepresentationofthetimecorrelationfunctioncanbeidentiedusingthekernelKgiveninEq.( 6{46 )tobe 3v21vi12 3(1+)g()`3nhZdX2(q12)(q12g12)(g12bq12)2(P12bq12)bqi12(v2)=2 3v21vi11+ Also,bydirectevaluation,itiseasytoseethat 3vi1v215 21+ Further,asnotedinChapter6,theadjointdensityofinteresthereis (1)(X)=qk1 2rv(vh(v))(I{63) Theadjointuxoccurringinthetimecorrelationfunctionisrelatedtothedensityby 2hrvv+vrqIEKhydT(1)(X):(I{64) Proceedingexactlyasinthecaseoftheshearviscosityearlier,itiseasytoseethat 2rv(vh(v))+1 2Qk[rv(vh(v))]+q1 2hrvvIEh=2'2(v) (I{65)

PAGE 208

ThesecondtermvanishesbytheeigenvaluepropertyoftheEnskoggeneratorestablishedinEq.( I{18 )earlierinthisappendix.Therefore 2rv(vh(v))+1 2Qk[rv(vh(v))](I{66) Next,theactionoftheprojectionoperatoronthisuxisevaluated.Forthispurposenotethat 2Qk[rv(vh(v))]=(rv)g()31 6Zdv1Zdv2Zdb(g12b)(g12b)(v1b)1 (I{67) Usethefactthat 2(1+)(g12b)(I{68) toget 2Qk[rv(vh(v))]=(rv)g()31 12(1+)Zdv1Zdv2Zdb(g12b)(g12b)2h(v1)rv2(v2h(v2)) (I{69) Integratebypartsandsymmetrizewithrespecttothedummyindices1and2toget 2Qk[rv(vh(v))]=(rv)g()31 12(1+)Zdv1Zdv2Zdb(g12b)(g12b)2h(v1)h(v2) (I{70) ComparingthiswiththedenitionofthecollisionalpartofthepressuregiveninChapter5leadstotheidentication 2Qk[rv(vh(v))]=(rvh)pC(I{71)

PAGE 209

Also 2rv(vh(v0))=(rvi)1 2Zdvvivkrv(vh(v))=rvihpk Therefore 2rv(vh(v))+1 2Qk[rv(vh(v))]prvkh(I{73) PuttingalloftheaboveresultstogethergivesthethermalconductivityintheEnskogapproximationas 3Zdv1vi1v215 2Ai(v)(I{74) whereinstisasidentiedearlierinEq.( I{60 )andA(v)isasolutiontoanintegralequationoftheform 2hrvvg()IBh 2rv(vh(v))+1 2Qk[rv(vh(v))]prvkh Inobtainingtheaboveresults,argumentsalongthelinesdetailedinChapter6inthecontextofshearviscositytoconverttheIEinthegeneratorofdynamicstotheg()IBoccuringintheaboveequation.Thiscompletestheformalanalysisofthethermalconductivity.

PAGE 210

thereducedonebodyuxis 3v21vi12 3(1+)g()`3nhZdX2(q12)(q12g12)(g12bq12)2(P12bq12)bqi12(v2) Next,notethattheadjointdensityofinteresthereis (1)(X)=qh(v)(I{76) Thustheinstantaneouspartofthistransportcoecientis 3v21vi1qi1h(v1)2 3(1+)g()`3nhZdX1ZdX2(q12)(q12g12)(g12bq12)2(P12bq12)bqi12(v2)1P(1)>qi1h(v1) (I{77) ClearlybothtermsvanishbecauseofthesphericalsymmetryoftheHCSdistribution(bothtermshaveanunaccompaniedpositioncoordinateintegratedoverallspace).Therefore Nextconsiderthetimecorrelationpartofthetransportcoecient.Inthiscase,aswasnotedinChapter5,therearetwotimecorrelationfunctions.Butisifthelinearcombination2@lnh (1)(X)=qh(v)2@lnh 2rv(vh(v)):(I{79) Sincetheaimhereistoobtaintheadjointuxasasolutiontoanintegralequation,itturnsouttobemoredirecttoconsiderthislinearcombination.Thedirectpartisthesameasdeterminedinthecaseofthermalconductivity.Hence,

PAGE 211

inwhatfollowsattentionisfocussedontheadjointpartofthetimecorrelationfunction.Firstnotethat 2hrvv+vrqIEqh(v)=q1 2hrv(vh(v))+vh(v)IE(qh(v))(I{80) Considerthelastterm.Itcanbeunfoldedas where (I{82) and Notethat @n

PAGE 212

Nowfocusontherstterm.ExpressthecoordinatesofthetwoparticlesintermsofthecenterofmassandtherelativevariablestogetZdq0(q0q1)F(q1+q0;q0q2)=Zdq0q0R+r @n ButfromthepreviousargumentitmustfollowthatFisanevenfunctionofbqbr.Thereforethistermmustbezero.Usingtheaboveresults,themeaneldterminEq.( I{81 )becomesZdq0

PAGE 213

CarryingouttheangularintegrationsintheaboveexpressionandsubstitutingtheresultinEq.( I{81 )abovegives 2hrvv+vrqIEqh(v)=qk1 2hrv(vh(v))+vh(v)qkg()IBh(v)qk@lng() 2@lng() (I{86) Fromtheanalysisdoneintheprevioussection,itfollowsthat 2hrvv+vrqIEq1 2rv(vh(v))=q1 2hrvvIE1 2rv(vh(v))+v1 2rv(vh(v))+1 2Qk[rv(vh(v))] (I{87) Therefore 2hrvv+vrqIEqh(v)2@lnh 2rv(vh(v))=vkh(v)+1+@lng() 2@lng() (I{88) Thisistheadjointuxinthetimecorrelationfunction.Further,theactionoftheprojectionoperatoronthisuxcanbeevaluatedexactlyasinthecaseofthethermalconductivitytogive 2@lng()

PAGE 214

Puttingalloftheaboveresultstogether,oneobtains 3Zdv1vk1v215 2Bk(v) (I{90) whereinstisasidentiedintheprevioussectionofthisappendixandBisasolutiontoanintegralequationoftheform 2hrvvIEBk(v)=vkh(v)+1+@lng() 2@lng() 2@lng() ThiscompletestheformalanalysisoftheEnskoglimitofthecoecient. Notethat,forallofthetransportcoecientsabove,inordertocarryoutatermbytermcomparisonoftheintegralwiththeresultsgivenin[ 68 ],anadditionalstepisnecessary.Theintegralequationsinthatworkaresolvedinasubspacethatisorthogonaltotheonedenedbyaprojectionoperatoroftheform wheretheA'sareasdenedinEq.( 6{64 )earlierandN'sareappropriatenormalizationconstants.Butnoticethatthisorthogonalprojectioncanbeintroducedwithoutlossofgeneralityintheintegralequationsaboveusingthereadilyveriableidentity (1Pk)1P(1)=1P(1):(I{93)

PAGE 215

ThatconcludesthedetailsoftheEnskogevaluationofthedierenttransportcoecients.

PAGE 216

[1] EnnisB.J.,J.GreenandR.Davis,ParticleTechnology90,32(1994). [2] KnowltonT.M.,J.W.Carson,G.E.Klinzing,andW.C.Yang,ParticleTechnology90,44(1994). [3] M.H.Cooke,D.J.StephensandJ.Bridgwater,PowderTechnology15,1(1976) [4] R.L.BrownandJ.C.Richards,Principlesofpowdermechanics,Pergamon,Oxford(1966). [5] J.Geng,E.Longhi,P.P.BehringerandE.W.Howell,Phys.Rev.E,64,60301(2001) [6] J.F.DavidsonandR.M.Nedderman,Trans.Inst.Chem.Eng.51,29(1973) [7] G.W.Baxter,R.LeoneandR.P.Behringer,Europhys.Lett.21,569(1993) [8] J.Duran,J.RajchenbachandE.Clement,Phys.Rev.Lett.70,2431(1993) [9] O.Zik,D.Levine,S.G.Lipson,S.ShtrikmanandJ.Stavans,Phys.Rev.Lett.73,644(1994) [10] G.Metcalfe,T.Shinbrot,M.M.McCarthyandJ.M.Ottino,Nature374,39(1995) [11] O.R.WaltonandR.L.Brown,J.Rheol.30,949(1986) [12] C.Liu,S.R.Nagel,Phys.Rev.B,48,15646(1993). [13] C.S.O'Hern,S.A.Langer,A.J.Liu,S.R.Nagel,Phys.Rev.Lett.86,111(2001). [14] S.F.EdwardsinGranularMatter-AnInterdisciplinaryapproach,A.Mehtaed.Springer,NewYork(1994). [15] S.F.Edwards,C.C.Mouneld,PhysicaA,226,25(1996). [16] P.Bak,C.TangandK.Wiesenfeld,Phys.Rev.Lett.59,381(1987). [17] A.J.LiuandS.R.Nagel,Nature396,21(1998). [18] M.E.Cates,J.P.Wittmer,J.P.BouchaudandP.Claudin,Phys.Rev.Lett.81,1841(1998). 207

PAGE 217

[19] HKing,RWhite,IMaxwell,NMenon,cond-matt/0209490. [20] L.E.Silbert,D.Ertas,G.S.Grest,T.C.Halsey,DLevineandS.J.Plimpton,Phys.RevE.64,051302(2001). [21] C.S.Campbell,AnnualReviewofFluidMechanics,22,57(1990). [22] I.Goldhirsch,AnnualReviewofFluidMechanics,35,267(2003). [23] [24] J.J.Brey,D.Cubero,Phys.Rev.E.,54,3664(1996). [25] J.T.JenkinsandM.W.Richman,Phys.Fluids28,3485(1985). [26] J.J.Brey,J.W.Dufty,C.S.Kim,A.Santos,Phys.Rev.E.,58,4638(1998). [27] T.P.C.vanNoije,M.H.Ernst,GranularMatter,1,57(1998). [28] M.H.ErnstandR.Brito,J.Stat.Phys.,109,407(2002). [29] M.P.AllenandD.J.Tildesley,Computersimulationofliquids,OxfordUniversityPress,USA(1988). [30] G.A.Bird,MolecularGasDynamicsandtheDirectSimulationofGasFlows,OxfordUniversityPress,USA(1994). [31] J.J.Brey,M.J.Ruiz-Montero,andF.Moreno,Phys.Rev.E55,2846(1997). [32] P.A.ThompsonandG.S.Grest,Phys.Rev.Lett.67,1751(1991) [33] S.McNamara,W.R.Young,Phys.Fluids.A,4,3(1998). [34] J.J.Brey,M.J.Ruiz-Montero,andF.Moreno,Phys.Rev.E.,63,061305(2001). [35] S.A.HillandG.F.Mazenko,Phys.Rev.E67,061302(2003) [36] C.Bizon,M.D.Shattuck,J.B.Swift,W.D.McCormick,andH.L.Swinney,Phys.Rev.Lett,80,57(1998). [37] J.A.McLennan,IntroductiontoNonequilibriumStatisticalMechanics,(Prentice-Hall,NewJersey,1989). [38] J.J.BreyandJ.W.Dufty,Phys.Rev.E72,011303(2005). [39] A.BaskaranandJ.DuftyinModellingandNumericsofKineticDissipativeSystemseditorsL.Pareschi,G.Russo,andG.Toscani,(NovaScience,NewYork,2005). [40] V.GarzoandJ.W.Dufty,Phys.Rev.E59,5895(1999).

PAGE 218

[41] C.Huan,X.Yang,D.Candela,R.W.Mair,andR.L.Walsworth,Phys.Rev.E69,041302(2004). [42] J.J.Brey,J.W.Dufty,andA.Santos,J.Stat.Phys.87,1051(1997). [43] J.Dufty,J.J.BreyandJ.Lutsko,Phys.Rev.E65,051303(2002). [44] J.Lutsko,J.Dufty,andJ.J.Brey,Phys.Rev.E65,051305(2002). [45] I.Goldhirsch,T.P.C.vanNoije,Phys.Rev.E61,32413244(2000). [46] B.J.Alder,D.M.Gass,andT.E.Wainwright,J.Chem.Phys.,53,3813(1970). [47] B.J.AlderandT.E.Wainwright,Phys.Rev.A1,18(1970). [48] J.P.BoonandS.Yip,MolecularHydrodynamics,McGraw-Hill,NewYork(1980). [49] T.PoschelandN.V.Brilliantov,Kinetictheoryofgranulargases,Oxford(2005). [50] T.P.C.vanNoijeandM.H.Ernst,inGranularGases,eds.T.PoschelandS.Luding(Springer,NY,2001). [51] J.Lutsko,J.Chem.Phys.120,6325(2004). [52] JamesW.Dufty,AparnaBaskaran,LorenaZogaib,Phys.Rev.E,69,051301(2004). [53] J.F.Lutsko,Phys.Rev.E63,061211(2001). [54] P.K.Ha,JournalofFluidMech.,134,401(1983). [55] L.Onsager,Phys.Rev.,37,405(1931). [56] C.Josserand,A.V.Tkachenko,D.M.Mueth,andH.M.Jaeger,Phys.Rev.Lett.,85,3632(2000). [57] APuglisi,ABaldassarri,VLoreto,Phys.Rev.E66,061305(2002). [58] G.D'Anna,P.Mayor,A.Barrat,V.LoretoandF.Nori,Nature,424,909(2003). [59] A.Baldassarri,A.Barrat,G.DAnna,V.Loreto,P.MayorandA.Puglisi,cond-matt/0501488. [60] N.Xu,C.S.O'Hern,Phys.Rev.Lett,94,055701(2005). [61] J.W.DuftyandV.Garzo,J.Stat.Phys.,105,723(2001).

PAGE 219

[62] P.M.Chaikin,T.C.Lubensky,PrinciplesofCondensedMatterPhysics,Cambridge(1995). [63] E.C.Rericha,C.Bizon,M.D.Shattuck,andH.L.Swinney,Phys.Rev.Lett.88,014302(2002). [64] S.McnamaraandW.R.Young,Phys.Rev.E,50,R28(1994). [65] I.Goldhirsch,G.Zanetti,Phys.Rev.Lett.,70,1619(1993). [66] J.J.Brey,M.J.Ruiz-Montero,Phys.Rev.E,69,011305(2004). [67] J.W.Dufty,MolecularPhysics,100,2331(2002). [68] V.Garzo,J.W.Dufty,Phys.Rev.E59,5895(1999). [69] J.H.FerzigerandH.G.Kaper,Mathematicaltheoryoftransportprocessessingases,Elsevier,NewYork(1972). [70] L.Groome,K.GubbinsandJ.W.Dufty,Phys.Rev.A,13,437(1976). [71] J.F.Lutsko,J.Chem.Phys,120,6325(2004) [72] P.ResiboisandM.DeLeener,ClassicalKineticTheoryofFluids,WileyInterscience,NewYork(1977). [73] J.W.DuftyandJ.J.Brey,inModellingandNumericsofKineticDissipativeSystemseditorsL.Pareschi,G.Russo,andG.Toscani,(NovaScience,NewYork,2005). [74] J.W.Dufty,A.BaskaranandJ.J.Brey(inpreparation,tobesubmittedforpublicationinPhys.Rev.E).

PAGE 220

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20110218_AAAACQ INGEST_TIME 2011-02-18T16:26:58Z PACKAGE UFE0013684_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 8423998 DFID F20110218_AABQQA ORIGIN DEPOSITOR PATH baskaran_a_Page_206.tif GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
df89a0df288516d36d05e1c88687a9f4
SHA-1
28aff385d8e3391a34637f5005c32596b99d1fe0
02c2bd0723866eaa73b8e69159df5309
5810939da3582f48fd0542db8d37c15f2bb7ef2d
9e8defe01580d7a482b16a56ffa64ef3
bf9c2298f4b72793a695aeb8831f7d942137bec0
cc240bab81a1afd2c88006328c81659ae83a2edf
914580339cdcdc686401ac5774dd7625
4f8d108a00ddfd6c470330aa778ca458a83e6633
4a1af0f7981b39ac85c88d55fcbee855bf11e005
e543e7d6e855c330a55c054140a6be5b
ffe82ab6cb0d02ee896a2da234b21c23e0aec2f3
9b567aa93e0e5746ea9bbd0ba40c6046
ab6439a43082a07f4e220a995ba2f6f0f3291753
2aaca68e6444751615464baf38ca1689
3af46937ed0b2be57d4425980c2598b576c3aaa9
2c8b31d7cc35a30a168fbf9826b5bbb5
3a30655ccc02fb08ca4f4e33aa07a48362137814
df21970debd0e52b749b07c778c30632
9673f950aa447075424cf8cac733561f06fe8e6d
417eff67f3f782392fecc08c4342ef47
4bfe01b156404f3cf390f26868e85ce071076778
87afd17a0e99f8380696b535eca0a75b5ee8cf80
142fc2bfe94c3f4aeff032dd11e57219
3d22619d647576b25f23692bfbf8288530d68f2c
e25ecbe73fa0916d9fc6502856dc5237
1dfaa421bc76acc267018557c47ebd0e8d6b57a7
018f628538ff7cfd8e185ce1e3129c3c
729eb250949f44f6eb21ec7466972c31284eb754
93e79b2f77bc8838b9e3e712852923fb
e5af8d175b16acc77de52afeb6636e57
9efe468bef2f942c842671491854cb2b4b3240f5
a84842044b21269569df1d45a9c0c487
af53e0c945bbc2a6ca91f48d900058d5f62d7d23
aa70328cf33c8148e1160da8a8fec9e0
8001da340c29054b74a328fca3e01fc2e73669df
f067d907d7cdf3efaaa6766544c14fa74ba5c7c6
33e10992315353a1de5ff97f6f5ec2c5
0e0072dc57425d4a052019dcdf346c56d2044d47
802eb94029e95248b0856aecf6ba66e3
59da2d67394155dc20b0227f1cf88f92cea91a84
1e5b607f703454556b234e1f62dd0d23
1bf9c03b50f1cbf7222d0e98895de10b1f05d809
3738ab10676f9b62f63afb304e1700de
28b1841d8f57ce36b33e87223286a3bf9028b662
8361832e1e4d2394887a3ba15e87b76bafd90753
01fb46e3d3c8b1dee14d796a38a8b84c
37c03b334829ba55b59c83cb44be7556de791228
75d0a26901c634b5f543d4a6e61c973f
15a69aef081d9ea5c0aa7c40930aaa04a6101ba0
97f368460e384b90d65846aeab26959b
abb9ee3833657f95de9bb53f1fe26756ddeccd89
a78d9b06a8ed974d3c6963b2989e593c
645d46e78aab4768bc5c2d2bfa3ba5178d61371c
7c1d891dc5d42caeaf1edd356e5eddd1
d66c64c2626af1fd402353093c149fb98b36881a
ab8f22c6d5b5302c6949d6f48b8b3796
c89afa8536be35d9b60e5b4e7b400fec2259a97e
35d804173151bf2ced6d3904afcc6b31
61e036290c73d104e6c251b5bcca658761159760
3e77f388a0378ed97116ab1f5bf13f19
af1c8d6063850762a89949212bb75292daa4a995
0698523a0b7ddb144b10613172e24e26
be1e524d0e3f2e57131797ec1e68a515d62ceff4
8b1d3833cdb2809a4deaec588e9c99da433475bc
a62e30afbca4a461a854e62371fe1147
b66d350a1d6dd4e4bbb67086f64c35d449813214
59dbab00cc6ff0f640893760b888023a
6d6c6483a547bc865f7b591f8c413534
c13ebec0c54bfc28afbd0fcbd22869d685cfbae8
87133456a1338c739e7b92a060d33260
a7f0070c6fe7605c07c641e2e711cafece4c4c49
373625e8a7358a511bb4b5d34243ac6b
a17652b047cabdca0a77254364a4dedff9f34e16
13f39b0c0018abd7bcb1b8e2953b55a3300943b5
ed1dacb12a158e60a429320e33b23d3d
aa4a426d8fa8e3c02951fc57d4e61ebd89767702
1ea794dab6a63c53b90a2abc183b7171
115ab0dcce4ff96772c1d52a7ac446b228c2ff2c
b0487f7f1c3c7123098e31dd3e239aee
4bfc8b3fd4471c2d1b7e80ce76d05fe370d8b66a
d2a9403b2fb06d19c06d35a0c0350355
2b13b9ec84952985c624124132e0898fdb6e4323
31e5ec33786c3a3e948a76fc497eb0f09d3fa802
5f489c562945da190e86ef16c913a001
249419d7cace0d0916550af34e12e45d32fdc8f8
a457f5fcbf858f78eae9c4fd2a523929
c21c4465e27b6b9e1512878e7aa6c076
790e3da32668e9a28555bae747cb4ab7ede26374
7901aea81ba39417b4a48230f9cefbf6
9feafe786ca65425d4feb77b35014cb9
e4d87f1da04eb45f282e78e551e12664
56542019324d814f80e4277a8632b54cd1eb7040
5fa9e5132f54bde4115a8b4e9de9478a
66ecb8023ed2203c440fa4470471505e
eac57a3a6617ed7f2d4a3ffea961482d7525aa52
24d1b84ce39c9d21bfa411cddde8f1a7
0e7a1f2a21c9fc59d7f9e9f7109b3d23
17f730e2bf4957d667421b4b6a570abc03bb057f
4feff826ea0836f7fa47d05902fb786d
82a0800655be0e036bbd1c29fbc72af60b5692b9
17ede53235c82165a0245375fd187bb3de2ac7bd
617975d4afc2971060536748886df53c
c2bb797db05096819bab1499002e9814f44c12d4
801b9220a8a9cc07161d54304a5d8eaf
7123ee56310ac9e09a0a55dc26ab5027ee3dcd6d
7ba6c73e3bdc8a8e4ca04cb1236519e977c5fa01
25bca775810924a510b6606718caca31
b43ea37dbf3399e83081c23852a6a5e81d258250
abc1b7824f2d21ca86f755a7a56b8b04
c9b091e30f828da7b841858ce8b0b721b9dd75bb
fc767d29bdd89ef7f12921f775c1018c
3227a7266420a8942499fa7cc77d4e8d73e9e3fa
882e4f725bf670434448bea674850b76
4018305669fa68c096b1715528be2da64381b03c
b28666983c601e008f3353caab90f84b
d2be39186592d83cbceb71b1c030a3e8ae87979f
9a842052b798c213111306fce7803410
8878ba0978ef16e66dbe103778b81c3041a26fd5
2b7811be1af7b08abd7bab30243510fa
4e643e0ac8683b96be2d4e40d2042030fbdd0421
8c2d06ff0ea7ba8f582fb58439020043
a06a34bd20d197ed37aed82cf0e93a37b818f7b9
5ec6414d9fecbd9c2c1b1a3e069c225b
06bd15a6eb605ba5df322967b7c8e1b9fe3c9198
d67fa7294e8ac8090b9458c0fcb5a63d
94ab3906c90f9168a818ba46810839e1d8bc2fdc
98af91e4dd7b3bc1464184552c8ace0c
159f45ba54d56cb7bdd8fcb00f1eedd32d0570e6
94cc8ecf3a98c424df5559c1ae25e024
bd2466972b999325f0bd86646e5a2320a05f6d32
55ba6a3a21069e8ba85b5832a8ddfa17
257e73a44bc07cd7abaa315af271060f
51f2715b53c3253f4469d9ff486f90ed1c06020d
37094eda2c963b3a9f847b5b0cf1ce6b
d19678ed8dbc21751e9b69ba2b0aa2fcc098c422
0f53d4dd527f4a33ea0bbbf38f1581b4
f2037164f21d5b10c1080c31919b5ffb
1f59bc009b32d93021a5edeb1314a9e2cd9b3e29
901c758c40f1c777b4139321e0631735
838f13c827610bd906d8505a53913a18eb861ba1
49f9437ec58c058f2333d746004987fe
5c36854bbaa3268f47294413a923ed6f23f5ec52
38538230b30926db63e4e4963d8d1bde
2e2ee47d017097acf6a0ef19d12e4ea316384734
88dcf582e85bd70d09434a416b516bd1f20f5dd0
3b2a23a0598d044e166c13c9c3da8cb3
be9edfa7b23d9c9a29d2ce6bbe2e6e7aea041c56
d8cf5c431917be34065f35e552d148c4
8fb5b245215ca50c6f2ddb6b6a1176daf067a5c4
fca524de4ce8dc8034d570ee0a1ce840
eb142c64ef42763897973e4aa1b5680c
854cd21eed6a9b523494192cbece6f75600e4fa1
23d0e8724afe61d724ab1e5dc11339b5
7f2842a51ae6a20c195e5de7bb5a53d6c7de2410
0072944366d3d1a48fcfb8b11e594a4d
5a2a079ca3e34ba47b36e2547349d6efe4c59a26
08771a0f9408cda75c8351f2fbd5f945
75c22b1d4cc158e45e329831437ae0ffba0b6851
5e001d66f639dcbe772134eb84d47fbe
140dae1281a6d290f59051146e74b3687fb01b59
fc8f1e38dfeb417690cdc8db340e1dbf
8cee4c32c8b0a9b4ab7808da7ceda137198f532f
8e0e87bcdb99c9960348079a9f5dd1e7
d700810e3854d75d6e1830e2176e5c6023e3af49
1a33b755b7699ab33dcfc5442df3575d
90a1d9140ed921c194c58b22f7156d51ac461396
f2e4f1148fe0eccacb7338508912218b
0a439a7db749abd818f9ce90d257c96365666c06
46fa8fe8a7f0eec6d366387bbc8528e2
a36d2691c16676b40503b65facbd5c3d
bb07e9e5a997e2e72edc14579e2a1a2fee832fb4
219efc67b420815b8df441d55f35df7b
f92d20839fcd29ae1e129867ef790145d87fefbb
064691ebcb3b809384b29293d918d02a
6e69b4266bffa046a1328844b1cc8a68b342ff08
45f4208efeed01a24dbeafb5faf740f3
826b0371413ded3b2a0454dc1ec04458fc7ece04
cc533e1c6be4e9929111cd1a21c82231
889081b4ed836cc20c0d2ce08c5595bd6bf338b8
2e0562aee78df00b7be35aef646c708ac125b7d1
cebc9079e8de3bd6cfbc00bd584a8d5be4bf7780
ba628b707558a785de296f0681de4751
6d89aefc23e4079c634dffe31843e3f672bf3c39
513cb9ca3a0e8152cd2d13f3b908f757
f1c90fb27e662cf47129f312a954f8848df89824
5588b829f7e8af64b58c809ac49724b3
14f137ae26c42aea408fa5cf6263a8b66a6278c8
3f72c2a7bf5ffba4afc3a10a6a0d3f38
9449a6822d311dda12de396281298ae1
30c85d572029d66763913b81ae2f8908
d5576d32361e93b09d06f645447d3c74095595dc
4acdfa2741925c2e7de0f2ed2aa6d10d
225d0c84de6af27fee6ef86decd0705eff7c6ef2
6067a934387c7f2a0bf6fa2bb4d1cf0e
05139c5ce9d18d857e690d747087deac1e45da17
878f704e1252d8bc5c03691fd71e3bcf
ef3f5d86ebc83cf2dc98eeb1af5d30f92d5b3d33
26ac57a13234b19353f7c41e948d5aa4555f96c3
ab4c0548d1ca9301ba118a2afe4d8e5a
30c1107fcf53460777b8d90c9949f84e43f69ea5
55944f285b55aef4bf746801c89dd017
c41e08efbdbb4f0801a7ba822311f861525df548
f7b816f211134ebabb1c29ace4754e9846db4b40
1c19d801f2a8b613661ab7262a57f33a20eb1afc
398c5f7766211d70e99d9e45a4c1f680
da13f90e1859b2c41d519cb497b549b869e8340a
37c47056c8614b7d29fa81316ed78817
d9a53a669ff650d40806b551bfe5e8a2aabecb09
8d6bc1efa61ae590390d957cf35f4513
a688bc2937ccde0787619f2c9d23ddcf17e6e4ee
923f888fa3e35cc1ff0b36127cca10c1e9cb0b9d
b97343102f25490beaeeab0f238d25fe
99b6aa89c1e2b690ac17054aa219026e
63276b44d99edb039a9f697977e59578990fd33a
ae587f813c577a43ac0474fa8762e14e
8b14b166f4ffd3f83441c3b8fbfd8686
9811ef9d67fd36d2e86e415a1cb08d1a5c71f8d7
f1599946fa02020c781627a3d24288dd
771760d19dacbd85fa9de2b0230bfd8695fcbd01
e73467f0fd9186b6256e27f63d923e1f
fbf274d533b6f94e48a8f74aac2591341d678902
ac1d9be6c184c52e6862eb788fa03628
8c351cf22144e67e826882affddab6169087d262
cb38e1ca5a9ec56e07a44134f78063a0
d3c14d8c42978d7cec7dc5f683df8802d50eaafb
37dfe6d351781827a3e49646989fc1d7
a4781e0a1fa63bf643ce5c67cee7b8019cd1a5d0
798a87a8d57411e60cfe5818c84e7fee
8c6a77c7f8513f97b6839413ca7874ebee377c03
0d718c2d2a7461f1a07d7a7cfbaefffb
e03c9472794902fb738dcd8cfd943bfcca8e2f66
a5a6be36d6b09aee1d4921b3624abfb8
17e99fe443714ae28ac7cae074e74cc3dd7131e3
1349667822d28df638963231e1b0195f
40735d040e158826b28e760559f826a0ebe0898a
f6cb53666aabe7447acc5d39c43da7d2
f24ba3b3a7043a44e4a67e58fa4b1b68a6b9eb03
4883dfc18f0d0dca1b09b609599be008
085d0bff9360a969d70148453c3f46f04160c514
589c391ba661171115b06aee54f1ebcc
d38afe9166f559853ce92b7a599f56ffdbe88393
4d4514b8c27a6a9214389021c5806a2f784d6b07
21d343bc5e1f5b58884745684d0cfa0318a9dfe5
52cd5dc9670aba07017264413606daaf9e02fe09
6c62441e12b51bb613d14dfac3dc5c92
acb6258fbd5ef605158b30f5c253f50bb415dd36
feac6f91e09cac3bab97850941c8401f62a2a09f
797391d3c801e0099535e2b8b45223bd
ac6683bdc1af0e5d4b40170d72a63b2650d2e745
38a2d002beb139c954bc66f4e159f0e6
df72bf53e4cd3647d54e7ef2f0abf41ce9316bcb
849a295221241265988fa6d038ac7f19
476aa02dedcc4e4723e13f859c389c1d728f7fc7
572d6a6c772b887ace3cbe69602695706551702d
d8ff81332204e909b57c63d238caf712
6ba27e7a27102890dab7156fb33197b3
b16d1ab834a4bf58b00cd3791c1e536b84f0f7dc
4373dc0fab4703f8fdbe46e991c35070
6f281ba76690cf0b78138736c556d96783359213
59447a8ea0e65a9d8a858ee04a56295d
beb99ab18fc473613d9a567a5c633853e7af1a12
c442ecd39bfdbb36e69ff5b1a8798acb
70b782963e3b7224921dd00743bd06e015d567ed
be73c620e8036744b5da1ea724fdc479
7086c9c79594ebc10fc3ba101ea7311e7d4eaa65
213e2997a33d14d6189554be4a12f9c2
74016dab2108dbf4a02694b299548370c6f91376
509422421a82b80d5562f3759532949b0ed0494b
c82031db2a8335138c5b83fdbc428f52
6047984736206fd97044955fd761c26e8c23a7a9
7ddc5b09ed4b2f3f79afc92075b8fa59
28028492a1b82105f3b2fdf2bedbb2dc
84fd1465cf8c1ff95e13d06de84a8169bf6cfe29
ef89bcc83b675dfe47fd790806454fbc
fd0cfdba126a8184dabe47090f721904ef3712d3
01d0b0287634726e61defb5480a54321
a98cd5824fddab568d109560074e8c44f67e0262
e72bdc77a4276fd967b7c1c6122b39cd
f85232090db95e8aaba5de4367a8946e97031086
8b954f754fa80e8d99d27e74bf2e1e7b
44acbe13b136fdd8fcc305b6b1312b7d8377f8f6
598b50b9dde3dec7426f748af84a8f3b
953733607bb0eda989c43b6d40da5566ea05732c
94359e3436afebcc3247b3a0376f16d9
e061e3a6d9d3e877168a97a4fbd21c7b0c427951
9ab9a0e39268f1a7ee0ba4c802262110
983c8cc379f74afb7b6f770bb3359f53dee7ca28
9113578df2610acb35fb3d4f6bf80366
dd9ec699a90a1d973a78c990d62f30b43631a5b3
86a3f2962f991bab782cf28a04052f76
7b63e70fd08b1dde2795aaa204ddb65510943652
c2188affc781921f372bed6578d54ea7
14131a72c79abb77cd4c2a1f294ed19ba5697e2e
68f4c739667921c969a804cd9f67e59f
e8a7ed17b7bc6cd2d210718446c1dfd3
622bb1e41fea3262dca9b7a077d3d92e52f69f02
aa3c88a2ecacf402de2a10531f06b414
0bf7bb21d57075994e1326a9e43d73d69cd46c3c
437e9c86d72c5181897d5f74b85a31fa
f87ee1a3e940d3610497af2dc6f510955498511a
c755919503101b48f2ce1115fa10363c
7f60d243040793b8e7086f897aaa096769179d33
350dbd76ee74a835996917278c8f3fb2
e08927d41e68363c845e51843cfddc42e3cd52e2
948017d5b985a90968f0edd778dbd5c2
aebe6cb75ca8bc733bd1b0d5125168ab
d1bf705da473fa8369a3fba17b2de77e6079bbd5
8ff4d54f7f3d919ee35919febfb7eec3
bc5f483746a2d1de68a82f42c59f5485e601c0f3
a6f62dafb8cb870fa26b9e0323eac5a1
5c83bd8fe49a601b0b7ecb12e21c89c8
8c7fc91d99c44cd720aa3a296508e7aa78554899
3555ca99e2b138afe2052a07538a74a6
68d0d047f10a1906cc1475df81e4b5cdc3c73d78
83e9399757f6127e4ac5029d66a27119
1b13b9a5a68369737fd66db3c55be21f1845200e
035d97f748a44878ae2ea2a763fd6573
7bc720c68b82e38c1a092e5319515a40
7b89ba02e596a19fd21a7d3a01ea6ab13bd508e0
95fdccf8e837d2121eec420f422da964
b4dae99b63f5788a038057abae70dbaabf33cae7
487a178a6e3f79e199a8aa948a6e1503
f2c160ef5755df0d83c2013d8fec14d2e93cfa35
308b3590117417e5fb307d4c2e17de78
448ab0bf7e80128ae3e2d84c0ba0b0d00dfcbbdb
0ffe415248eb6192696fd3f839a7ecea
931e6f274cf4f30e3b7e87d6354a92d1
c0e449d9234b590f7896540ff39aee3151681033
87ff32aba4ea0244abc5bdfb823ffef6
d12649eaf60624da3cf271bcde5ef2fa
3012152c6b102d14d175881ab0fac37d8d442db8
183bf263e80a9fb516550bfca8eaba382769902c
b66874cf875164c3fd1a8f5cef9572f9
994e8982216a651aa634d66e6c77319fc01dfb90
8ffe476934613aef97b1f1b6e05cb2b6
79faeddecd6ed59ac3622be38fe4f768
43ab22ec01e365d6e59a258bb38ea28febee4d52
9dcec0ef4fe8e2d590b7d252200017d6
ccfb040960255086fe379a5aef08f868360301a0
7448a85363be3a23e647e76aaeda1f08
38b5c4ba158425283be7ffc09c912c1baecd2c74
c9bcd0577c7910a4ddd6e0567fd20e24
2c2ca868dbc2ced48b15a8459308cd9c
dfea1845aa2a38efbeea31eb01095974089b2706
024632a88a22e77d85da478b8d9d6194
943fac06d0bdfc45a2c8403f5e9e64e6f891b4cb
8c3686c497eecb94b3974312a9320623
02661c0ba039e2fc61ffcae8393f1067b3428ab4
14b3d455b1ba8e345c4c479cf3ff173e
4ba058e2e4b272459c5989805f5192f93392536e
514ed1af6728c51f92168b6bc0395d56
e29676914ebff821bd75dace849b1b7f20f746ce
d906f3a5ffcffe7b13d5cdcced055421
7077b645979899f1ec82623bd0146cbef843d941
17aaab6dbb42ba8995564f90ec99b7d871183c8a
a0b33c5da8e777d5a54f2ba46650bdd5
e8d94c74715c26c54b9651f00a13bdd4f0110318
50acd53b81c5f93a38e9b9a8b4af0913
2a6a1635a5f2866019f033f50c123797c8ab1373
a3c0fa665257507935a6ed7b80f762cf
9e7ef72580252552901c769685e9c0de960afbcc
9bd7842fb12054a168a08bc41249fe44
70153963b57ef9d29f9b07be592234357154dd9c
b87df9d91f92a372e26da91c99bd83a243521f7b
886c787385a7bac2e4e41e614f83ae44
741b57968e9af2e534be62fed94529a9b697f390
051d14579ababb8e50f67c6d8baa4061
34ab5e10d7fd6f039f40eaf0a16f8c8509f7a23d
1940702e7a3a5c03c8fc1b7d34029211
24fbfae3b37771dcb5fb1b5c65b7b6229b6e1f67
c95962875a95b2a77b1ce44af56a0fa1
d7fcec19059b01d5524d6e531126511ff2d8d7f1
a9ec3c3c74a0a3defe10e3ca3ffbf2c8
0903642dc5d1abdbe36695eaf0dfdb9e2f233699
570246f5c6018b6ac2796d2bd3698ccc
c784c935b89b199f82dec20d5a1618b5
c81d399b0a0d594d898c8234bbcf3b8ebe1591da
356735 F20110218_AABQJU UFE0013684_00001.xml FULL
a3ff2cba76bb63e5f77ed0e26921aa0b
c7546ab1932129b2c3dbf358c27bcfe91acd4116
89cc9fa63dbeea06cc1793916f941c5a
123aa62c387482cbe821caac797570a956078803
57ef76a61cc7c4c9038078e82520a5a2
38e5881794359b76ece424a728beb4357ba849cb
c6e32fe126fd24745c1cf969d166bf71
3dda5c23bf68e1322f558cee5925cde1ee392708
8b979890e65fed71ab49ffdbe6d2be03
0ea0afcb248854a7bdd3269c67a68b35
e0b9f6eb370d8cb66e8cef50e2d123dc6da35469
675f1099a12df6e38278acc1a7390330
0e5c8c1e3e2fddb8773e9ef1026746cbfd86f070
55de7ac616dd5003835b738b9c724107d572d34f
8dca6c12aee2d9e8a00a544d130c9ea7
7dc851d15df3f9d08d33a1a0e2875183f793ee21
6f88f672fa9d62e3441feec40ddf69a2
e30e9ba91deef04d203351a20b614372
b9ac25a65856deacb5b5481e56bd13a5544eb915
0264b928ecfb4fd7438a1c008ab7d560
33f9123493d3ba3d904854173ce51e09516797b3
2a3fd3c6edfa6d37b28bbda99cfae5a6
fd508f8c7580325ab52eb154cefc1a9157241372
38fd99d8b74a224027e5002b1a081502
dcbba827971dbba78fb77939d68fc5de56e1d401
de0365d4a0b443f7ffd5531d8b1c70d0
54b92a44c8dbb6e5d65abb28364bc03cde285913
118b34bbe4bbb4243c5361f5a0d895b6
3f7de17c317d0fb498d5243e0cb4709358bba407
48e7fd5b69e934c17890f59085fdf1df
03d3e7860e352c81c1348730930b55520006a776
b48f9bb785215ddee70eb491d4912837
f6dd53973a65895c04b7f7f2a05b285c1bfaf855
b85d60d3bcda261bd5a4d2b091b21422
050a63633919ac9a466e78ccbeb2a7561a522723
54d3d429d7b28e5730ff85affb20137f
28a339d03cd27c8b0465469d24c7a7382a1e4810
615ecf0948db207b487decaea06eb66b
71dc9ff47d0f1ec9b646871d027be1ea5de79626
f327b9b01ddd72cded5f1362640ed606
f989f08c8b8b168c7d8442f9d355083273297365
c422d7109e55bfed28dc7694f41cdf40
b6058a5782e1971f207e5173e6f458dbde71c35d
97495d20cb1e3dce510d47ff0dbc0fa3
d6d65e7376d8fe4d4a89d28f0c09ef48b43b6ede
a4b9dde056dbfcac399fefec76a33454
823bfaeb5326f7bf7840f6323097b5315874a515
c37f370d5ce89011888b96f2fac3d765
9b329ebc8e30dc4b7998705fd2e7603a83b1f585
18727fd466222b148ca9ba827f232a28
6f1436b204a08c86b3fdb33de97948ed211eedf0
775e784bb0488ff9c41a0c6468fa6798
c906aae8c22d7b9131489ba7ba3bf452668ac49b
1f372151860e377cba026e90be4c0d96
53c6cd7d6d38cec9b0afec379c1586c58792fcfc
26f04589901326c9eecb21558e008a31
597a17c446f8d2c1e009e21c30d66ea76898bc3f
7e0bebe528ff76cd12d6759b027300a7
a7e35f0515681133a73f18e2eda62780bfc1e81e
d328dbffd329991c595850385f7a7c174ebe2be6
9c69b8c4901632fd5055a2ee7d732573
aec06ed09b9e77b4f37656bcc67beb10
d500946e27328d1e97701245082cc8a984ac9726
84317f593f9a825f928b823f833cba53
b8e52aebe253b9051687c919303e9ab2
1c1b85c095aa1400f60404afb02af34ede072485
da1650f3d9e335822e9e8f4b315e9484
d47f70fac1e900bb19dc1f1c986f634f3de9f0a2
6bb6070c1d549b72549063838d95fbca
fecf58b56048fc2bf3e016d4235fcbc1608ef60c
0e185bf2ef620c6495b035cc3cf7b442
a667841eb523fe0c4d26061f3738eed2ed5edf05
707577798a580ab8f33e17de2d0104a7
af00d15591586814467d13ab066c62f72213e10e
a083cf8997c61b177f3b0a78ffb9034b
6e856fddd97d93e565ab650feaa38250c37932b5
6ee675d92b8e765d4d4c08cbf247c691
9d4c9dff1191832ed1aa36f061ac25f0cdb388c1
5acebc44b278f9ec85970c5ee13a8781
f41ed72c951c4c2bf24be7ba6da9419c1025dd28
c112a9a65e6c4d6a9f26224916ae3e4d
56ab20e315d4fe5ca06f675a6609e6fc484e9207
25485bf0dbbba6fe831f4c3715f0c936
b579905e800182b63e35b3b1ee15f0357540781a
438f2df95f992cc2edb87af1f9383ba4
f634378c92b587c24893a84a87eedc90ee6ff608
531277680f50bfb725fc80fb156b232f
28fa1533becfe1e41060ceb6958fe518f4a391b3
047710837769879a97b3c5615132dd31
0c23471423bb016e1699fcd6d78e63eb32f095d9
73edccb646d8589baa9f73fb7584eebe
5f07389b1149b28ff2245c9f0de2ea9d1e7b0d7c
c476677b3afe7899ebca460c1734169b
6118246a1984b17c216280dd347839e9c28ccebf
50c23a1851dc8cf9a257219ba83520482bc683c8
d0a9244594af618416eb4a42ef37470f
8e3bc58d1c7ae40a56641ec425fc13b2565fa496
d90611f0b1e05dcbe3ae7a007a56cf3e
e03c4fe51a1a6f250b3350fd419cf1ba
f3a2fd25388d69a8934344ac17e30b68f43ae19d
96f76bdb64ba4ee67cd95180fa4ff120fd7ffa20
f1349356c9226b25c1f64075c22e645d18915177
f020a89ce94f7437b48832663ce5ea32
e45546a017c92faf4460aee33e5510632f9d3475
2d8772314dbd13e250f57b7d230994894a6a04dd
7d61186179a6bc7c28df547bd9813af5
c9abbf5f2af542017c91ba14cc2184d670b8ef20
5531c2831fbf700c5386fc11a6e123be
4412ac14809c4b7baa0332b066c0a99fa6f141f7
cc81b871407c830854d1b6839c192e95
41268259aeda7652179126634b586269f764a6bd
63415e10a09bc29d9278473018d9bc61
b48fe5e3c0ac020f8ddc21f9f3488b08735fe3b1
57888788ddaf66c2d4f0fc5cdda3ddbe
2a6dd04dea4d2fe21d4933dde02a066f72199403
388308e042ecc2079f705f6a0040fa93
ba26237563bdc784e8cc7f83d6eef7e24ff0eb7d
bd5160c2f0eb2c802961bfe93cda04c1
51ddc2def94c745b3bc31bd995800ff31840ce63
1bdb2f57728a3104cb65c187417153b9
41f5dfdf3c3fd10cf3099981bdd63d4ea16cc51f
561800a8ff2845b9b617f4431561c74c
773d8d701d1fa8c9a5669c24fe2f7292676a04cf
d1a3a0093f6250e0a493007519dce4ac
62733677df10db7199c87a42583718525222af1b
fab4e538455f3c10648696a1914a6348
489b4afc4e0533caa53de0abcc4a8925c7aa7c56
db6494d19c47d1128dfa154b236b712f
6174a081b539753b57dcab5c55632473df87e23d
cd0301579608d585477e33fffa31d38f
75b578f2892eb02f30b4b0bcd8d89a6602d9a391
fe785444bc4659d64129a091d3d18798
72ab1585737ea3023187b9e4ccb6eb3ed23751ef
fc53c33b5c2e5b26f503a0b11112d2e68fae88c2
2bd4a4eca8da25d88452883c3f77431d
b475d733d82c1ab22ae16cfea1077bd06b18336e
54a5bc254a660a3eb558ab510f3abda6
09ca53231924d6a6bcb0951e4d606cf2a4959dbd
20309971b9e11dbd853ec98db0f9e8d7
ceca5c3c8755e613867d79339d232e21b427bf2b
ce37b6f80ea2820e534347643aa8b207cacc1c5d
e44e43f8cea9195dc0197e1a119ec56e
dfa5001276cb20f7c404b6b8f7aa16c8f5ef44c0
2ec20a2e441f8150f3c004803a47dfa7d5956936
c2801dbe2251a151862470d7088703c2
4133e58400e49c4bc054695017f15a767cec28e5
9a15941e983bf518c0ff2dd51706a2b3
22d916d0d88d16c93d71f3caf678a08448f061e7
29da918c63fa7af1356f1d1e1d25e4d6
1d5aae690a6928f8cb13189c4f6b16b349ddbe63
71e7cc960130b0bfcd1b22449151d2eb
c5f0f0517dbc474a1ebc498ea1900937c9be9075
7490cd4f7ec4fde498c04b72d8deba21
029b3379a336697a49d7fc42c8af58852734bccc
863589842021b4fcb6b22edbf444afb2
91af422327f06a101fb6a7067a1d6f52ce97a0f9
07311a60c54cb8f738fb0c59c16bee4f
e428dd03f9f1f646eb2a6a56b9b97a5beacc0d87
b1dfd553aafd332d20a53ef56efe6287
8bf948afd9f32348e960d5ae31b68ce39c48c150
ab43513362d2294f670edf97338fe877
0d79161201729b919140749f4b50dfa4d85110b2
55e937120b369f425c7b946ec61f19d5
4ecbded2e6d649e1dae353db2d624141e1b6d0f2
3a2724477744c2fe75de48dc03f92368
a2b0a8b4ddcd5dfc6062c78c623b75a156834e35
bb93a7d207663f856d739485ec815c70
a7ce05f85dd4dea25d0d1a78854779b4b0281975
df916d98327fe70aef59a1be3a434afb
f6b2b4912645d38191d983afdc098d81500ea9c1
e0f028a121ea6ab0411bf01f7ea8b1e8
b375edebcba339686ab0176fb3b010d3b10cdb45
24b95829510bbe15f49273665a914326
005139741fe37b7a492c713fb7997160eacbfbff
9c3ee3c11ece8052b0e86a4a9f806811
20815f64c49143c14d7e73a1377c5541fa99d8fc
0bd3ab618186c15faf8647c28242af2d
08144b76ed5fcc5c84e1e2a8b0609edffa0a984b
4d3117926869b248bd837cfc8ba9d6f6886fe959
1fe2656267ccf6115e498596c2a55c27
e2c7c9f5a2797e8c544a03c45edeea1d906458ab
94ed39201fb2a2d680574913c6bcbf86
0bd2f5836ff0a3cdaeac526c2b220a228f2286f5
e654856632bf51f6b8b76937a5ceeeb3
2f742335af23c2a497310060b93e985c5c5d4723
f12ffca74be0ec74957d40307079bb3e
651ec7e227283a21a431eddf9a0c01e4296052eb
573c765500ab8922581efb519b902f6645d69ed4
68d75c1e68c9bb178ccfe7d92250ab45
6a4cd852c36245d5548bebeeb5c5897c9419c195
d39e6876178a1a9144babb00cfcfaa2c
0c8c97600f289dede2658a78012c71f55ac2d4b0
553dbe7efeb176db272e7491336457f6
218bfe1876d0c4fa6b5d469b487725bac423f1a0
a20ebd4802fc89c8a92dcc4b2a1121b6
f3f1ca2fe1329e908ee04463ae97c7a38c15a5d6
2229f0cf2acb160619418d6444e14f35
24ac087c8e805a20f00435abd853e4b745bac6ac
7e60b3443c06f2a59ee9406d51e06899
41b9871572e82ce66e02eaa22a7ee85a
72af37a30cd7f7a1a4169d04a7f0eea0
082469ed4c86b25de445de6272dacd855c391a90
952e6ffb566f83966987cece56352f07
0619a9c20676fc633bc3747eebf29be194f133e7
d5ea6f92b7d9af14c6517e88778724d5
4895ed4f9be64a70440a678b973f35a6216c9766
98f11729c6311f567444e7560410dc46
dfafac67a816b2862f296d4213ee58fcbc04ba35
874f201b24aa4e0110478023aebdfedefa9d56a9
a73abcdab8e68974afa36f354c8dc31d
c780cc7a8336e937462e09f803c8bbc447274736
81e5530cb94a5e2d3bb814302d0f770e
ce1b82590952e76edafd9962613f425806b8f94b
a4d8dc5109ffefba26910609f9cea64c37478d42
ca2b2e2c0b9b6484f784b045bdf27d89
0dd529fe2c4b99b9faa8fd146a53a09388432110
8aedb2699e937f943a1ec857aaaf74b3
38ed031eb74553121c47ff0ac304802b8053e542
58ee3dfee7b406e151d1f8099065ab24
2d0abc1c17ac6089a1b13832c3b297cfa58fed7a
730208628b6ac35292c443302380b0ab
7bf8a7c793d96b2e139f5832c9772dce7a9f5535
8cb3ff1f71f95a40730272cbf110d569
e8bcde188a7696c2fe2eae49594a02e37d315f37
a7c3e94709efaba76ee249b055d5017b
ea52e14f22153b6428afaecda1f73b3d7181eda1
b0f5216cf0b23f1cd2d27db6e833f417
d1778bc50d5084b76ec158cc5dec92d29ec9f3ab
6a29783be2b8b5f426966e07ac45bffb
fbb3e45662546106bed4e3213a99567876368d94
b0f60e95e3e426f5e65cc9c3ca71d12ddbc96d74
d5e3e6a23628d0ccdbd5cd500cf55ac2
08fc6a8e4f6e787068065c0fda09da6f2c7a3ce0
8c2263da2144c173feb19a5ed2fce5d8
935d9ac7304db62c736eb0140d403afb653d9772
77b4828b941e9d20b7d75ff602607e57
fbe98cd8c8893af48509649d221acac644353c0a
ceaa5c3de847f067e1d0d288c232e7ff
29dfeae5d0ff148a2e4dca763968d4fe63977296
456e57f55bdbc81cb3ce7863c82c9f1b
5a0b65cabc8494eb631d463a1e1f43827850bcf2
2a3a017c14ac6d546f6f17bbc12c4b30
b3476fbbaeee666b4f736e664854ffc89346c184
b6f21f6e379e838eba83933858d9e804bc77493c
d0a85787063082756734566c34c6de9c
68b98f9d5e4ef73ba2a0b7e89f2569f454787d8f
d256fe81cdf5e4f804cf39de950cb309
b5fee828aea414a2281d3733caed1f842f275b37
0c03fc29fb7b248b40e40d2e59dde845
11ff07d7a651c76365ea2bded7966e0c9e8fec7c
b6df871836f3dec11f8513654b449350
f306cdf0bd317b3f4cdec40ab50a0dd8cbb9b9a0
e3775ccaf004d38fef357be169b09dc2
de2a1ff5f12576ee1862d3a31421512fe73648b9
9393a74abb2dda6bfcebf7a77a9f954b
7f9016e9cf1c57f5721c3216136e940762ee2958
d8a9c0362b29785796a2b2cab2bfc62f
0189067e221344d037d94a2836bff4969953cec2
dc4691193b79e4022919f81f644d58fa
752366aa06a2faa01fba61271a39712c785987c6
d4b10a057f4f4445fed4912d5687880e
f834780f319275cd84494d086a5c1e58
e9499bdb5421fb0bde310f8cc0b930edac9bfef0
611a94be7c49897b44ffd382e50c9135
6d03332d33ddbed8ec6378911a5b0512c4726bf2
3f2f01ef4040def61e95f7b11c5d60de
f86faf54d252c5daff46ac4e5b40d5d6fe4dd42b
29779d487b261387999aa51b5a53d87d
0a5281729ce7f159dfa876d011df3ee030865b7d
3f86650ef63b55fc395f4f9e7b824b45a8183b5d
914bc55a07e9a939052089c4abb79954
65b1574d0550e062858a17799db2250c8d2261a2
5f86b6d5b8a12fbd10565b4fe3f30d8493eee390
7ca0b4a3dca4cfc120e138885f5052c6
fbdf2577d7233cda6cb862c46a0cfd131d4e94c4
87d4cde51d61205501b15be017c92cc8
f4a8404888e59ae3593b9b6380911e3a9dd75ac6
f2e89a2008b5196f9d6942428cda6847
79606691345d3ba06fe072aa85a093270c93c473
ed8650b3f081809dee0cf99babf0f5a3
514e3be9cb1c2d4453c93fcb99ec782c1dfe2e92
331ce1d9b43e7808749ea8f60217a404
bef6267f70a4f50344edb4edc61b3f9a62d8a11f
5891cebc1106b2bba8e2b959a9583fef
2f25831b9eeb1bbdf5303a43cf3e789d
df66db841a8a893dedaa14b038159bd8efa06468
d92700baa2390566ecc7bc9d5174a6e0
8b58456ebaf2b31a0ab1c740c7294852f089c5b2
98a44e5c82b3b51da1613e9ee8b37cff
3bdd6fca32b18452dd50c869c8602909
697fbc6fbfe42a8e55895e938db0fdc6ee9e77a6
025d567293a5b75fcf715468d2ee3f2b
779dc5f52b95c5121a74a92f8dceee74
16b5a01e0c4b94207d14975041256ed24f3d1d3c
53eb10dd3de9e59d14d3cedeef41b070b309dcf1
6efafb2b1e335747db23940aac605cbc
6a6ca4a0c07ef00ec550dbfc40041f3a57085067
2c21c40f15d2d255d65a4f90f82851282306e067
3af775482ef93b42af288497095af348
5bc03957bf4c57b774e2501b4cd10b1314490fa1
5223e8dbe902147a88809db425b748c0
c18896f37b7c47816526f83f02aab247874b7fea
f30a5b36169854d81f9358de853264f2
06dd3b50094cede00b7232f3f358489b
9f36063e32b4e151fb0c1f8d6a159370580851e4
6ef7d88515e152b679f2fe48891b19fe
53be48f150a652a41408159958782b2f7f0837b3
7311a27b20c493068d49c942318d9b88
209c062447f1aae11a974525854eb0399523e10e
496c1ac6f9ce872203324732b41bfd48
667fdf27de53a6ea7e8dbe5dd9747783acb345f5
9d62fffd137d8e6605c219e8696754dd
8f3f2039efafe202482c6650d2cd3f7acc52cb4b
0c8390fa07aaa8258244bf3269c39b0c23a1897a
ab058c5e130e4d8e7a073de76100b5b805be2459
855e15ec718df53f47e23ec5638c5bae634fa01a
70890703a7a30c3677faecd7aa6300a7
af7bb55e0278270a01505e3cd70b719d24c5237a
167dbe6bb6306d7ed7deecf42af5b6517b8476c8
38884c2889a2d726d708c1b0394635d7
72e8ab9a5be70042b2357c567d9dbfd8eb092d41
3f0e01ca4fff913f0569c962b6076965
454dbc79a843caed77e0ac06465d935b3ff1d01f
7c91644a980ccc7f5ce2b505e7a668a8
003c314e5a42783c200d95cf4053e1d4c2d2631b
15b0ff84ff6806b623297fc0aa4d1750
7ced365668bd330c6c116c6dda5680020e58087f
b47ae66213f58fed782b687f88446f39
d30282011ac349da20eff2a160b5db7ed0d3672a
2895b8280906e635aa22aa976e589853
37984a3d08d3412eca1d15e4666e4c48c1323e79
84d2b461306a5ef64cbe865b351523c8
75de46a8401fe4179e37a483d2199568c7743fc6
6d2958e716aabb47832f12064d99e85d
bc1038e5eca15c8ec96e8cc95ee5502c
d8c82163b10fd2984a418603ff7854f3c3672468
b19b7627cc44ccc5455ec2cdf9c78195
1224b256ac3c1fa69023c00ab8472e09
cdde9610b5f34f5a6c636b0dd30e78dc80506556
e9737370d46d548a86cbd2882a6b8e6a
b64fa1476439ae5e3b6b87c002e333d511bf7fbf
e56c213ecd31be6bd3b64da247339030
db2829b225190a2ef9679a2024b6daf7deb34aec
784d63e0d6e46a029ae2ebd2ed7fde52
1a088ddcbb5d739170a870a5cda86aaf
005db3227cf5ac1818bee9af584e9641ba089117
f6c74b79232c2b4f8e082266c2819078
e85aa4f1c0c7076d4d79e96181b9f5435fa824a8
05a651365210bdffb145d4422626d5f3
6b2d4b34450647b8394e0a11fd35560ea56a1f2d
37120de7a0a46399c340e7cb863c2a6c
4b8f9736fd8375602808a0b28fa2ac75a795e25f
f344574bc283a3a45ed511bda6082a83
307fa54e4a613d395e1e6473a212990de65557bb
f7a744000a74c3b3edd72e94421e73dd
920e8d5ca2c52cd1bcdc1a745270ee4a4ba10d37
6ca89cf8435b0af58e0d08a21a49cff400ebaf6f
80d4ebcecc52ef861ce6638830e1ef4b
ea2b083f3b3ee7f12ef1cf9b3a3a31cfca9567c0
2ec51097f616c529313260d788a2a489
8b9082b1d7c1b86dfa0fceaa7777cfd80dcd63c0
d7ee9fc8daa0385b4b2c43cfb157fe9f84c5cdae
b49500a361ed7602fe20398b2c27b016
c2f6d2faf2f67f55c1105d443b24336c2f15c75c
3e9296ae2c35117195d58745536babe9
b83079eaa85e8b2e49221d1935b645f0c4332660
45a9198f829b8de0c0acaee30cbb56e9
40816c643f94107a3b3ea4008d50e1d4d0b728ca
8a2767592515471a7a4e8cf0b1b63582
991e2ef34d4c49aeaca434f4c8c0df1bf65074c3
4ea9cfd5f0e6689895f3eca2435c5492
31964c150dfb79c1ebccb4f16c04feacfda3bdc0
e7ef78c125d2cef22d00620e634856e9
aefba186f7150f5a27d55ba2695bab30d30db3eb
cdfe8493314001d8e7a2d43ab32e82d7
282815bdc600a99be5560aedb8e7651851de73d8
1cfb1db9c6ff627b1a8cfca0b94559a9
995bd578236d5b235f5d3cffdd6a5b9076956839
caa4d44b30ff1b5db5f0d984ab278139
1cf4c2a6799166762b16f62d3005f905878694b5
bc61df3764478de9b520d108007a90af
4bb2131a2801beb627d8664db514e8b3
d22f9a9ea6df4705ac713c547f4b49d3f285a7ac
2a9fe41ed5db2c55341fae0fff2fc0c7
864b9e8662fc75fbb7936c46da37648be6b9cfc2
23545e25af4bca991c8bc89a497479b4
a5491484e5b180d47f019b25cf337b8344e32c91
5c46d9d316f3cfab96031ffb44a6ea62
04eeac37c25040548267cccae9e23b969a9f60cc
780d5e0ebbbea52a40b107a8dc2cc3fa
830c7daec0050def288950be5c26477df795e32a
29e798ef555fc90374118be38b8645b5
16a8873a5933c1b96df5c68aa96a8393
ea51a0055a26edacf2e164dd6a5abff6b8503cec
68980227dee772f8042d96f7179d616c
3b13965b438e64217c57e1a094fff378cccb9985
118c9c5d00a0fc65e3135523df9788677b75f986
5a5084faa9c60b70f93df711feae9cf0
1d19d05cd7bac73db66ef93581c2268ef48a45f3
233f3bca03ec2f26b5faab72e6f30d5d
2d8f60fa033b79eb4cd44a8d1995f5cbcc745696
897e1e7c07092c6d5d3a064c73a0f182
e2654bcb8d04bc05e001441654d16791
e2d39ba7246955a34943c937c64536d4d7f4f81d
8c9a4527e35861181b13755689f1a524
08ce2b1a263d2522617eb1dfcc0dcb8cc65005ac
7f8652869b62579ff43864e337808fc7f53b5d5d
ff757f058f03d985e604e6da298d1d38
ac1075a1af526167f065749062b5da88ef08e2b0
f32a6cd18c07873df5bbd7da9955ef57f701c88d
ce27be2b965f79bdd9f3660e78410c3428f02361
5a6d6994d66f67804e8be830c7c747b0
5e0ff5fbd404b5eca9320e68bcda32032c1c3c47
8f24e8226de865096e99cd490b98058e
baa9b0fc78e3c455c0149cf88f37e0dc06397efc
80c19c19cacba187c804f7b2130e1035
36606d1d58867923a445b3c5ffd4e80c6c37b276
0ff2728137763a193cee3a9f212ae5c6
a5353263b4ec6c733e394ce6187b9e0ffc058cfd
5e8cfd50b64b2e1248a6ca8183def7cf
e92d99138b89e256eb65479dc269e7d0
f46b3cee2f6c6d7d39142eaea186c339391a66d1
4d1b71e405c8a3f171b4540eb88e41e7
2597c82b340a5d7ccf80396d2a6241b84fcc2df7
7fc1a619da135eaae9f05c2be6a256e6
8ed77f7d1c92c7422d4c6ddc4ef60bc874c47afd
53569b35bc8c5e3d12af81e7f0f6cf04
941d76ba4c44b05da31b319af11b67751a2f282e
91520c88b899196084ce0dbdc0657dab
3bbf7b3025da6a92bec8a9e139efc43d
df94363709c94ea9fd74c2ab65f57f3a1f478740
940cfe04c20a01dc14a02f9da3bcc6485d0ab2f3
13554ab67961024e77528fa0e971ea9c40b4c340
a876dc8ca318c8147fc964c2c8eceb16
a0b8df957f883cf61a5e55bd68219eac0111022a
87761ef88bac119829f3262c7850e511
a638c70b1425c46aa343546614274fb85ce40ce9
bb91607d1f1970bd5e68a2bdc5627ed4
b3cf826ec70710d207870a176e3d68142e986914
f1520a25c8b717857964dde5af4c507a
c6701646461a37b4db545bf574efa93165f00abb
c5857a4d750133143f73e52ff38db895d52c4656
2810c12b0a41ef721c952c1298889598
5723fa33229e86da3bc3e87a8026da5bcf7ae2b2
3097879efde5164743cc2887ca1d06ba
a2557b9a2c18a2738f8cafcd5c20ae99e478b5b6
6df58c5901a33ca6da66bb2bc7ab0bff
b270e153fbe9c1771454b93041fd2300
284a3a7d4700aae9137a2b32b33f34153ab610db
1a761eab1a1f0a451164361911accda1
74f02f757fb9fea5c4e320ae234278c1
4ed8dda4e9fdc5f6cb5de1e41a4f70c9cc35bc97
80900bf21eea3fcbe8fdd0a6eba91c76
aa614a8b99ac6e01dc98fd29fa4772acce2b25f0
b932e8f46272688de7ac85f43851d857
fee26b0bca789cdc30f8fedbe5b3da827a0db14d
992e55394bdaa208c757353c553903d7
6a4e15385414861b1cbc370ac31557b9efb61fd7
af113e16d1d0e409ed7606d7c416507c051745e1
38b1b5785e55230ec8110c4f2d73608a
d2d78518f1e60ff2cb2b889fd03380eb
5d408a6a7720c534237d7e68f1e02bb9b042b353
49f09d836429a55e8c3056de19648d04
468696aec3c9c03c352f53c43348ab5360c48bcf
d88204e7af9d6807405a416553bf07c1
88eeef9763c8d9099d82dd6192ba885fa8f85931
3d7aa11b3120f98f879cc0dd19db4f47
353a8b4af847acccca5826fde092e0cea9e4d21f
d57f3ab951b4110bd010698ae359637a
8460af6486d3761ff8d71a4da369172cddf86247
cf3790e519af860d01e72f9c9c005e7439346b9c
11c3ee0045a52dc9eca918bfc0e754d3
c4cc079077c5016f1c6d87ceda97b6963dcebd1c
6e8e8cfd926fe9409706307a4205110e
b8f99182b96512d41fbd3983b212128e30983ac1
af3ec255d564e03b14a4b548bf1c6be9bd700a61
58d9e4c9196b1936190d7e4cc1321c26
1f49179884a327fcd174c65fe62708442269360e
7db575e920b5fc3aaa23878872855d63
7cd1370881bec47ed2f48397b1b0428a0a6cebd6
5bb8e4effb9f77816822bd184a1baaf7
d213d12dd105773eb103552e87cc00f22c278c38
14a369c04221960ebd99b967824c954a
d35102a9452600b4de3fcd040da469915e2f5a1c
e7f32a80d4b81afa61b414c1f4a37488
30c4bcf2cc9388abb2d6f83fde1b1eb9bb92d1f4
e2f1fa624265fb1e95b43db44fa82468
0de6ce5729159e0452f47480912c76915e6a84b3
ecc09d5b09dfe592842d56580fe7916b5b0593e2
92942c126708310c12b40d8363177cfb
96eeafa4e5f14b932619b7d528762316774f17a4
ab3d73c25cb0aba7f8d5f99ef55655bf
cd267c22e935a377278d372dba5140c27034fe4a
cafc69f4c4eea65637ceca81773ec43a7638686a
b9871279738216edffce7b2e5db1f8cb
dccabdc9e7c994363855895e56104e53a29fe429
db25b95922bf17629e3c9c6a839419e6
f30f58dde322ac2a5f4fcba52db2246565dc25f3
e9ee871ba9678cac430110f32c641e95
1d2a0b13ec8261e5a8fe5cbc58a4c2666a411d90
f542d3aac81459b7a78f0bf53ff0a677
8d31a84221723b98196e5e9a2c0fb726b4610816
6154819b86b369d0571bfc2d025926c4
c67c26df6a69bc42330cf1cf8a85de3acdf101c4
744e17c2421f6be0c043b47c2ec87554
326783c3525d39a705576f907bfb6774
c3524995af41a39328c72061dd879900b151d8e2
6121d80f00ce7588c54e20e17e4fa9c2
dfcd23b08a93c4576b9ffbe87eecdba3ff91e581
ce31ae75b938debc435e14d924633a95
08e761ac4540f74714877045549678e0223af6c2
a18c0a3762dd2d0acaa0b4d5135d9fc8
a9e809ab6261759168b4d43921c82332104d009b
5421bee2d81914188a467b1105332f63
f09768855474168fd2b63bab3d0eab54a69d3cf9
888b909a34e1912d411da75ffbacfd93
72263874eb62d133ae083e43eb680ff4c3697567
82c7636b9fa774ea63119f9ab8c7cb4e
06a7ddec38bb1d528456ee75d765a724a2105a0f
d64f666fcb8b2931258ea5d79938a162
1ace2fd8c5008e977bfae0a20aee372daeaaa27b
9e9459d2913849732820f6ced963e673
e9f863e55567ee01d72b5dbed9f6e5a86ee68ab5
a08af9820a2bd0f23406c15917aaccce
cfeacccbe0788dab6840b9ec2839d1b1
44333567960f9d86427282c08e62c81cd011e1a0
d8dfc5c3d97031662edc41688177d11bf2e865a1
b69249c5e7077ec288777dbf2c12aaa9
dcda2ff5fe3e9e59c1f6aedf3672c341303c68b3
2c6e47750ba4e0dcbb0a33c36ba3c721
b0cb3e7e11c990d3c369a7c1036dd126
1e6701ce28878e8be5f9b003e07741fbf95f039e
014073b91da678b033f7240cf95710c1
38e03a7985e36e8ecd32821075520623
322d98da59cee92c8b549d64d9b8e74820f5660c
8851f52da1db336831550f238cffa7c4
ed3dafda980b6377be05dddb42efbd2b0efc8ba9
a5c376f1cf65c7be45e040eae02eeea1
8982d8576bd679337e34a716b0bf4c8ecc961086
4f67525f855a59357157f787f5333f38
d286f2ab7492d8e0b1a1a3334694a53b539b5296
42a46fe53442248cba8e1040f1a4b4bf
891014cc07428d692007547f6f61c7426014584a
e9857db01f53d957e15dd9278204d2f3
fb15303ae17d21277e5f6796137fb67f761ae870
e671b5f8cf0c5da9e9c2e458f7bc4d16
196b303d588c90627e375af3ff38d1f01fd09bbe
851411c572c44b1a1ecd0925a8e19c45
62842669cf0a5057663051275d538616
c5d49089689480ba2ac987c6b6a0aa30a14ed3bc
e9611c274ae4be88d8b04694d11d1788
7f6e9d8779c1031d4cbcb95709ecee9edf8290ea
27299575ea74e5b93bbeff04f1bf4bb6
017560c10fc4f0cf40e3ef5062e776e9f9cbfcca
f363cf7f8991ec3a09d57e06cd63fbc8e2462b5d
275670b4b1e557a03d25749fe00d22f4
b50fc38d5fbd2595aaee776485c2511b
185538786098e9f818fb15db243c6d8e5713783e
08f457e2617e48f1d2fa6135f3d6cf9d
9f3a6b05be0f8f14d7914e7933a1d81f
1500c7cdfc6572afeb9999642ddb6889
182c40c7a476bcd0f4b779a9857b8fa54182be52
e6375677e98837480ba48db98a5c565e
1344755d6d6cb3664ab3b46cf8a6df1995592fe0
23e3366eb71df2ddf8b0dbd487715a00
9644b058cb8a0822e750272cd8f62fd9
488419c21bae660fb432b5fb7b76ce5f
b947137c4404c626b1204b1ddf8295c53b8c8873
a6711b826194b4d9e604cbc8f325afff
0564c7150fda3f359b07c19efffa407fae2ecc47
ccbae0ddefd3073292a815078e5a92e2
6db6b94b398ba9af94dd0bcbde19427d11092b3c
10bae994e098c320008b7e4e5ec801d9
d8879b431575930e9118d987a83100c0e1a59a23
6174683fca9d3840f7ba3ecec5e31d3b
e6453c98f494b6f600922c9989b24a77e87775e2
691e9086e0d8b711020495ebeddf0e57
67dc21eb4205b6959ccbf038d5251d8e68fd82d9
f6ca306e53919d0fb35e63b7f46018f0
8038c87211ed58aee5fee8acae773d2c
956ef429a6833f9c54c019c53ea197e5468bbdb6
51e5eb7212b933053340bb22ef620646
19b51e38d8eae3b76624b1ea4c44e5407380bf6c
44e422dab8f88462dbcabcc38495e6f7
b678c9ca7efdb537aaafeaa5957232a0ce625ae5
4ea369569b06c9293caa1f6a419682cf
7b3ab36123474779767b607859e07be47c369b14
336acd72f3d559854a06d6b2ea3f319e
8416e3a61bb58791e68a24387db3ab96b74fd674
77c611c029cf8fb1fffb5561671921c7
7beba8f383fe04fe8188a80e42b824a6
25122a62b79e0a77b8c7a111c45ca758ca1ebc29
aee959b2a3edf9dc4553a3f3c7bebe58
e2b9ac81ae575d4133e6294bed472c1269341069
11a8c720510102a716ef79133b00394fa3b386f8
04134f442c97d6d69ed41a2c7a48e298
6de405c341bb2e2d5220d1477188bc6332d8662c
ac18e3deeba2015a15a7b457e2bd917a
2dc69f1511ca9b9bfe3f6c687dd37477fec8f4c2
50596f49921abab5e4ea820708256548
e34dab88f0ecd10e00010aeb4a3aaf04
c812e22e7b3b46a7e412f649561b55014963c50e
4dffb317c824c42bb9684cf222cbabdb
bd3aaf41454d635d6b60ce5f04e18c758c7b68e0
d04de5578e39e656cce3a2a85e474c28
97d76d4269f349625395fbf90022081bcf6b3d6a
0e446411f0843e43e4ba60703cc2e1ce
66f9771cd79c17569d5f22edd5f934154300c2da
52f5a1a1ce22f562cf8fcc0925c29d10
770123c2e5d532bf21a0b87891fbb6aea21c32c9
d1b0ab505ea77a14b2556ea7ea946368
d9fc6567f22f34b7f115fa995a20cd28
7c84d64b6892f619012be6e0f3d19e1e
3542c7a3065feb66fa429715ca70192875607fbc
a252ca86b6ea72ba36b89a6ef5c7127c
6b37c6222f0b2466eb28d75215178eb13bcac281
32e63c9a1fc0d76956980966f3c0b951
447ce59042663d24d3a38dce0d5b1a8140b4c673
550ac011efc2ef276016d9ab422251ea
541aeac23ed4d562ac22a8c12ac4250263569cbf
f85491b7a9c918296af1cd57d9c3b440
25efac0d99e25b10c8a7226eb56f38cd09da927b
f8dddf32db95a1db0a921777ac587704
a8e44f206c6d4c2b156a2d229e967752afd1dd11
05bff3fae314ef4fa1b4453560aabaaa
21e2bc7a51bc292f955b402c6895615b48e1456f
bde551979bb7e288ef00619811ba3552
25c8fe3909c6834133a90f688631f765fcb69163
94bbc54ee3b383b906e8f963b43a5018
9918f53f46517693292a4abea0219fd2f3e76b52
1cebd7d4d50d4006073eab9d532ab8f6
6803d6dd1cae79ae6c9861dd44362ef9d3511dd9
500ce7c90ec6c00d56e400a347552faa2e18e80c
71ec258a05663caebbf0e4d8658f7334
d4c5e93d75855bae282f3ac040c37fc58e855f8a
08c967b26986e2faa540feb0886a8065
5eb160817a4e7e2fc572f9ab7ec2c1089ea42054
6fd4cb4b4fdbe4a8eba02fd30839dd96
666aef806d122ab3296b6537490284710527dcd3
d6801ca831f9b5eff70f037bf29c447b
e79b652d5e1baaf9f5787b4eab1428386d078cc4
ba55f6be6053e3cef23f38a512bda345
87daf4cdb1f69be53d3cf0fdc2b246f3d3934c21
386e47dd5e9b6885052ef110b0cd59b6
e7de1b29e292ae285bba69c5b9a0da71d7193b98
b55dca2c1c5579237246612303a66b9d
74b7d45e9cb762ba1285abe66a5a13066e6b1870
0d962ec0c6b1d2232256337c2a82cd76
be6cd6a54e2465b0e0e552e55d9295423443f9dd
0101c9d47774510896bc400fa9f9c6b4
ccac4f7c9474e2e5b306cf3a6ab5d61e29e2e558
887dea3ba347e76699dc005c536cccc6
3ffd82d557665a37941dc1eb02ea546d75f44690
64b1763996c7b0ec1b8ec13e25f58962
fb669be64be6642ef0082be2e3186302d4306d70
57bc89e4043e70e21632eb18a42bf3df
b098bdc98331d0f5c281f8e1d0d2146b2ca8fd1f
6f4ddafb262723eecdd784a161e83fd2
c70589d385fc5ae4b9b15bcd7242918af38e66f5
9e258611f29a7fdd8582c0acb8bf4bfb06059c8d
3583451ff515019141a03953fd92b9c4
2ff996aa34da81586375f66b0e0f9173ec537686
82971ce6a6545353d6aba57daac59abb
ac069995a12b3afdefd2000414ff7c21ca3e6048
8ba40bbb75c6738d178d0a3aab7a48de
f9fff2d50ababb590efd9fd43666600a8822e6e9
3f3d68e317d6648eb5879c10f86920d0
c24cfa4eac95756b44d9230b7600719126b38f5c
b045f050182600dc3e17f656e9cf2efd
534edd40f148a4ee62b09cb0b6795eb709fbac13
3b56e50c5fac7948dabcdcb2789b53bc
68e6a7fa2bb9f5005e61df879ebf649858cc1815
bba65f44ed60847129c1865477d24e93
3ca2fa6cb2e2d6228f1fc31cd14ce5606aa0760a
3f22e332010d9262f3b2d3203bd5c751
4cd74f8fc4544045b79d399071426bff6a1b7e6d
b19fa336cc99aa1410ef3454ba1d8bfd
e7cc34927b9678146a0c1e23d242e96121b62dea
0c0b895cc8b455af9d2d86ae4ee6d7ca
251febcb366cdce6e0a34c0be011acd161bf3bde
2dbe3dd1b18227f3bdd3198260aaf8d4
4093bc716168d4450b49d29d9e3fc959917b776c
c1bb63543d7f0e884c99bbeab068a61d
413625271984846e1c766e8bf66c77a9
c4f17e3295ae6ddaf59e39178d5ae3c5afdc2ee7
2e49ebf23fbbafaa56ca66db8e595ee3
54475bf5990f6ac5fa1f391faec1c91c4f5b13c4
ffdee8d2947ba30a6e217e42294f8e3f
a0b4548b773802ae54bba55d6563645110e66f95
9c147081b9b01d81fd1c011434a8563a
488824f50bd61470315e17f8ae9697f1
bbf7626136e5bfd9770d5213943dd128b3219798
d010239b17e8e7c35735e67ddfc2854f
b18930913269b0deeedd1438222df546b49160ca
a0c5ddf4b7ce70d020b5cf3d1787d341
ba8d5145003475613d8d8a516091466fdc5f01b4
0e8cbbaac37415d17a8864e6fe713726
4a5397668a62d777cf14e0cab7e21b3b25590b13
6de26aba5ff7935ce8eaba4821ed63d2
6675f618c56f4843d719267d730c46d837347f80
aa7827a9eb719baaa17892a3eeaab9de
33d336a6acab2773530df3910870721329b847cb
6dab4841319f6ab680750bf6325cd2e4
56c1d3c7eeae07e1e37f9f25d99238f9
f226d50ee7ba8fe50865e5ea350219a744f0d01f
1d9f0264f68cbdcac807ec07cc6394a1
7d91295168419e3251b72d27282e009a1006976b
6a067665c4096d3a69fdca38f58e60a3
e67a584242dcd9370203919a68df720aa160a60d
b4557bbdfcf74d5b6d7f6910d5dcb2018e694624
fc3a572895592fbe4b15c08ab3159e3a
0aa9d1093c60f89861c8c7c0159c8c9c845ac2f2
87af0f469d7e8bfce38a79e752627fe1
ec375d794e709398fdbf885e1d93cfea
bf979081cd3f784120d8d78f1ac23d947a1db08d
2d25ce8190fc95a99b58ee9db9385384
be03da26b9887b5b620a0e8f7185b831074b81ae
a918a3df9b9811fe98d4fc937bbbecc1
9e50ef4717598174df940ab797e19e614cc744e9
a75db39efdcf9f89fa41e600fef15f24
fe33eb170a2da5b47741cdac0ae2ae28a0612f36
975ea3629a5e5d0f21e7d4d801d8043a
b6de29115127698b19d54ac450ef0733d95925c9
337051ab72a2b6ed6d20a36a8c2d300a
c85d0981351e7d0321a9410657797cbbeab07af1
ef9d9708ef586b56317787239cec2978
b0d9c6b8333383b1035f9c151175f546ae2627b6
94de57697a9c72e7b0c21f19789f12552620d8eb
42b1676917eeefc6d1129c68750e0ac4
87b3fe7eb757f08d6373fa13b5af8094f1966814
90472cdbc9aed1414d1762344762979e916b9be4
05e81fe611dedab34bc9f984b4e8912f
fb4941bde49c636b0ef277d5816084bd1f852ee5
f67b72e85b11ca335e072c8e6c611861f5999d7f
a2e2003befbd045580d642ba70d81328
cc5627ecd43065deddd55b6a82272e87fa045243
55b3583d4d0643c6247dc897263ffb98
91b5312abb3650392c0659e186ab6861b30d8e40
6cca65c9828b52303e38260bedb2a167
0cb210b344ddb16fe6c5128f8fc5cf9c8d86a6e6
161df0eaba71429167ca3fe4ecb8565d
fa74f741a0a118c7c91dafc6d5bcef4a8868a6a7
d6528840b498437885af881d756e064c
6cf10dfaefc596e47e215aff06af1054
1dc15c7e4d29cc187c6588c6ff43e2e592881505
4b36ec096435b6a8e43402eb17599966
117d63a83c09335cf005fb1d056e4aea666bbf74
792a8718794386db976089438d29b11fbf0c6848
de3672854d18d937f9f3c07eb884e9ca
52982f0466aa44046af384be9b92b440081a08fa
6bc7bbe0a7c87b504bae3194eda8ec01
cf035e86d9537a58eecd06ed4826b8edbb201b3b
d7f1211f84187d6fa6c166ded841cfc7
5bba0afeba2ec67633247e2973cb66b09c9e5815
006b4909f21060b94a1b5a8186f43025
f4d92d8eb5ba6138807be1e3961fbe63b2b7d818
4e97976627ff5cf433875ff7bf1b951d
f2d4ec60038b54a50b92c09ee08620a3
248fac757fe7d37542514426037e0d8514dedb77
801f0b2c7ae90d0bbb45187b684948f7
02b07d192012a7c00d82db5d4b3b5cdac59e7772
b7620b8f90d8acd308e596106084958e
faf81413895f6654102dbe56d6c6028dd02cafd0
570ed1178dcf0a902f5e837571216da0b587fcd0
b36067eb19e38503ee60ac6897e928ef
d2cdfd1f20b18fb1c4c0b4de06d203b5
301c4ff2371cd23247612decdb0f2126
31e22bf89f69237d143cb294c3bab874bb29fe86
34f2c56a77f3cf16d2c86895b27e2606ca237011
d85b6113491afcb06f95be80da0d2f33
45057eb4bf9870abcc3dd825ebbfc0658d6b7a33
ea6423a5b3d411ecf781d20e590d78d1
438c0b06370a3b2dbcd7cc770e26b050
e40edb93e88940a6c6dca956291b9005193323a0
ec5819ce1c7efb618fbd6cb0d8895f00
66fa934d2151348b106bb16d87ab4c555d7a45e2
fbf69a2c45110bce32b1497ea6209354
07cb4f11175e8b0d61f9c6f9f99b299d
ea48b8afdc77e508be47f19714264cbdc5c09727
1a75f2cc5941144cd17e380461b60331dd5bee98
9a921e7ae8bdcf33b5998d6bfdd7c9df
18c2e77db8cd1854e2998c1e5c2bfab24656a334
92a6de9fb97c2317bce5bd7b54e12505
559cd8da4ddfd02001e574497464ebb7b713c407
9cd57bd7ac0a63b0482b628964c543a7
f5d17008820fb6e8857f6b7ee3b11cb801c65afd
a8c41a16f4ef96d6e5ed5308b06ef83e
a88751d45faeef71e9e9d0cb44bb688eddb76beb
49e68035fd918368313c7e0001bb4e8f
f3cfc6e99f7cc0384b2be568a6aac77012733125
210b81f10f542465f8c7dbafaa1002d9
78716a8c0efc9cb609527d8da723f9f58ce33341
72b3a2955aec0a45277cb076fe3f8b52
184be3271f1467cdde93bedce957294ab7c067d2
e597c191116fec15d745193848fc3349
5ba2cd2f25f72739852004cf101b4f35
c40fae34fd27efc4a4c387592e4124ce0e748cf0
234f9c307291c2a1a81e9230086a98ca
c1d01d358535bd47875efb38beb531d47d88c8dc
0da3318bacd1170c77227e72c612c2b0
8633c89c53e93df11f6718058a6814541bb055a1
b3e5454d73f51f44fc1ce35d13435e52
47992e2103c05b2fe133a0c23c83c62288801a1b
5336588bb51a43931b14da300595f9a9
c19cf0e6a56b997741a7d946f431a1bbde40d5a0
68edf276636c75784f722ff17e9d1c47
e961300002685960a24c934a1211a49d34b8a6cc
9078fb9d0a18593b4a2bf1b15c4171ec
7a76a76e34c6a4fd33516af5f03cce6e7c9f8a18
fa7f82078f5766902d0f7cd27fcda9a0
fae580c695426bce53955c2c655ccf2b8c39ff4a
71d644277bddc512bb137fb0d81dce0c
8ac18a65e27d64e22deca172a3dda69ed92ef365
cbc5d77fd52c2fdc37bd57bbe0229e9a
0d57e4e129dae343c4a5a0cc1107da73
e65dacc6c10866f220ba4b91714006f4f9e0b8dc
308c455a6ed68e80de1742e8f9b96291
f20e23be65025d65abd2190c1fc13474b2890b85
24a1b93203abd47618bc161dc6a5b77a33c5d5bb
9b025e0e0d1dda531ecab9b8352968fd
49668e46779ce53177730d64d24099c143a18aba
03bbf1e3c54ee0536d5b3a679646d23e
0fa413660c2b3386ae2713436f690a68cb0c33d9
b34919b25224982a3047a2efbdfbaa13
78a21382b3394a71e9694b169c0da193442d5656
390a599e70a2bd46736ef080d90b7193
f8d185fdc434505755d41221a474586f2c60a78d
a6577dc27778a87347bc9ea13c1808a1a058124b
b94811a459f3c9193a21b2f9d1efa7be
c9c96971c670cc2e801b4ea6f54c93bf02748ed6
cfd55c0c223f9e2b7be120716fa5b1bd
13bc8ff9eec91e4b124d7f5399c281c686ea8b20
1f4151bb3f526de674d7a3d883c0cf80
231901548f35eb8788832fabce42229492149957
af9c686ce1085946d6235827e27d5799
5f7777777dd5b4f50d1c6baa58a286c9b497134a
4f4a6c28ca0b289542fa7d4002f029ed
746905a426e8b0b718262673df4152ca
cb607aa80fcdcbb7f87c11aa2f9cb05d6e8fe9af
4c841d67dfb8bc45d1af6870633c570b
25228a1c68199f198e1c0b04c8f2521177e8e63f
b9fca72421e757a026115ae7180fcc9dcdcfc3a6
f03ba7c4486bb4c31319554a3a3f374a
4f5ba758b9a36c98a52fbd3e14f93afa8e88aa23
c30d02c344e1eb6f01263d9a4a95ffc0
192045eb5bd6b5ec7814640fcc07ebfab288f6dc
e5b04ef57e6cae1d213163670bfe1b9d
180ddf656465f7d8bb61d5463f0e7596
80933fa2387d0079ac6e77605be954c52684f68b
0c1ecbfd4aedd37ecd6a9e9a255de7dd
6ac057e8314b7b9048d475bd677f10b543abfb7b
2971bd2f1b2974ac09a34fcf7c721b0c
197faff49d3811c28074803f230794fd69dffd72
7bd63478faf62fe2e616bf9a5e6c1d09
08c35ee5922db8ff2e39894b1d5ab95d9922e4f7
25d34867e68971920e73208dec745327
35bd9567f9d1794f28387c5cd29c110c5628ef29
8ee45b78096d2a0763367440d92c0cc8
8c53883ea08fd8bb716166c8b02e1b22
6e62b4a8e70af8886649c4e88f3630068a932d1f
9cc12c768f99a01221941c4be7143e98
5f11d76855e9a22941c345367ac1a4903f4c4e5f
22957878ecd87e7e4be2f4c8e71379c7
de3e0b135a6fdfd2486d696c4d36dd3aedd3e12a
61bc93976c68db479b2e6dbabdf007ac
58af0525b27ee9e2f4183930d9cbe2e5
c5401c8f76a6094b0e97b20a51fdd81c82dfc6d4
c1b72bb119c0d55851296289f2b6ac2d
032a383469cbd15053b40a188912834d
45e556f76c6654da3821cd49e35dce74bf1bbe17
df3798df244f77bf9987684e4fe9bf9c
e5f4452d84380d8eb8a139c4911214709a7e8b05
f4fcfd3af5def4347279d707f87c436d
5d62778457936fafa675108ae9efc82f
fcee35a8958ef25f7d47cb1015573aa9d45d0e82
3480b6e636af641335dd673e5088f96f
5be5420016cd47c1f2e51436d4519de7
3cbb45f279a45b4a0e2909fe72e9787c1f66ab42
94d93191201cf3e6c1f724e90459627d
2d74e8a6b7b040935491c96977e35d02936b18a3
5a6291b1dac145ae0588f1c5eee517ca
16942b8596015e59fccb2cc70489b654110b010d
9649e0b8514fe6a8d82e34af449defc2
ee61b3afda64c2639ed2cc388638e5689bc8f400
f76c4ae6b79181e1f54f3ef0968cc714
486a1d3a31483b515ebd638e4db4c7186334aa42
7e8ff741266d70ca39938fb3834b285168d264e3
28907a1a1affa030ec6892db85f2b5ea
27f25f3cbb515179813b49a86c7135279f40f6e7
d526801ca5a860e5ab365fb3b40bcbb1bb170c5b
456573231ae16ce27dab6d5ac0dd9403
f9006e65defa5e478919aa512cdb1ed13b1e42d8
71b08a7841b30d5928a208635107348f
fe0c1c6d022e71a8877e00db29538e550911f213
b39c9f6f724b7bdb11b65351c2ec9c2d
bc658c74744986471498bf199edce9aab845efab
db8915e54351044fd656dcf6f3987c6d
40638089f5de0a278f60d426478bee4cb7aa6572
54234f29339e3986bee9847295efbf09
964ac34330a8d2fc2c3268a54beb2df9f9288d22
be53ab8df1f5f54a84ecebdd09b1bc48
5fae45deb8360b9ef73d425e73cf8d9b5ba090d9
55f9a05dcfa5765e66f6905fd53ba92b
6a37cdb187e096a27d827495c38083172f2f3d33
bd9c13d11124725d533a93fbb251bf4b
fde6ae3664b5e13156561d1a3c8c49c3cf5f5ab3
4e86eb62037d1ea7efb45d4c9a3103cb
31648d053908b9306c95ebb19dfd30ca6c305c93
0609a5de94b02b059db523f75d3657d1
5d3ff8f8198dc3db51bd0c39543cd0343fd74bc4
4d45b9bd96fe8412f5e5142a0bf0554f
35b2713ab77c205ecc88191a99cfd7a7390a603b
8c37698f1f30fb3e8276f0069b52a176
b3e972d57930a17d33f6bf0889a809518e114609
dbb80a93ba6cc55ce45c74b4453dac6c
381c82fd6e79801a300f5fd3f93cc4dd35afd760
c8c8f7016dbc1a40d6e9dac8dcab0720
c2eb7cca899d83511debca63b302240f75b47a7e
ccedb9bd56258eafbbc3474ab4248176
8ef62cc916c975b1ed1c87ed893c31066753ee7d
1b3aa2790d38c4900a1a7e3587fdeedc
346283ed0fe2447e8c4dbb30f6ac817fe99a7a98
687110dd063d2d0ec5fa9ed9501795df
7395cfccd311aac7f6e54a2deb858fef4f88fd85
2a1659a9a0524eb7d3722a5bfd3bceff0b2bb91f
eff78104b6a753b65cb1fca885f02465
19187ac04128fecdd2ca3d0dd1ed35fd0c8a1e6b
b91676b6378c568e6448a7edf17af841
15d0a45efc2fb8862c416d2b7e50818e8f858507
a731e0bb730247275c36578670c93532d0d903b0
8d8197beb1dbc6eda547b58edd137a6e
0153b0b87d7159d2259f7c0551482db1d65f2e16
0771d59e3abd61e20908eef633c327bd
eb1d9435afc9b3aef21a42ac5b95435da5706ebe
ae17c1e67375bb3c60b3671aa29a4b86
56698ff6b167f2eb302a9be1f20f5a66d5903ec0
284ba5aa890d6c75b7131a12167b37c4
101a94b2443c094d070ddb56a062fcbc
d143b565fa8aa6c30afbd92d3f3525a96eec05c3
a3a70bc7342e348c000fc63a34931a70
0ae059880f9bb118a9ed2fb514e8ec480b12c43d
c351664f64a1582d83821074607b21fe53f11b48
245429aea1c24a98ed1ebc2e3de7a07c
13983b08b972913926a53db34c5cc1c97e5a57ee
15917ef382831757bbdb20f637431fd4
51264e44fcc9d1a78b77acc1a8a937c8f05b230c
0f9d3ce413e4c908fd48be08e5af0df44fa49598
1ca778fc0bcc9a398a74b82cd127656b
218c408b18bec19729a32ea8b9812ac229742be6
3da7dbe138e221322f1b49fa788d13f9
8b605c13e287c0bb974d0f8c8162ee3f5523be78
ff8a6da93a851a16f789725e17ee1248
6c8e6cea713c7a94c376b0b014ddd1a70a3bb007
4e0ac29b7624d7b226fc2a2eb626f23f
09a7bc151c0ac63f9d142dd32706323a
9cdd71c53360702c4f1eded85f3b7493
59fb0887a1af558bf80cec0abbe904dd68870764
4f278bc678751a5a78aab1b16e437ffb
baeedfc57139199272c99afc0bfd4025be9b3905
9406d25dce7e45f858de40126b6627740dff48d1
0844bf168004d8a2b0b56e7710d96974
6ab1d94f2496fd449e88e0e008f0a3b4c2c49050
cf82670d00fc1e6fde1097fd54edec9e
7353f609c5895e756241f5995078f188b64ab6e3
4ede6fc1b91b9892277c300965fa8c66
bc99bae5498eec1cc44b4cdce13611ff
533556dfdeaa772d45904464db67303285e42080
21529fc3aab40d60394c8cbd0641c5e2
1fe34d069ddfc7382634b46b543ecac62fda6f25
25cf71458cc8c37f2d7c93c4d73658ee
191abdc8107c11306375c7f3ea2472a920d85be2
aae99b0f12f1cdd33e4dd95d06201ca2
414a588aefb99056d05c7404e3aa9476cc868e6a
738e333e44633b931a685b6a531eed13fcc207a9
eba47d5a541a39541fe96279a312fd56
2482fd9d7c4667e36a27c27bbf8ee0f3b2226501
8b62b5dbc34b35f213a2227551cc2691
4c8f073d8b88496dfd068b7d7ff17097596fcc81
eac9849a95073b55efc84158ffe64a0a
08f58ff30543f7e929d7869a16edbe39f72e2924
5ca94c53698f05f503c0e4a55c939311
481e9e5814c4710bdbcf3fc7e05100b642afc546
87e1ba3d63d526af6a7055eed373ec88
1011444b7c942a2585f24ab0947e486cd7198bf7
f1c63108924a86ecbcda337b88cf95cd
0475c9ccce42ef9750a21e8eb415d1b76040e0d9
a3030e0f59d3ba3ae6db79409b828ebd
ee9c5bb5141a5507937dd225555b5b0c9639ef0d
9c542c57e5fffcedd810b9e6de8a740b
3dfe83649b7026498f75daa4bd7974715fa26ca3
60855fdfabd86785759bf811a22d2055
f0b6720ef1c27ee6b0a9bde947d8ec156684585e
86943525bd3be3621e7e7d46626f09fb
5bb277a48eb64840d43cee2b0e9c06f163babe69
bc1f9ace755c8b1cb965128aa5156e7e
7d3e1d8d30ec1150a099191db221005f6c1a1833
b58edd6ebef2a7ee89d665801c5f4006
caefa2a8e2e7875208671e15043571862f7cf7c7
59157c37aeecdd02ede1c36aee30313a
9d32815d592d368b10580270c3cecb9fe9051992
2db07a17901b9953ab212d8e26078fb7
7eef873b6a13b190fe3bea7c3b0e1a98fbf859b5
0eb7d3d374931d5ac9f07cc5cc8d11e2
fa3c86ba31929fc908c09ee7f3f8153f26a5fbec
2d0d662b4ef0b637ecc180edfb4fec0ebe0b469b
17794f13e4c2f0e9d6da7d3998745126
9b67ddd02af72cb86cb5a7a99c608985d4e6c7c7
7da0cf5e83a134df57c6be08f8c67c81
d7b65c111a0ef6da7902dc2cb4ede2430eae786c
711b14a2ca721cc56eae373a6d1a9e741fa72abb
d4601b47de24f85bb24857766a662ecf
ec8ae54d3bd7c3e75c0bcc21499fc5a0
b165af51d483654a15b2aeb1576fb038f7b6442b
0ff77f6b8b1b14c685f1306c60768327
49057d985e92f5aa370b65e6db8b6171a3124647
91528c145094f2e5711dbbd49fff312b
ccafc2c5e011216ea47b05bd49bb26ab7c662b4a
11b413073b8eda0fcd9bb0cfb77d12ff60cf13a5
dfff57328bab11f33907e82b8c526fc4
71fae85f1458cafbacae41655ae3eae35ff8db68
644685ab164c1d9a0c15a4f8d3188a7e
66299ea1815d68e15960d31d847b133e6e43faed
4c65a4b0c850a5e6e833b81623f94d3b
dcdcc8de385cc07b71ea507d0b26f5dfd1275ebd
082b23d2678ae20317f909d4e8ed0bab
1a6476007d66ab646523c93460a195696f99656d
6d5bf8c69f21752f406fca6d4352e1ee
47aa40bd6371b315ae786e7c0d8e4757f4fac0a6
3488c73fddc6c5f9d26c8b37c318562a
750e92bb0a391a1ca986208c522aae63bd5e4168
8e004d3c7629330a03a031c09ecfa397
f422214e73b96040b1ba4152d6ba67468023902a
feac20979556e50d012d066b8db9b98f
a78598d5b44aa95bcdf004f75ae51826
60639797e42dc97a14c3a88b6fb52c76aea4000b
15ec987ec4c0882745fcc20776e0aba5
f40432d80deffb443e1af111cac3a79f
245bf8eb4dd56ef08b47136d055694c8
1dc7845e2b0a8138ec859b9cb2a29cc0
9f9e894e92c63c04a01c87125728c82920506725
eec7ea6d14965cea97df082b9b3d3c0b08f61a7e
f5c5bc052b99e1d4d83227fc9da91983
8ae6f7135cdb40c0acebd97d5c7eed5020f574a2
36c1c85141c579e2c3ee54edc6923b13
2f68eaaf0fe548bba7bd3c61725675e6
5f63ab73638439e71fe145c5755523ab51f171db
b27a11318311c8f9c590903591956a1e
468ee84551a7bcd212d38c3f59cbe9a2fd5e8fc6
0cf84c84684609bc204735316944607f
6f25f8b713b6758a96fb9eb9dddba430a9c7e813
7b8bcdce73244eee5b660f45e0737379f2620587
ddf0b13ac31bcf898108b5b1f042495f
c48a1b841f9c1fcb542ceb3a7591e944db74cfa2
b0e240220d02eae92c66e7946e52d24d
2e56c4fd0886ca05b968f2d5d188df6b8a7d222e
3038a056766ea43e0acb566c2de0b968
4afbb15745f42bb9feffc3bd0525c5f2
ca2ed9f6d7d114935efdcd6488f65c40
f496f5ceb3536c2175628b84db2a9a5a47eff318
9a2f98c08cd15a293489a5f86678323c
e78cceff30059d6e4c9166aa25b6acb6
8ffb8f43cc990852ba331dfc926e09e4aa30f021
2fde7880101ec4e26d99ec99b3d35203
3ed8be5ff5086a6a45b321a74f5b58463a04efd0
4b71d15d10c65650f4300b7618755178
8a705f397796763afc970efb18fe34aa5674d43d
8dab6e14c034d4ea86a13514c4f6bfba
cab88ea5a40388f1127e1a917775164f0b20868a
3d43c016356e88eb7f9e5ac6f4c966f9
4975721bcc7d31fd03e0766b112d003ded95a147
470de13121429aa218a49c8bf7acabee
3a5f894e10e13442bc00b818c6805730c12bdccc
6d1635ffb26eb5018a2fea2a1fa01a39
c7e0ae028585aece27567abcc55108c3a87bb9b5
89970b1c668255e410cf5bc067c7c65a2638ae09
c1e097683d8a9fb141c44137cf74fb16
727d18e21029169f908110820df89cfb907fa950
a52b3cb5bec5874a32e408639c3981653087344a
0514bfa81d897ccfa8ef20f380c56566
62616664e9432bb687d7499263200d2ff3c0bf00
a434934580375605662d54b90e683ea4
3b3b43f9f3024a7f2e43a1757efab824
e5a765b73cac878e622b17508ab3837f5cd274bb
9367bc0a1dc8d595e0d08fb2fdbe7e4d
d4bdcc258c5ecc6cbd8a980261ce299260723716
18be0352c8901f7c9e347e086b6e1262650e8230
189a4367de0bd95e404fb5de3d3111a7
6e9366eb5a4a7fb757cf6e247a137b9649251e1a
8606e042d3e272cd409ab90bef4098d8
3cc9426e9b02779607d976fea486633874db63b7
0b753b79080e5a687874ea18bd306991
1650a20a964c10732e743be29156ac5ac670cab6
29d448028681ec08a96d9a5222c32f6a7c179256
9e7b2a5f482ec12dc7217abcc06cd7ed
8310b4b6be99b78b53ba83ca8d66207ee1393a0d
aa01924e4250dd015418ffa917d7cf18
3c26fac1139ce630c79ffd089515632a8fc0c3ff
ae1acbd678494c3f7cd4970eb0a0b934
a260e90c620c8185b6ca4bcedc440a5d65f0daa1
1ff94fe87966c3f3c8ce4609f70b593f
f1c93349c8473d9b9055fb48c5680a9a76c98389
3b82f25305f54c6405ee4b68cf68026b
3a8913f1fc084b547cf8e63be7affdedf1e0f70f
521c9ea9f5d0818768fe11e217c9a015
646bca34f1587c7477deb60be4eafea9ac98a911
5d68a4f99e5d50e9a9f07d1ca4d3d48e
191996253f5d08f6d079428532417178
46ecfa6a7ca97f34a99bd930843ae646751ef3bf
65828fd880d298f1aff303a83f62983a
f586eeaec69c94c3db2bf875dd370eaa713bbc33
9169914d8a3d97bdba32e42711b43035
5a94d3e0ac93ec097816c117e3a9f6893888e071
6f5cc994d71a1cbb64fd8487e8d5f0dc
31b30275ff0140e3731a881535d85186
47744cc95145decc0c5a3d706ce6cd8d27b8b77a
dfe945f046ebb57e40126161a6ac1770
d4b98144fa3e54776869e8ff41fc1d32411dd89a
7ed943f70e8c9788921c22ba1aecc4a3
9028c8f83715a2720a7ea4b4b62236d895113cb7
acb4bc3a1568fb8371f5a04fea0b60c0
4ea96679db1e3b528a93c7938e90a9586c7dc5aa
0868aaef7a402b9dcd43296ff249865b9cce7c2b
cfd3b69aa1c6525833a7c58daf49099983359cb2
4cc49262fd1f4a56a2c1436f4de092989bd0f890
f5564c43dcc0ff5ca5ded89a2238b019
1fc10e27b3a3429f930fe0a364621dfb252be518
a30d057a54642cd3c21cba696dc55196
9dcdcc3e48db33c3f3333d745223d1fc04706382
e590b656f6f8cf0e837ae904aaea7825d59d54d5
3eb27bffbf4df8e335f49c7bab356e17
8ef24c1334214af34e6d810ff47872d1c8aa468c
da584664d1e95459484a8af9241c757a
2d91f749fc3bbee45303fd5a250763b814d26c43
928ae48dd7bf2fb9408759f62a2be5a0
8d2a6b55e745ef8f65c8799b25b3f584d44f8b60
6b4cabf16c7b3e63fe8e6aa287c8e155
d35bfbcc27a437d570fd4b165fd95122c1d418df
0de24f700ec4086ba04f0aae82c8eb7048478810
2773c3966a0af0da3165d51185831a72
47535952df71b14dabf60d963a78cf2e43ddf5b4
c15395b43a30a61ab009c163c6a8cdfd
e407424330a4294f350e994a56e2dc39ca2cd2e5
151c8aa1c7c477d7ce9ec26da079da5c
1db28f212e55d324e09410825834dde123f3d1e9
d2b364f1546d479998ebbde8d5980f24
40f38431da405e46048d17f27364373a4b59296c
6044c490b7aab33a6e15a9780d7df0159f46a58f
5f43d02961cf8d43665580c2b7ecf67c
f3845e3a4940748e605daa14380c93b073de44d4
2fcbeaebb5e6654b9467cf4f1b5b2a13
d0313d2bb48d43fd6d2e8b6acd27fb27c18d8fb6
0739e90f23126bd2fa115acb59147b9d
221f0ef95918d87bcb07266686243af702c62b3a
d03da81a94d65776c944c3b756ccae59
919b9f0fdf0ecfda8ef154d9543515d11130c47c
91037b94a0071f8d3f49f5649524ab40
eb36799774a19a581b07d0e44288bf5c7b3f0a44
c7f3016cfc94785228b26802cf599bc6ab2c8e35
869cbecca4e1c6c0e1362538d65ce6fa
f554d803f2b6b433c703c5ebcff516be858201ed
55e4cb075149d63bae660a82c7b699f4
9bc116d1acecb974a9a7733542cd702c1cbde8e1
9cd9db4239fc7d2b8682a8d3014fc9cd
b23cf08269076ec41cd2954ba1e3969ca5021cc4
23d12ea274c9945754710b8c079260c5
8bc385118dd3f009d595473a8210b15bf9ffc423
3ee97a53e3fea4324d253f0dd1d424dc
a07d3875f40ac091a8f068df113046b8b8141dcc
cbdff01670821d7055719b4e762c48c0
c7cb1e3718db6a2d0bee21bdf17fbd259b112ffe
958f4baed9a0750d412c58aa0598973f
db93e3d4b23a311b91ecb53e79e97d0c86b2e4af
e0d2f2187c45a8685635ae0e4f0ffb8d
79bebdabc281b80a31b3d75d4177ee4b
4251880cc20020e3b7dfe8436e23b1f54e4358a9
dbffb4ee3d41498bea31ceaae57feaf9b5738d01
9557929d8cd885157bc8d2da0a9df9d0
7d68a76014782d2c0a9ec293a9e65c741e75db95
a03ab32c91e68e9ef1b8d2314b5874d1
68d96c476da77c11e6be6f58db81085922a45bd0
3421b412f7b4b44cac93a4408dba5f3f
c4b9a0d9f5d93b316c6a22c969861ed3
d0a9b3c2e3fca4c9e23874d566c65cac
b3ce0fedcc7ce23867e33e44084bdc11fdf498c9
92318fe841b66b83d6c299a73b769dbe298682f2
0bf908d873317a740db0b871a03d5ffc
82e40ec9d821bd40c762901e26e6599c1af594f8
b7f50426751f980421f43e73e7c10187
f0c972eabe6945b780b6c5e4e029e766ee559eb0
9184c30d84ab573cfc0996c2eba05aac
27b8e9654ebcb297019135c368086f8b
de05cc998aec510310e33816fb10538d6e47ed18
58485486b8f4a657d1138a212b9be7da
0d38edd28228f17d823831763409dd34de030d79
664d7f9ea468fd7496d57421ce84a78b
6b96cd29d9a823fea53cb41c57678a66ddca75cf
6f96260ab021aa781170c5d53d98538d
e398d47359a935b8862b721cde3667749c047e21
3789725794f342db9c455bc3dd69ffb8
066baccff5ddcd2ffc861bf33511dfab282b1534
7f8c5c6f954f53867f2e4d482515f10927189840
320736528f5f2b31752f741a896831b6
f45a58533291683afd5f0f96e0b3d818e568cc22
84cfba1d12839846cc74cae84451509e
fb7e301da32613afd473fe4682d437332f6689ff
a0644911b85fbd79c906081ac6b36450
f310d06665b88a3fc314c0e0fd099d3dec757a36
eb7069743b5c2c9f669777c04c563a3c
21bc4193e18431ddc0a41021579f3f358320a8b2
3ea87a8b9421a2cd2878be40d762ea6c
58f87c9606da3b9cc03eb45b526aec21b8d5c3fa
fecf0f142cc03c98523dac9cedfae067
ce1352b09eb8ee60cf00830d6794afcc
a3ca301d62c4b55138040491cef0e2ff72c383b6
01152a8d10c70416fa0a393c1f72d206f731b380
2ae0bb45a3ce7c618224b49ce76e434b
010e20f90f1de4ebf35f6aac829a133f592ccd38
a65d5d2331f53b5a48746eac047b5d9d
df32cd0cc1f9ff56875633fdc590779b640a303b
d85a046eac739100f8a2910a39609b19
64b7e5a5ea2a929e3a558d45e46f4c4e
099d177326d0153d3f30c0b56fa79d306c6fabd7
778ed2189e57b86c26a2c5274656eb64
0fabd0381ef98f28d4eb2cc3e3f8950ba6378cd4
fd6ca6c142513fb47143918cbd9692f4
6912d545f189a9e944f63b2436ac6f0c77feb2dc
56c8d370d04b45f1cc78a1bfe0928530
06b21c8bd6e8794ffa60f18d1f580ff91ae55f95
8ae20e30099a834acb485c486fa43bea
1cc93cba53bd5c554e7162ea4028c57d02cb31fa
4f20a381c59a285baf5909f5b9c9b551
5cd4f6117b715b421ece0a5f90718f3a24a5dcd2
74d408a6d7379d30d9a06d5f210a2b5e
cec1a9524049f5416ef849df2c40ab60e61f0bc9
f4d298b1d6e80da8db24712387d44d94
72e9674c4762716098f6649d1d57de7de2cd8ee7
5b75937ae1d5da1d6ee1ddd3997c7cd2
062f3abf7efd4ce8433a6181d1b9a4f33c4070a9
1c936b90429b507ae271c7061c4b7231
cf7077cc67c88260edd3daee8862ab88
087819750563602ee6ee582bed8d870a032ab192
6d322f402d3e3da66eef0c1ff9cfde79
79febb215160348cf40288c23c1949173e2a1e04
c5e65fef5f19ea8eca9d3ab882784a19
8b9aa95f6759ff0618b57900b6edf995d9582ba3
43eca3c1f8c604d6bde5833aaca5f8a4
4bd5bba7a60629edd1ccaf34bfa335e612ca39af
f2d3c805094283b0839aa24403f72283
392035c891500dea05e7b244cb971235c2100907
79b0bc25cb85ba6a5d763d00967d5d92
1c1d520323ab35cac8131dc49596c3447dee288e
018f64ac6a450a2914cabda7f26412ec
59c0a0ffd56c95cac50e82b7a0ec7a57b31161f4
dc28c93f6421c95833d1bb62926b93c7
75f49a9df8cb84b207a565eeb23f1eb6a1de00bd
6a91d0d97d1e7f4ddd11eee21abb0b1f
9a27bbc1cf080ba3a71b4984409756619c7dff41
3136163e23a9c9e590ec5f6e2f51c066
29be5ae775a12eec2009810b3358236c01347d38
f6a0fffd526bcf3244aca6025b66a77f
dc905f6636b73170b98923e1cf8277143c66cec5
9554ca62e223963f9b07c72f0608966922a92979
d4dbbb838bda8c0755637648773f4aee
f3d6efd5303531388b58c81a047ba996c857d0bb
f30958bf2b8c67073c62f409f9c6a46c
723de9e742e2105516e116b6d5994d5efc29cde8
cd56a737f6d5d626ea30b8d4d1b5c826
b8d4624ed114e7a9540dc7530e22d17ed2d6c1ec
62a74fb0f917a585ee602bcd7b49a373
98ec570be22c99a099759b94f73a67e5
5e83185ca547fa5be51986222fb683fd5d8e372a
c93bc733d9769f57b34c534eae4e69fc
cdc85b35f0d3b9e3d5699887543ba108821cc386
c71092b29dc52b0c340c06973948875e
c116a3df6642b07f1fa27863b48b2f942855f34e
67975c3c5e60b24397e2e716ffbca601
c56b201265c09dfb8c8133a5acf7fb5e
5be6a48c80150bf48f802e25f98424bc
c5bb37aed7462bfce85e6c4b04cd767e
142c8b29930b507fa3edc7edca66b443e9ee1ee3
4a71fe447a18de54176d59ca641eeebd
0d1c57c4ab4eb0037a97a8b6cb4f8e15b54a181e
ba2dc1c9912e84e5731d7ab6ce7c8034
250b6d4f9d82647667c53099d83aea6a251cd360
f27b6a523d8d04c9467494211228a8b8
fdb489d1d69718840bb5e4f3b26719e4dde9886c
9e68bb60a309ee08a2191c61ea25d08e
4462a909814bc3516a5a0245da54c3db79200b23
343c94a589f60ae5951782cf09e2d63e
75ba9eff4cb2c4640efd0691b1b4f87c5bf450aa
c4b1644ce799e08e652dbb50f2b1e1c4
a13ea875d5eeaa73e541fdc6d4400a4521fe6729
076f49b8f300c90595417f1d54c80656
75799808c5f0c8628b504d1b68113df823c0f42d
cdae2c50871d5126bbcc11e54fe4170a
71b1db0ee5ca6534d8cff4c0cecb4b4d0a7a66da
72ab92b424cb58b17b5457ac38512918
ec2c67928ed1daed0ae2b35324e932f856b1e40d
c22c5c87646da0d0aeaa385bd1ae0fa9
20040869d89ef81c8f0112e4b26f5689170b3789
82a3cdbdbbd9ce256a64cd122c997207
2a208891dec5e0c031a6a2afa4ba3cfa3199b9e1
7d2e0fd36da63a8199d8626832170a04dd4cb769
e7c9b07f4de76edbfd2821a073acfc04
0252e19767ee24175448f5dc65e77b9ba9dde4bf
9cea8f6ced317b3827f82ecdfc107f1e
1efbc03d9beb1c593433989693d6134900b25b25
2875eb4dd2c2bd959c77c9268fd8fce7
240afc7f6287263bef2c6dda50c79c0cf4b55ed0
2e33269f02bb083d86c5a4e8bc83c805
68346fb6c1f61a9111b0d1e43562d3db6a730394
ee0bfd78a0e8688fe8d73c017397635d
07570f1c793da68ccd748cf2bd626ce1
4be4fcf94bac555cec62cbb70ae954ef98578480
e2faf7d8b2959ddc5cbb5c001bdab2c9
0999601fd53fa2c6e39c04ac7dcc932b3715d2d9
8aea8836d54ffafba56785605d6001f6e0c8df02
61145113510fef9fcaf9d84740650a3a
705c0fe8a5dcbe621f931f76343a970135298dbf
b8c83b6e77559b8d6613381ec56abe47
70ab62824588c818940deae2f90d4996404f2071
93d1d19541218d67e31b19ab68a95ccc
fe2b737db07d6e0fc5abd4eec38f418defbcff72
14ba41e6acbd9bf092fa3fe5af724ca5
e13a00d32b3b8a84cb6ae7a72cc541def80ca532
4f817864f0b8ae830a54976856a3aa5b
762d11bb8a59f56e610c80ca025bc84a5ac35884
3ebc7e0cb442bac1c81a5cc83f618ce687fb02bd
7ba5ba56516427acf550a65926193d4f
64224fd2a07f1d692177eeb542a8b14c487eb4f2
ae804d23a0af31bfb95a5b87c6face27
43834f14db0900f1dd28944a5b972eed1c28ddba
79c1273887974ef297bf434e99ddb803
fbc6d2416f0b56e7165a006185f094c9423cb6d3
6a19acf1811160fe9909781d0957b8c3
0a5be9e92eeb1b7d468d72a4859547b9506229e7
c09ff388fcca9ca1c9ee9dde10cabba3
da7e6052e26a621daedcd47178867cae07cc6572
44d4a82ce68d56098c62b7d285834af0
97ee273ab02a0037fe4c72a9e831e41ebcd6b07c
6568d6811f5a6cebf0a32f4858565a23a6770519
d56b16afa9a3e260239db7337457748b
f4126eff4197c7cbe1d93ffdc79f91d5
8f7aea645e3327313cdffaf3a6501845eb286e92
cf0858c52200aef16e9c901f3db37041
70381586ae3f69f5694469a1533255f40124e9cb
08c7bc583098117a4a28340f6b83d7ef
18caa17bfc163a85ddf1cfcc5f1159d4c81982c1
1850aa6efa273fae45c6ba5d54d3e628
17b446842a7f977ac2881e92800cfda76ec496e2
1078e2251ec73d5b76024e691762d1fae24c4fcd
1ceac9cf8d814e1993556cdb574e9e8a
c80d5a69da8844ac113a678d197ecd64
ed95b663a5666b2046c9097b475bd92d
0b5592d0be3a518dda917d72bae3d2839a5fcb20
4edcdabcea1c938381466cd75b1bbf86
03de1625cb567683695006f9dcf74fe9b9c7c43e
b7f31c19695e241f3711e8241f274094
55d240dbfefe840c2024356e873a6022a5912925
d16652cd00617f5475c9ce4e31ff7a65
4ff9b1749a809ee99807cbc5ee53d5ea61c7cb15
c498912c95b48feca376cc19db82140b
1e22fa0b918aafa6af28fa5d9b21c5055335c1ff
63c2187c804a0ffe80c21783af2ee70d
84076c086ebc7e70d26aae6a43e048a4e658349f
261bebce4d5021956c7bdf1279688cebe40f952d
78a90b58dd20f4df8a41510a1f017091
9854b060c5c4229b9032636acf1b44d7175165b5
d78fc3103f1642caa3492fc9efe18a9d
26ba6fe2a241097e8fceafa35b64bd32bea45f33
ddcaa9f9f15bfb3fd8a673f1ea98b48a
939edac13eba33e3db0ce6b592c323a0
4b81be22811f916e6d16e53971d0203fa071ebe3
0e20579189bd6ab106b5a7bf917ace7b
50a9c19f0bb82ec04707d7c0e48dee9017b5d77e
88c29989e0ed207f058c5b82eb12cef8
2a5e65f0c547a4b3d5ac231532dcf212b1a18517
d8593d478a2e9f33091dcd6198e12b71
0306ee06e71db35e1f09a618c5df4eb56bcce8a0
595b291340e90faa82259cc5b47b605abeddb4d0
320e8e01d17326120ce81417d95c75b4
237f652bfbc18f1d884e3697ed9da69e40c19361
29cfd02568e59a81bb07083f92c6d1ac
e2fa751090d0aac0d5865a6fb52033c74e82c498
835220cc11dce406fccbe71017de9733
cb72e6dea349046d6afb64dd8332ebf37d3e906c
6ca64105008d9a9d3659dea6e4cc93ff
1f941ea6b251665cf54e3a23bcd25f197cb83c3f
3f959f13d92475221b8a30daeff8f1d78cf4858c
f08d4627ac6047c133bb5d5901b078d2
dd0ed9344e0ee40b518d782d06089a20fc2b6978
62ab31b5f6591d53c04505fe06549973
ccb3ccb4c7b64cb77b14d266bfbbeff908317780
dd8a6af2fc0c8697a852c064d0df3bcd
4f68ec79e77a608a99e9da1dbd780f195df1714e
3b2ccc9958b6322bb4e10b4057032eab
8cbdb0c0b3477f313ba1d760656deb8fee0e42a1
253eea3277b5bafce6008c221c13d332
8cd6b8dec5c95c42b5eca4c2e35af0aa
e98070b5b2105b71644f33d7c99db829
fb283a2a2cf6ddec585b57089be81901c7af6880
3045a2b28fb0ff3d01ca7eb699b7bea5
03f68655819c2d2ddeacf814aa8587ace0a762fb
460247ff04c8e029188555d8d3444ba4
e67fce327c3a6722d059455a21ace95ef5dac8d8
5b313e977cbfdb2ed38cdc3859868a91
632dae3df787f4811948b230d54a043aa2f5f91d
496fe56b7e04c19cd838b99e4c6f2f8b
21f54a53f38bd38ac81076e9571622471464b821
effd671b39ecbe33552ccf38a9f98bf3
02f2f05cb3babf0bd99fa2a4124bc57caa1a2c75
232919c0bfe57b74699c68a8470b670a
2b0a928d95b9c5346c2cb8c154c67a92a2a0c11b
e343a3920e361102d2e71960aa9aa066
6260d3a6deee1d1fd34752f4549270fbdf09fc2f
a819984750004074678af8787d0ca2a3
28c42cd7dabd0e03d386631b8b229676
5dc834b7162927ba666f31284ac88cdb1297961f
8ffa66800211488b1b989cec8208ed61
94891d672745d1bf72c24fa97964755a41eed0da
dfcd77b0e9aeb1c2afd69aaa3b2287a7
cecd1250b39ae42b31fca56de3cc67964ae2b959
794031d1f999c109369d2a67ab2a8f9c
c314b8c043c9363b17eecf8d18290e85b90ca77d
ed83e8d66f4d0771929f5ec7797d3a7b
af689b8fdb049e434df7acf7f7af4b2f2b70cfe9
ab37c7a9ce6aed47f2419c497618e58aebd28593
8936acb901242d700e70d6446034cb47
e673490d6319636b8c16b0899c0824016f6a9e94
7bcf0a20f4c3c4b84203cf278feda71b
6ca3117b4760c8fbb63da2fdaefef29ab11fbfcb
7c356a981e69f602e210e43424c7c6d7
e57c5a513a28129896685a6a3ee23087
02685b78699f3fddf655cdaf564899493a93752d
f1c4fd19dd7b5458622eccde2cd405d7
366640ec6f002effeb14a23f861ec052429fef4c
62bc4630a91789a2b88f3f83afc051ea
88edfc4dcaaebb95229c7b2129d4a3c4dfb82938
ee93dd0b70ba4d3982c96709f32f2881
6f32242423bde6e26790ca79d6fa9fc048851858
654683faafd28463b10d029278cf6e49
fc4a95feda7088cbd204732334021648e768490c
06dccfece37a8b62a639717da7a354c8
e9e3dacff284730a469c91bd135a021de428ce8f
a1bdd759397430e0751df99e8c31e3c8
4b9aa9d49fc7fc246cbd326c7f296e50ef9d0ea2
d55f1c1b2753c91299fa117150f29400
33acc2eb38c9989bb7e7f4d5bd3e143d
45c50e0b6ab03c105ddb85c1ee4aab041e5d5a72
e0471c1c5fe84f7029527b6b507d97e1
190c016d40d08f5d2f24d3d5f029b65e
9072ae9502e2c11f04c11034258c3387
28be95a3f69877bc078f250a7888d33b6a621491
101260b21d336af1f51a32d242987eab
0196dc2e6289531d2be909e992b93e0b683b05db
c39449733842efd81bd6aa2d26102dc9
4629e4709f9a987df0363eb500b8430fea0a7674
9c86ebb36d973bed6338f2ca6aa00c68
d9c31037624aebdf92dbec1911ecdcf2e2e0f1d8
fbf6fb6a40c71024c8929b9bd79f8067
05b164222e33cde7e97a5eeb0a28da5163dbe366
cc43dd30a72200303a6cd1389e7b7c31
36a7109c48a6c6b0190fdfc8d0118d41
35581223f2a5a40f8fc6759335b89febc39e43b4
0823d80475a1ff8bb07014a743cf0823
250692394f7a9662e4f8353725567962443d4153
59be6623617fd48b31427c4949d60a3d
d83564015278b2157f9c8cb77b1e1ba4f8ba9e36
c07acd34170c4645057feb03ceb113f1
bcac6ba7ff2e3c929d612f143f45f2cc99f9f8eb
f55daa1a3201619bb70fb86bca252209
85af5ae6f24b7cfbf03b5c383333e5292c427157
70314d105f65bb843c6bc210dd7d0965
64f613f8c4d4fca0b98dee6abe207c5ec0a52dbd
7cbe464ded2781c9a78fccb23252d219
1953b2d94bdfc72f15c62596a61f37526551b4c7
6223bd86092ae136a011c53f90481415
507bfd02a3272474de9581b73d77c1953c8eaa3e
602972265bc80827d60f4e0b922b93d1
36b4e2d6abe8457b76564ba4cbd6272a0c0a31bd
1a8d8df8f41c4aef34f1ff10c412202c
591b6c070bd97f31532c0773346ca6df67cd1f30
8245275ac606aa586b65956cb1b87096
d4c35f8e69bf8b1d5a1f2bb789d500b1666dfd46
13f58ff2310dbcef3c8c404c49539d8c
3e6b4e2f3d24e005f2b7303609d769e9b05dd498
679bde28c6ceaf67dbb3a0ff3b53d78b
11c0e13b0a1dc5767b3fa85a3c09f2b6
06d0219df2284c1cd7f1680057cacc93458e13e5
f2c5271f36bd3aee4bc55232eec01aaf17e9e9f6
5697578d0707f1180661a513b6029bd2
cb8d3a0b4cab7ac7f64e46a9fd91dc31c0f4ebe5
f9233003d5a4ed74b48842924402ba78
694ab7a2c4253123d839b007cb757be901ce3239
0ea9156bb13278710a53fd819e801bc100e11ba8
69c5b50bf58b8813cbd7c82fb43d3767
22d5c8e59361bd0946839aab787f514518fcc12a
1552537626ed76cdcab0fc7a4ffba49e
7c1373fe5b5827bd647d959676b2b598f06bec26
66c7429ab2e51c439289993db4930bd8
461d84df90a37da696537461b240452511251564
abd61a51f46e0a06af86d795151d857e
3bca8ca64e3791591aa165aae6d6dc6ff77e8ac6
4a68a96bdcc5d3c874a98d812a5c26f0
43c6bf7d1d7f0929651a33b397b8d3fe24c71f0a
b2f88eab84178548eb8003436f2df462
99f6fdb4c48b21fdee40b91ff7298ecfd1be62bc
63bd789714d7f37bac52c11fd884e910
d1df3c554cfe734ed8ce89c4db42c90db8dfe69e
31a6b5c6e50983fb5ae428c222636108
1ec2bf2a2f7bcce8b1524bf1e8e8185dcef8906e
fa7c07905675fc120b7c9aee9c5cc3cd
ffd81ab84947cffaab810ca3a768aec08e0aa608
f3072438249e4e0320701eda91796500
0975cddc97251169f51163bb13107d82427e87e8
a922bb6ca1f39d4ae29b4d00b9838177
3ec33f8ef69aa5477a90ab7256bb953f04b46cdb
a97f50a822d5ab4617d509d1f24dee4d
6d849f2f49001d0e969c52707636db2c6b640474
62ebb1b61f52d16f17d82c947e5f7954aeddae5c
3c5e9b9881fc70aa7f50c95eb50a2c55
d16a098cf55413011cb5c9d3c9c0e287ae06784a
6387cce37907b4ef680a77a8d946b58f
5123be80d267d9069b09b13cd3dd00c63b70bc3d
74122aa48766815eb3d6f44998d55798
5697e915e785c9581ea56df713ac77dfbf719765
e111c087dd16cb459a5903c824309074
486ed5ac24438d0059e9c5a003dac1013edf6cd6
c9c174d0a3645da1fe630af70bb008e6
4da23cfbf3741689632441138dcfa40c2a5d275a
a971681b072ea0d01a4f2c3cb14c05ed
e41e7e3b55e6170d5bdf23372e6df80673498826
edd48daa1be3582e77c0a3ce6774ce32
be17bce449567e18ba9639d9c1671db0ed1f5141
6aff3159772b651b8d1def7888bdd80e
9ab76c1c2ee0474b240ed88b2678822c698657b6
b032940c6f081e5d5a19a9c8a659f1b6
6c7d65d35001febe48dcef303dde9f2cef7b309b
9d15b451f7746f56d18ecb19381d5613
b92bae4e2c8e042035ff15862366d80c010a1323
3b2d7ccc362a2d903c0f4f580c302b79717800f2
66b5a887969e077a23b7144bb21f0dfc
f88a833d5f9b4daeb8172bdfe16c952791185143
7d0d1c46c63cdee3e61cb4883cb3bfba
8bc841a72ce3db364b30fc1cdae3da065da4230b
aa5084b4988c1aec0dd2fafd6cd3e792
2a516ea82f95e30b1ffebef6d5f8b491c47cdbe2
05be3d8f745ac022a5ba3f699aafabc1
20c881f7cd4eeda7994b03737fa5ff08c97a2ef7
1b5c92dd6d95017797a92c6798f706f4
bde933c14b4f1cfe3d17e206567a9d8493c94b13
c65f0c5161e9023b1b6a8b871be178b6
cb394f390b0ea55f39b0c0e6a3e5785f9168e1e9
57aba4cef61a3b5358e4211916664139
88a836e4e706ec918e286b6628570aaf02e70f86
7f86fab3c3aef4a3a74f3aa1a1112a87
7a27ab0e8bc969a42e2ee4f39d50c8ff8f989a5f
acc448479b86dc2dbcec911729049276
e2c840bee6f18912af7f15fc82f5998cb69b8b1f
720a736be667b7b6beb9be6e0ef0ce6a
4243b88ff8ce2e505502d10aaea5a6387cf56407
7ce7ab39ef1b51ef507c3f86e86b7a14
e46d8a82929e546a38a7053edf2ae203758c15dc
9f1d0ef62cb96876ff8bf1a34975d65b
584d05d5591c14a26033cb2a858a86efa4de30af
c9d4ea433195ecc1a005d58137d3d878
32caae9261a0c3a02267dd4de2c4203b45618378
b512ab171c7ef358871004c9c9741335
cda280468f5b4920a6961b44a241537f99e7beb4
53d68a8777e4fed8b423e5d154845c11
03479ea69db088bd2c54734ccb7249c898166b93
fb554d8fcc30c0aa7258fd2c2bba24e2
88bcefea8652f3cf56555ca13aac7cd7970a6197
b9c92c552580bbe3157290f7df29082d
be952af6fd869b417980d65e0dd73847acc52ba1
aa177c16b6d06f83a242e252e0438d04
b6cc3809cf982eb277cc5c26190e5a1b7225268f
6ab1dd13f2032b5eb2ee87436671ca7e
9dff4611f5e2eb25fec9501e870bd93184ebaa04
07e1911ffa741e4563b17f6aec084ccf
bf0513efc7ab9b41841c8c10fd069df2250a2c05
6b3bf3b15335b453695577f058fbe3b1
fd58e46b23648337f8e2025a15b66859a0959428
9a0a23c78ec036afce360612c5293f9a
5e196131bd829989d9ce8b04ab6b90c299f0f381
6065fc6f2f7a924256601664b712878b
748962f34f66ce87637c7d494a2c9e28
9caa95a4d0032c85c0a63f9943c08dcd46a84a73
3bebbde614c9e053eb3707623c5120eb
e8172be0dc54d93f223452bd23b60d6937dfd152
c5dd3643812191101e4a2a38bc089a49
d0a942ab219ceb82a0cfa6d3e1a077155d2577a4
5d290464a8077c826459f88ff7a1e020
b9caa0db7a7fea84409b205808d90639c640df43
6dbfc2edd9826dc9854eebca3e0a52d4
5f65a75ebe94e1ea84460411342a270b3dc6a8a9
40133da247a2587c3a42bf7107db786d
d63520b90ae3923efcb23928a8d3ee1d08b0124d
77cbf99cb7acc9a0b69cbf789bbca693
cbcf59be5bb76118247c313cdd05e221cbc14a45
75d91d802d50a78c28b426ac6fb5e270
a155b9b7da6086706d21514da9549b7af95cab2c
4f0eb1bd98f686db6ba05b9eaa1697d4
4e6e028b09d12f4a3f75215b26bf3f563e9d90f0
c250c157044957fe7e5feba267bd1ac4
c0be0339595640785948c103fe76ee1e
39aa6bf728548c50786bceefb0e4303169c858c6
cd1fa279167f106086f92e4a6a2b52aa
90d8ca5ab1af5a6316c77b65afa1e239
bfc276de54aecdc171b521c473342b7e62f4e598
e1775fd0d9ae58539862c58d5cbe0940
d149800f20f3459eaa0fcfa1aa608683
8d0843d86361d4bd1f0ff583dfb39fde
e424ba072b6d3460edc0754068c03a421645a515
df03b5942d20a78e610343ba9b709149
44f1a8d9df610b3856dc3af21f8e745caed7d850
e92605a263564f6f04a06f1833d526c7
15f627960d0f5d05c5d85c9a0acd418d06a7ce8b
84b4160d7c5573a73ca6c069805c27bf
79c648b0377f615858754106261649a5c0c756dd
b8599b21f140b7dd357938528c7d44c7
74fc7ced1885ff111810bce7c084e2ddff3d1630
3849a79566133009c1f9c7098d284eb6
4e2f81d726e490b3f2044a7713e5ec4b5056c474
2fe9739a214c93e803e8036961f5feed
a23d243149d540bb2bf7f0b3201f2bf0e97c9704
bd617ddf928ffd776af32fd90fef93c6
2339fde70ec44228cc51f11528e605fe
a8327c19c333107cd6ab2039dba7a6d25d4d354c
239dd47a14202c9fdc1fe87031f11063
5332d24456b98aca68e44fc85accac49edb4bd4c
c2b08497e3c7c6e496f50b9fe14a0e52f548b5eb
dc3e35b9c01fe8c93cc57647cee7fe7e
18d1ca3c3f7fd315bb58441c5bdfdab3
8ff5eae1673b449dce4c0a51b8ddda48
60a6805e222bfc800492d42749c3242dc7cda667
71b0812d3392c3f2961060803de36f19
33c3e77c8dc9da61214c1d4e8c280bbc
3bd2fd243b5291f5bfdb741e05cb9c05deffe9a4
d9fb101a2b135f432cd8a4c46d2d4b81
41d2ebdd6e2813e5a03ccf98ae719d5f1147bd74
db0133d1a248739b0819b43471d272f6
1ddeee1635d4a4bda71abb935443836e8f79c76b
6454e145873b808f82fff49613562724485e8255
70e8ffa54c87a2707b5c6212ab95a416
5501dcee646c842d2a10e104f56afef2feed206d
404e74558f907326c5cc8987865c2bc6
d0f0eb8a001f1e5629feb1ae9d443010911cdaf1
88c11faa2e358c17fa93a69b5aa05d25
d208e8fcf3a9446dc29aae4469a11d5869281097
104825547496583af11e7bfc2a30723c
c340328d08bab057dd433581654c7b15ec8b2a49
d18405996d4a8bcc73195dc64952d2fbf8ce2d41
7dd704278005ac62f1ef04ca23fc7d98
1065924181cb454720955fc2782cded3a305be89
bedb42182890f145c5ef2bf5c5acb48b
b8a26f149a8e44b01eaeb232b7572bd7b42344c8
0106fd276b7fdccf8ff758c3c79e2f8b5c5c12bb
2142ac2ef04ba6198fa2fb8457a903ae
f605726dbba0273fa9f628d2e99d621278dcb27f
b1500dc61ab6b46d4e2434036a5d7112
2ba0ab56c28f7b79239d555b7325ab4dd4d0cc45
2e93329203ccc58a10deac4618e1a65c78560125
e0139d6f9604b25925955c76d2b9ab1d
572442c6972da2f5a9dcfa7174a9be3448011946
073864fed1efdd563a16f5b91d31c2a8
8bcbf7d45d4825792a767bbfc2c4228a213319a1
871714840b24a2193af307b36272fe3a
329a672e91a66373127a219c6ec6ea2cdddf23b2
4681bb4a698624e38f83c8ff6b9b115a
d65c307c482d5ce478ac0c357636d325cb2fc7a2
06227faa720f560ccfe5d6a19a433fbf
156463a3a9c89c7fe9dc086008af4df9
d208e486c1e09f550c75b113462bc700e92dc340
e49574c8bb0916b3aaa04692ab744c22
608970944eaa77127336a6820be454a1f9ac4dc8
5bbb7c1b47c377366da9fb8895d2f839
d16006a3e27889206bbda2698f18731377ff51e3
ba9c8c9a2f620b102a1ea778ca459b67
523b296db54a77d0fc0ee4f671fc3a49
214f14bb875ef720595a46b894dc904aaacc382c
f531d59321110da23a3045025b765fe9eb80cc03
a48d8634d01a3175c41ee86ae0a172eb
a5678bb836699dd6edcfdc38edd41de2e28de78b
2a202570e8c4ab07fd2d21bbd015ddea
c2d32712ed7f1751127f02bb6d1934df244a61ab
4be0c46de7cd149a3fe46a0a4d1eb4796e8e5d44
fc72c50afc55a7d7649231cee0ba7e2a
e24c216df384b285e580b130cd78f3fc97fb4610
5a37abc0d9f3ae4bd23cce4c73fc2238
3bca565155b307d6f151702183d1fc44ca76ab9c
eb0779de2a47e0233bd8fd5213dcb668
f16d65bd15187d61625f1467d627fd4794f2a667
0331f019b0faf9b5a7f5bbbb148461f9
7de32c32405f38a2186af7a56c4e7b594a7c47f5
8d5a45053a2ab7b466a935e1cb55d988
644970b22b3cf1d2727a78052791d94798812326
a002e699e434056f44b98862bc366b20
186a17369b279c5a8f0109ab0bbd47ccdbd9ae88
c57965d47612f998c157d04764600bbf
71e7365c9d95c041598ab4dc531bd2519c8bc568
117e09db9013bae486d81478c9e70739
b23e04b13514675f8a2ea309eeff4c5d23d276e3
0f1c7c5380af83c81fcabdcefbbed88c
8fba41104e7c6641f39395f2f8e926caf902dd6f
2011ca018674cf1155f68f2acd666bee3805234f
f02ca315e7f26d41bc473392dbb6ba6c
30b7ac626a8f40bcc0eae80f0a6571221f86146f
110d1793b5a90725e7c0ef4d21e2c08f
654b84379a1bc7f21ec3911c2c9e832c08100b47
b08c6d42f983400e4a154c3642b0ccf6
702b71a4dd671985cec0f1d0c3bcbd01c5115ce7
0999f373206085bf9b3824a0febc352b
0d10c425b20508097a3cfdd195d604762b244374
d1d6b7c3b1c1053bd6b0520fe54f4574
d3d93a1f54e95b09c6648e3c5084776ff39e26f3
79840a88c30cf5a29113cd5a14908ff6
743ebb3f9a756f359db5192556b96aa0
bd920858532fbccc372f890b4f1e3a22e1511815
3f44ea26c63b6e6035553035c64a279b
f792c305462403585427679a0ed51fa4
956c9ecce24783943c544826b27c1352e57243e0
001d733c26e272d54aa4cb55fdf63deb
02ac92fe75ea8e32bdfd0e006b5a74a637da791a
e5f1b471b1f9e067bc52eba13b9e7261
2ba309ab351f36d06aa0355aa28827b241a1f468
aec5342cf2d090049bebb8e2a9666417
54d36a7160d143753f54268dbac38c3b964ef803
bbd77cbafe43cc2a0c5dc73147c782be
e040ff1d9afecaa3e5cf88fd66f11e9053e74042
c461a0541c0e3b63039f3eac2c466bd3
1341d26f16fa43c5de443b29b8b3ee87b0091d1a
4c5a5bea27531c350eb78c2ae35f4e79
8a854a846b982ac47ee77ee99011256e1501134d
f5e19ea80eb1efac949dbb8719c3fd3c
99c223581b099b9f7c837eb4c54365ca6ae4dabd
b1bcf32fb2cfff892b7a0705079093d5
f92807758baf4bec6b83c5ce4747d6a0
f457e0b61844efd3cf9036e4da3f9121806fea58
cf349d03f1afb5ff88b62f4befd8941d
2e906d39c11fa6c1be467cb6f650868e5e3cd0b6
fbe17a43d918968817d501baffd167ff
ab2fcdb6a9b6c527f41aaf7c32da37615a916eb6
42d697a3e7655a2c69e1859c93ce09ff
18323063b5ff994fce2aa25d5a32c0c1cab83c07
d48c0fd5fb4bec006711827629623036
fcd0d0b425a08cbab2601b4895ae5faf98d29b4e
eef12abe7872346cc708d7c49abcb479
acd00fc489724ca6eabfc61f6845e911
2df6301d552e3e73054dd3d14f28200043383bdb
b2a1cd8899169630ebc857cbcb2ffff6
c8cde64e4e7831afb0d06be890887c2bf2c21123
2a402e761f681a2b7ea43107a3a0a962
902c7ce369d93f8598293739a8791a134b6f176c
ac4ac6192ff640e04f5c05e18d178226
3997439b915f6a13e45969868bcbfc686854e4eb
9b73bd9d09da272e28c34cc306e53c13
4c9f0a8d31baea046ff4d21a99781a25482eeefb
1397b0a5edabeee43e73d4fbab160081
1accfc0063fd683118a86741915aeed9195afd44
fc798a99170e0420eb7f54eed1cefa19
f4512d8725be969703ae7d8d7210e9044b5e1c26
4c8fbcfbcf0cc3e4f9fbbaeed871847d
bb3ab69da4161c44f5e990e6694fd20873d9cd0c
86956cba46e17de74920db270c57b6a7
c351a1d99aeb3d96de1cbcc9f249bf4de0c8bc94
d42ab0a041c81278660de61a1afca793
45716e8dc817214d2da538104ab1f8ce
cf0d0cd004664b666187a2c6dea95380
c1b4b2e14c49d370237bd5813009a9abdb8e6e92
b4398aaab5a8d031859a2482bc0b7440
ae18991c34013b8d0de40e343afd6b696850f546
7f336e4999a5346cbae31878f8637bd9
6b5bfdd1770ef764de1747337e21e091e885a2a3
5ecb8690d9d1f7afa6eddb647f54b825
bbe4c2bcf2f75cf69ebea8f77598bdd352fc2e16
0771890ec9f89e8196cd7dd4b94c4d0f
ca1f1850221c3c6d8f6a40e3075fce7f922cd706
18bbf2f0f626c81fbccf7ae3b3f1fc6c
93ea58fb45ccce4903ca3bef733f551015161785
693b93ef869ae61af192e53be533ecb8
9dfaeb3fab81304017f1ffd895a0938f103109b1
5b3a4365d5170480a3d22e7bcf57f8c9
83a26204bc3174014326b32bd4d4638ebaf30061
9a5a28c5c59fa55851c350a93da7deca
232f4806d811bcaf4a06fb2e4b304dc6
6506e43bd5e700f145b6543e2288fe7d31bfcebb
3883477bb5935b7f33727d86f0328eac
536ceaa80d7e65f60fcfda02fff5256f673b0ab0
637b5a3e6517270f035f5e953f3f2465
4c159e518f6d45b446c939f54f6ae4d9f193d7e6
a51bf578a80dbc1afecfcb9f77561f66
ebea33f23653eaa43b18fe4b25c1f627ab93fa93
7ee9fbc34e393cf5e6069648c52d7f31
c69cc51a5109342a4bfba51755ee176e47f00b0c
134d267bb73eac529d3eb8e61fcecd75
5c56241dba5c640887dbe5bb27e82f8a20616386
7388809dab1cc258d16e218483819d5e660fc236
56ab799a8c17bb7753b05c5f30a26962
d2a0a3f11db65efd7ec530deb0f170b59ece4749
ec3cbe634154a984d6bb8690918a28cb
67de930d6f3deeaa1b0180676eb0aec1f74536fe
d54a02e863d0a55fe5895187891f74eb
de3e8bc6a1be8c9737f7f48e6e53ea17845709cd
ff36e1b4f612fdcb951190a2b7a3b004
d13501f6a648d685b17d0f6ff3c8fff48ca4645a
3e86808e3c2cd262ca94e10affcae2e0
9761cde2cdec3611785b538a5ef4f115594c8b68
4bcb647ebcb8aee0ed2b39b6d63709fb
470c754400b602bb12feb4cd6fc59d5ab387a541
a50b08e2240fa44c481bc066fdaae985
7acc19b1b10ee51091786de8fe0273c0a421542a
d0057d3ca42a2f7c22728c6ed415060a
cd39a03fbdec8a6ac46474bd066b5636a2a58fd3
d0848cf2681f89a3c013e8d78f859cdb
5f13cc7d4f2d28ac960fca47ddcc287770828d18
2e03eaa7da33c4a2e616344587c4b0a9
0de08f81a2200e61a67d91520ff7392d
a7e9af514965bc07354782367cc30fc7c5485712
5a19f757d817aa46e51078809449345e
76c1c40a2992d932514f3ca41b852609bdc06897
9dbd4c82a66cbef81616c5a6a7c1dc4c
87a7aa72e2a24a7f282ff6ea52f01b278f17087e
65f5f3732d03995c1c21ed14e9bf2017
dba4c8c440c81af3f0622ae46329dee433ef2ace
71830c72854fdef0f49f38f3ff827d6d
99936ae1e756af99e248e93a1e2011a597f36d8b
513b857da3948ce18aa67b982e1e8d5c
826a5c336aba7729af3d8a7462c20eaf576b8816
2d23bd4eb7e2801a9e93cbb7a6d8829004d8e186
44b2da42151ba78aaa0867b5836c63e7
8c2dbf0a1ecb1084caaec081f7b97d5bf893e1e2
191ab7c076901b8fefe2f7fe4d5b32dc
dfcf8337b8c2a4e09389ca55ddb974e03594ccf9
aa32c00846c6f8e04e85b555ff42be54
938468ff6b7e2d4cfc208e09430cd39943be1a4e
5af813b96fcb904d08ded318009131b6
d4e3ea9b8c64b1e42c64d0d46d034961e2b8829f
b185bc8417f326be61718eb6f469f7d0
f946f5489490d4fb38124c454b12f1bfafe965e1
12bf7ec8b28b62943cb96524fcf2ea39
e5cf09b893076e1af528dd8728949c0dc1a0c050
56458b225c9e23a64415a16d3079f727
f49e40c7f96081843e186327ae56cfaaf0b801b4
f4f8f80fbee27fb37631dde22edebe22
396e155dd71f789c554612d6a6be7fb85383cbe1
4c041d2e29ffd2fb76265c41e3c0ee3f
267b62bdcea572f70bec9f53bc1fcdb72761037f
f01fce1ed663bdd453b5363ab093799d
2c2ac2634fd82c8f49cf16da469a5c177356e68f
4af1b2e86d18b4d702254a27dc397ea9
d2986551c8211b06c9213d74903f960bec67b234
7cb43e72a3d71a0da420cf226a3e2dd3
9fcac42fc53cb7aeb44739a6ec58e67bea80ec27
bd5ed38274480728ce52fe3b9e60c361
5445b848eb948fa6a43ae129b5f5142a042b769f
62e4896c09e51480c65713f7a758137c
0aa8da8358ea238a25de10d17e5e07944f57b7f5
591c85319330abfbdaf73a196472262b
379d235a16479822c1689673eafea07c2f76d356
252889160d0d34be1988e8645a727268eb095ec4
3fc72a94a6345a4ea6d0b35a7d5691be
e3d74d20c1540631a7e4df6b6b08b8d3
b38b5b6061c8e6e0a6fe424de56d97c21eaa3b34
226cd5bfa0670294e9a90eedf2d8d663
736d09cc9d68c8b7d504ae425a32f7ae0828a48f
03698ee77998f083cb8f86c28f047e6c
71250e79c68b55077167e99dfe8b6fd13b9142a1
611a08fff40d68e3f917d0a9a58763d8
f05176b877b1f8f8d9a7d132b8ffaae6e3b4767e
cb1731aec593e59f4d435aede40b296a
62bb6bdf84b60e336dea4a8daf7ab2d1
71f0268c9b0f16be997c3f3e98a08c13ca4794f0
05ae74abf9d24de0fcedbc6da81f3bc1
2be2e9ef452498c0f0186b0ccbd6cf3c66e66764
77affdf3fcab5936b646a23c38d84319
c2182213dfd0d95822a731ccb0c7c9cb031e83ec
23313b61c75c2f9b5746765a0ba5a3e844fdfe6f
508c473b78a7065ebc6a230b6b486633
13bfb8723bb3c2d2bd10f5120fa5a9417abfa82d
eb7a7e52caf25ed24ab7becf0cda7b3c
e91157d90ebd1d8cb8ea94184bb79e5662bba10d
0248af8ce516ccbf1c43ed2b182872dc
544895418677643b38720ca49fb91d6738e22394
4c4aa0b17e4d61f4f8306c86084ed9ea
98698600b3ce2c7842e955fae1eef09ae8ab6b2c
84aa499793c2232170c33f83e66e09bd
69ecff236b44848854b9dcb31be1c5be3d9468a3
698c7e75109bbb0bfc164ec6959ae104
b26862c5f3053477c70e821a7f904532777ed89d
ef6199d0df00ac8f1b2bc8dab6505caa07933ca1
97c25fc344c2b1134e7d8fbefaebe493
551b09cc56590a9007b54375dbc23020374f1c74
236c32a75b48825f0e41725d7ac4716d
2b39ce9bd8105a5bc50cf11ae3239307
f4f9891de5b8a157397bca1d72721eef4e8f6263
58999709efae7cdf165efc7164e7e210a21e2d0b
b953948aec8962e9f610aa9cb2aed34d027d9fcd
2275fd9a4ef61bbfec62334f397dfafb
f7496f320ffd18f55432c47c3819efb7
d90da3c9cbd62d8805d347ab2f25bd27
d26eb7c7ba49d4d2af59c26bb2581d23
e9698b15266c7677c3d51bb39085bacf3bae35da
3df484761b017ef836e08a74cd8a7c54
fccf6a6cbd77c3ba1a31bb67274f31de2fc83407
ba928f8b286a8f400a2acf72b2bb5098
a172ca03381dcc267f2bb5a297b2194c927cd730
2ce90ffc0664d90a875fbfd6e5685e9d
4c8f5797defc8f4c1aeb34683b7dd462c9974d0c
356e30e9b09a66cb4cccaa17f7015215
2d297ff242ef1ee9de751829147cd3ebf43397d4
91fb9e39c89b07f621ff4f33c8d19b31
18c076081453f60cb9d70bca92901227a9aef1e6
c00eb6fa35288628959e1978b7794a5e
28000ea6ce5fccbd532890a454c05bf9
97570fc6c6c9e71366ea3cf6dc43f9e6c6885d62
11561d1c3ab54c6292e0b72522707c1f
5c01f53ff23ff27a1a62b6be7f24e0f619fb6a39
99f9be4a89262a29c0e8dafd03148c1a
6495de33634321ef275b6cb785f357012593b6d9
4d5c8385666e116ee415c9502f3c1a1e
fb279930a88935f0f08e50342acb0dc36a68f3f8
5ed43496a80e7cbdb6d8dd608f4bea12
8c2137200ca81eba43c8973e1fe79843d36fc4a1
9d32f0f52ceec162cc014a2bd7ecd418
a3f32db97a1e4c15b2c0fdc19ff74017d278d4d8
32c72a5be1fd7836198f42a6695a8b8b
ae5293b33620f7d226352e1e9ee52da17d3a71aa
dcd43f262e7e7dac17074dd5414c405d
661477790f511bafd8a64253223111e5d222d4d3
5505756b6becf0b7198ea39281de89e6
bcfa0b904292d336b3527d440a35e88edd891e4c
4f62f978d090dddf63f003e3f4c320c7
e4a09a87a3754e1fcfe6a92804ab8ec9157c9575
3c898766c871a88c4b460fc4436f56b1
ea0de9d3055f7f87b91c42380a4318fe12823fc8
af4d67b95c3e3b9d8a0b25323f9124d7
9a6d5535c5a3938c36ef9500989baa75591a8b29
77aced408cc19b6cbe941485528a2c4d
d4bddc3d411653ba8963960279bc9dc59d826a90
310061942ca54eb8ec794daf9b56319f
709b2d95da3d38755a276f62f9c0c2754a44a438
cacf632aba1644ba8f10878574852458
6ba5d9d014e6fafbae4fe05a46f1c227
661b5be7eee37a9bb38eaf9f1bf8c0c2362267ff
53061dcf2a7cc8d63fd6ff93f75ed5cf
76e53a6659eba9148af86301dcb955db2ceeaaa8
affac943b1f067d569becc86b0b5900e
10942a8ed29cd8fe4d0282215f883fc25f367fd5
194653de46c4756bbceeb3fd1bdd11b4
3f131201955df262ecd05cf464687605e36c554e
7188a30c121083861d853f67d0c2abcf
47383d539902d193169d3d30223fd56b91928870
89f9b4b77936ca88edc40e747cdcf0e7
05b34700cbef84d6f1b620d679f63d1ff965fe88
e080c459dbf17d23c1daca83083f586e
54b3dc1101a99b972a87d180f1dc78c704f70a1b
0926ddf42a5753800d15bf4c3231666a
44744cea2ca0e2daa2996cda16ed9ec7b5c82fff
b52e8d6dc1a3e97e18cf8cbe4c80b2d746f84f7d
69910693621f86f1dc5f6a9f7cf32bd4
434dbb037a79c74618c6544c0950b56fa3c4a0b5
b863105525458f4bf66c5fb640a9d0c7
2804affcde5d3cc1652279792e152763e95530d3
7ea7e3b6e19f1166d5af7092436e2231
9c782ea61af30611536e237139095fd520f6e20b
4483605e7a685d95acc28a2f9770674d
c8096ca810407d5eaa211843575737be60216bc0
fa7ed3e2fa11f2e7c236e5f42cb1f5ea
3d7ab10f3f1a3232d81079ebea2d28091dca7176
a7a3bc5bba057dc3897a2ee84352ae6c
3d4160a5e8df3cdde6b93e9cc353c7fd4d295cb1
1689b802445e1bd4d60e776fc578cef3
e5115b3f0ac9afde4426e1b0ea0c70f4f99a08c5
b2531a8e62a84d62b0c3e88b57f0f4d5
e9b214db42c5cc402b11737e52d563e96174e4c7
d2f799098f46d4fec4540b9e8300e74f5cf18166
44dea088a515aa664a22583b98bdc620
324cdb1c2acb892c0059909c1cb2a59e2b76e58a
0d4c65d4ef06b3dfd06c27c9c0afc01a
6f5f0dc948029be25308f8318bbc09e7fa10c992
9f047481990bc947235514ddb7e34339
9787e90ea5c0ddb9a7d0b435b42d64234765a47c
57998387f9a5712e4e790576ba990223
c5e5f93c65967e73b0e6e806f41a62c935f2dcff
0ec1277d0599628514ca04f0959e9074
16fb5065c20fbd855ae2cec0da49c0174eabc37a
409c67140a7de1eb423886a60e19ec57
847eac1daa841b68b41d43bf071834cd76075184
c9354e107084db3e3eec8468ea9ce5fc
5a03911998fdf6ff09c57fac9c2b62a7e4249706
9077a94c1b40b28665e31eeb148887d0
ab1092e768bfe6a8eaf7e0df43317d94e42af487
1d8131cfcb0d6fcfe295be95835b581a
ea249f1c8a341f01fc9428b800646c6d1f86bed4
4e1b784f7df0183f38ac18a2512b638a
93b2524e8fe3fd1e4255a14587a89e6f8647cba4
375e69ffeee0c033cbcbf50109750c4f
2fa803a2851accbfce9c39247cdf8c425f7c320a
7d8d8db9b7dfeaf41a51607ccb9d2eca
871911c72673bb7500d93026144de5b86242f04e
f7ececde5137504144d973a6bec73941
72343e9812749b2d5c6fe362982494df88efe873
517d02e8595ed7783095c42a6f220653
b9e8b725095b776466187667e7c0978d63a4f09d
e4d96009e6e971f5edfb9779e63ea5f1
e1a38b3ddeec22c0d9a98fe9b17b12d4218409c9
e89246d412738ca6d6301e03bb7cc599
6f9941751fd98735bc97178d4948bcdd7683d239
1604d8256d5ac4e0b105a2c78d7a5066
9c861cda45be77746ccdaf401b3a2a268c9a4a27
035dd1f73bbf846159d7ae3fc77c126f
e4defc7366ab16387d4edb2ef0902067
da9783d87c2ab712a21bb145b0de3ac9
878bc3a0b7336a611f8787f95cd2905e3df55dd4
0148c70e5575b2de1413f0fa0f69a409
60a1d771943fa568dc4df3b21660b5366c2f3616
b29cc074423e37e90a0d4bc3df4dae4e
98d888f84fef1e14b56c8d2e03bdacc4
87eaa5ca3f36165d5572fdd8d0ea8110e3856dac
0b150edce6f3809a9a15a01db51ddc63
333bf72c6ff4f3234f809d2296bd5332bb57b50c
cb4f63807c25812b95fbb45967f77270
d91078f3970ff8c1476ec1733cb508d6f1196392
9137348c51eeb054d2058ce3ef54f5f8
f761314fa7354cd35a393032945c31cdbd8a17db
7e3cae7f762108985a6f49f73a44482c
dc9b52670f1ac855c8983e9f3faeae4226264be9
707118b70d9a16e135cb8be5412089fb
a06b5f2094529501f12365f453384bda9026ef7c
71f0481d7af1c105c50c6da66bf2ecbb
2a8e8fe31bc88c5884e3c19b9f30f79b603ea444
7a74cd59c00b3f6e6abeccaf7b46753e
baf00ab99e7af6228cced6e9a80f23b152457a4c
5ac703104d9f1bf7e9b86c195b03d100
9a850ddc51dca260ed443fc37500eaf51fbeeb70
f7d103c1ce7d8c12ac6dea7d6da4983e
a866a12cf5711d5e0b1f3a4b5ec03745d6b8937f
a869c38e75ff4a4a9223b764b77e8a4f
06a081b37a2a4efcba2b4670fe6023f3
0e33a6b92db460c66e22a7987f79291f
652ae7e42599a549640a28e4c6db3b9d8f708625
06c3856acf906eab1ec7f1e9a9508aa7
f3ae2fd8656ed511b53edba9499c7546a992de91
e38bc63f7fee4f072c1ec735c3cd921e
9e56655c375a455b36b3e0a8b6700b734874edf2
b3ddebfa59359945595acfe8a4ea402f
06f1fa1abd0f331b8c4742b9a747ccfaf6b7eb50
4b646cb7f885f41303a07ba9354cd187
60258a042af629840508b07c13e0277126a15dcb
4f3d5a87875063af7ee68d37b608d0dd
1ba3e9b68231ce6680962aca671fb0b7690bd962
005e0c61109a57d5616b258d3d2e3b2a
45f1919ee879f17817300bed1713eb6f9ed1e3d6
5828c4e1170c260880dbf12456bf8291
3a22b516468402188278b8b67e7534fc
266bfc90a919be4e3e2d3cd2e295a4313e48a412
487e74bd5b501243c6c656bc31eb29fd
5db5d1361a3cefc26be4f838ed262f32a3b8697c
88d58f9b3a3a4902e40154400134ca3b
2d5a88bc193db3595ea51ed8e6292477eca2f3ea
43253f4a4f1ce505fa2f58bea3a6da3e
d41f730f20a382610dce27b92b4bbbb7b2aa38a0
91ae98867129315c90c3a0ab5266e947
247499c4c87545efd1342b255fb4b6d82974950e
bdfc554590cd9aceec4c570d7d61b297
b94defd5052cc57a0d03d75ac7614c599866b475
cd5623e679c1035d8370c7825c200013
92b9b9d2c8434e03f34c62a6f21a185ace780630
fb26dee678922df47b0c1a109dab3820
06380eb7e391c5f6066cfb2f786c8cc5
33d77816d87769d38aef0c6190c27da6ff1f0803
bbe5e36f3ece6df21888f5d09b7633a6
a9267d6654863a1e67795ec9ee176607ee785c64
9b241b21741e44b9be840bc3e02ffcf1
779a8b93001ed35a556a40b17077f4f7
14014bc471327cfd47076d32fe14faa506f39ffa
1d7fa255c537c496580a621798409688
6f809ff1c4bcf9b7a8b62c947000490f3abfda5a
dfb6434668cbcbd9ea87de98d7d2e14d
82ce41da7fd6978cacf9bdf9cef600fc481bce23
de321b9ec5b8894ed2e51ab43ab37b5c
bd071c5885d7f0e12cbb4d11985ace43
76d8d143f7676e4eb22a92a40c7046fff1febc90
abfcc1423353d1f4cf14ec3d556a2aea
1c93c3dae9678bf8d5b7c53a6b94f9083bfd0e02
b6d709e5f9d3df8d99b56caa574987d70939a605
3214643850e32ab218e462478007e2c36b626cc8
e65e9212cbf6fdc1db65cb54fcfb5a4a
3d95528d008759197451fd2ce55482e0663d4c74
1b0029701abb28899c5db54f08126a94
8e4fd4d548518f6e51eebb72c1a8654479b7a02c
1f2f640097253dcb762848bc1cb7bb7f
574d8c4731c65abc74b9b940038f28fd
cfd26292bf460e3d0635b09648a5a67835ab39f0
9d352ab6b98a16c921f1be995e9283ec
e5ce47ea71725a85f09c832370ff2e61
522df989772c744b56460be577d1f6fe612f09b1
1a2357f719f8ac47f56e22cfb46ae23b
93f2af1b878eb896e9b680b376786248392a169e
8bb3a687e98b99da0f94b3f7d5c199dc
68c54b381e5960d1cdbc4b9754e3285f1222ac28
b695e6d8f829e98e49e8b1f1eee2d029
d44fd201a445982e5f5733b8996f76361ccc020a
75ca74889a033cf1088be72f492b6459be19f464
6867dc000967244280e11d74bd9c77cb
b564ec40657d585f98d4f449dbe543ee16c3f4e3
d667ec2721c8b5370849f24eb95e2278
24f53dde579714b4835edf361bbcd4d33f6aef0f
d831cbf1c6f5cd0ce07cdfb05e13e5aa
84e25f0aa91a029100563daab4c12d1df133cba0
53947e38c92a7f29abdd97c6fa352121
e18a707c7509cdc2c8e22ab18e48877a83043d9a
a396cbe6811458655511079e378246d7
9c2ef1b46ee563b20c294e0cec17c08a8e4ab554
21d1c5ce5566a5b399b59a4ebc077202
512a0a26bb2702b51eb587ac5b9ac37e516cee36
9a70612bd912c84b056c1ac6e32c6ba7
e08a3f0170083a98c6e8ef91bb423d9a
4acc8b576921397b8fde4d2e5af49027d784a385
3994c09be1ec94fac1951c3420f847cb
98bffbc1bca7d11221ce466247cba82f42c2b2bb
48f74869cf5c8d3526dae73e704808a5
007373db405ff052d2ce1b8b0af2c5c1c5c38bf5
afe626ffe3ea80ffb4fa6dce8a8d1b26
72293b100ca70c009e2f4368522536e9
3302e7edee3bc03df3d442360b7b9982975c41c9
2ca47a92a8a061337c89d3ac6cee8440d071ca71
97b62b1707f310756bac60c3bd7bd975
0778d2f0e721893a28eec98692a16a41276bc6de
b931b21ecec8faf4380fa38faec81b03
1ee706b14a46f656d8aedd58b5df4f74899f6e35
4e24ed2d73946521374cfb42f0db404059a021cb
007825da70c54058d108c455f71fe21f
f2f6b53d013b0d3d1b25bc4602bbf887cae4ae85
f762b1884e91556c53e36a9faf322c9d
78cc5d523595dbcd42683fc9d7a3729310716a45
a72cb32f5b96276f72eac434c2ac834d
afc17a7170128771fc73a9293a61f2fb5c4dc62f
da08b624e93fab14413743794d710d9c
1e19329da5234b9f3b18efec171345b1
2f01cf19aa3cdc1dd021cd100b43718797564130
a3b9b287555c39e12e9bf7e80beb7beb
9ff5c58248a3e7714f6aa25db3705520c829268a
050eef93d11015b634c1c37b57415942
03c78f71ea02a86496e30a8bff971570fc450a0d
4e67508e3ac3057c924d5b45c87787d3
4e9d671e7ff5966a6611bf8bd1be2386e0519d52
2e97a2b6d6764f913610157082260ef0
fe5b3b9b10fb5633c6a68002e57b7e33ed699bfa
24325a3cd30f46053c7ef3ee61a6ea9b
edeab6abfc90b671a5fb148a020c61972991374b
807912aa58d6213700ebcf01cc393915
5315c2b7a9e6ff55aba297778dd801c3757e029d
0139f8e0ff52c0fecd09da4044f5925c
8a3d318b5b33b5860e009319be6667e0886c11ec
96ccf544ed2eea074e2cc8d368c93d41
dd001873da2134d5af244123ff6d3ec8cbbd0ae4
7758865f1f5817a970f203181558e76e07b0a78d
fc3d83bf26a0e811852c987341911554
720e23c1e406eb695456c5273a10e87cf4192c98
84b339a52ded8326651a44fa67b290a98782bc79
ced73f7aca8d9381e182bd9ffa5a3a354c0c9ca2
e48375c4169f2d40220736144370a024
2732c3cb28c94006b824f137cf37a965f24069b5
0dd93d16f58452b9a9b4bccf1179d68a
e9cf8ffd2b6aedd59a0a3b18296b449012f6385d
83646b40815fa62a2e8e7f20f30abaeb
9064c8a7be9ea4840cffa08cb11b9b00ac4d1159
5c4362ec42ace0cab9d1fe96bd3c085b
2c77494ff47f94391b0923d49fa1737b896b8a62
49bc9af0cd83b2760de82577f4fe8521
580c587b6fe8392efdd7495df2acae9a48898fe1
1f770a419eb969e0c89d18b80a878101
06f4dbfea6574c734cce72e8e39645d75d3b2b83
bb84ab51d6363ceb8f6034d1cfbbc468
140d962593502f668076e1f90bf0284b811f0203
ae80219344a848b20bc34cf7fbca1353
1dcee947953519b94a16a86a9dcba98189b618a8
78667922257c513ed39472ec8f52358a
370ec680d2d749ab5d5378e085a958fa46eb64d0
233729507924ce939a1109a5fc3e6b1f
81341556f5a99957cc02050408c89f0050394338
6ce8206f91319101387afa096449e56d9d203348
5b00c885b0d850468cb2500b39c00420
31b56fa79aa9376c1be2e7a401c334946a61afec
f4869b9956a33105987ec0d481421ab7
732ea51b931b8a28d2a1df363888869a6869ba5d
9c242a8fea08e1d265906aa0208595ab
b6301ddfe45894b803af4f62a732400c11d9b157
318a8ddea6d815e9f65b3c01e116783a
e92fe7431797edbfb56a8147cf64db79
ac5eff435d477e69e3aefc1a7a967793d293f74c
d13402ef8e555d18b77828cd41a8ef1d
e3e654e98d1ea9f9bb8ae50b5bf18c76e5d392a9
76192db7b294908f239911bc87442b9f
63099d7c578b7d8983f44962a5738d1856f96a82
3547aff88fee0edf6ec15fea77f6b3b1
b254a8470f8cf929eed8b2e914c4bca9b4c22725
05193e712a22b9578498ef55191fcf1d
2d31116ac496a6cc4e0666ab03a6a9eefc0d337b
f0b6fbb27cd11b06300d5b20c7214dc1
f509e9a71a564367afd2a17a47430e550b114834
dccbe1485714ac3d84b7aec4bdea9342
68f5d612e47abdcab063246d043d6ca43b2db0f3
a284d0f9d5b7aeb0cef451e575402b42
13d3df7d78ae8142f1cf4df52d1ac173ba7a049e
81a7319682fe86ef588fd27955d69406
2f2b0e19abf73ac7a27f6b1a002ff996c6dddf13
a8d3fb17bbc8e488811c95f233f9fe26
4b8fc68f5db9a80291fb9b0d95f9aa65a2a90c88
62de6836d189fe949dcdb76b8333823e
701deab2b2518463324971b857afe0a9c3041f73
acd0f6beaf2315876393a21b939e537e
7c7916450b3065d010226172d636648b4f5a20b5
17b789b8b62c3f45f6c1cbe09a89a648
5932a54233984e2da4c90f3ffd8858acfce77760
0a314dda41644c8c399f6ba086d7497b
930be6efd1e390b13d686c24d518c8ba0a5e897b
ee46f8499c25ef1c9626dae23a0941cd
ee1dfc3d766660e957955a07e97d0b44df041440
d69ed86d6c2812c2258eec0b8187dda4
af431305f5dbdefa45fb5a6c9991dea7f360ffd4
95883f79d78ec37b2941a64c12322222
49f6488fe1901c52642667d1b5a9e89a6d206260
2cae02f94d5b07f0094195cc767b3a01
ff47bc6e46a6f5b5c1d06d0506a48a2e61a6f6f1
eb30bc427299ae343a27dde1670c9dd7
2c2b0c89992052737f1a63ef5f6d902a
935fa89a46dea81b699502c65ffa22fb822edb83
188b7578ccff23f67a953e385d2db144
fbd7200ddce118387478610331655824326914dc
bd8386979f47d2b9102948e437d03027
a921432de9abb66173fbe418d962725793d580a7
6ef0c3d2bf83ce9d2fc7e5d765cdf45c
b7e7319f9813dd8567ef94804cb0d7f31a88ef7a
332095024196c674d911bb30d55dc3756d1c1afe
2ae47013949176222e792ba162107d2f
9e87954b5f9cf17ca698763dee91c4ee
1bb2773b6a83b7d1777057b6d3800dd4ffc90d6e
86414f50576bccd31471242310d6e8a7
033bc510239328f71a78d91f4b1cccd861d977ac
47412eeff1a45aa39b8a2a578fb669b5
e11da518d1a216eb6e2da8eb484940db904d7d5e
72a72b2a5a40576c95cb4b4e688e6aa0
e37ac4dfbe7702119ccfc04694ee22897a85c24f
147ff387d8e3dd61cc7d4e5b17d32885
c3a79234e237c22096e69934a63054e00b3f4e5d
5f52ecafb26b896517a4c7e3d4c9bacc
e6435b2217ab004a0b61f5acfebb317e092a1c74
a1fcbfe56414a96a023dba604d40def8
a72dcf0443721ca58f599fcf6a437552ac9fd785
27ae93019f01700162150e96028ea3f3
ff57fef8edafa3932a1a6532ed262599d257c37d
a77cce4716c2281038e1d0c1fca3a49a
0eac1950bc6b8367cca69a164489de001b0e355d
fcd3da0c7ae87a36cb5ec40ae98ba343
100f219fdefc45b36c7c76e061739855bac50d1a
3dc780a6b62d921c374195271739f3dc
ecf4af388fd51232bcc2a70bfaf9738b
77406838574210f93db41715d868b324
fac35bd22c91c1dc9b6f33387aa35b4cc13c991f
4d5487e07b70eb3837aca88c1fd8c3e8
eb7279c7035bfb02390428e0cd6726d038b32279
b57a2473e6b76be0e54ac8ce8e6023d8
547b276de9100287eb01fc4e2e677ac4763763df
58125068e64052a0bb5890d2318a5b84
902463e793dd679c50fc6c1041de0d1c
5b0fd2bdb238ec51e2ceb92c6584b3de53eb30d1
3718da56965cf9f381f024df0fee3110b31e2daa
aced542accc3f81d4666634fd14d4b24
2cacf23a1cb548e700fd31e3c9d5770e70f25288
ee94978e49eb07b482d51193eb7f5267
b88ff0f4706561b3023542845b06eb3f01411348
259367 F20110218_AABSEM UFE0013684_00001.mets
150a16929b7968cd5ec6195ab66193f0
0c76786504948cd9435bb942baa24b81abbca0cb
daf06767a077e0c8ded24da965a952ef
f675c3ef031bba170765fd695cf98e53
7eee9bc0f080f7b60b712e3036099d93
3aef9fc1cfd4359ae574362f6d12c103
82846febb7b53edf346c2f873841633a6000198b
de412769e954919e021158711b91d3b69538f137
49bc5ef1eeedaaa7890d91f48a01f6d1
0114a3df29a73e6ed87051e38cf88e9cc2304574

## Material Information

Title: Statistical Mechanics and Linear Response for a Granular Fluid
Physical Description: Mixed Material

## Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013684:00001

## Material Information

Title: Statistical Mechanics and Linear Response for a Granular Fluid
Physical Description: Mixed Material

## Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013684:00001

## This item has the following downloads:

Full Text

STATISTICAL MECHANICS AND

LINEAR RESPONSE FOR A GRANULAR
FLUID

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006

by

To Amma and Appa.

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor, mentor and friend Jim

Dufty without whose guidance and encouragement this work would not have been

possible. I would also like to acknowledge the many stimulating conversations with

Professor Javier Brey of Universidad de Sevilla that pl i, d an important part in

giving shape to this work. Further, I would like to thank my committee members

Professor Hirschfeld, Professor Ladd, Professor Muttalib and Professor Reitze

for their support. Finally, thanks go to Karthik for being my sternest critic and

Pradeep for his uncritical support. This work was made possible by a McGinty

Dissertation Fellowship award from the College of Liberal Arts and Sciences,

University of Florida, and was supported by grants from NSF and DOE.

page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ...................... ......... vii

ABSTRACT . . . . . . . . viii

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Granular Materials as Objects of Theoretical Study ......... 1
1.2 Granular Fluids ............................. 3
1.3 Theoretical Developments in the Study of Granular Fluids . 5
1.4 Objective of this W ork .................. .... 8
1.5 Structure of the Presentation ................ .. .. 10

2 STATISTICAL MECHANICS OF A GRANULAR FLUID . ... 12

2.1 Microscopic Models for Granular Fluids . . 13
2.2 Generators of Dynamics for Inelastic Hard Spheres . ... 15
2.3 Isolated Homogeneous System ...... . . . 20
2.4 Stationary Representation and Time Correlation Functions . 23
2.5 Fluctuation and Response in a Granular Fluid: Some Observations 28

3 PHENOMENOLOGICAL HYDRODYNAMICS . . ..... 34

3.1 Nonlinear Navier-Stokes Hydrodynamic Equations . ... 35
3.2 Linearized Hydrodynamics .................. .. 40
3.3 Hydrodynamic Modes and Stability ................ 43

4 LINEAR RESPONSE .................. ......... .. 48

4.1 General Procedure of Linear Response ............... .. 49
4.2 Special Initial Preparation ....... . . .... 54
4.2.1 Special Homogeneous Solution to the Liouville Equation 54
4.2.2 Local Homogeneous Cooling State Preparation . ... 56
4.2.3 Simplifying Properties of the LHCS Preparation ...... ..57
4.3 k-Expansion of the Transport Matrix ................ .. 60
4.3.1 KCF at Euler Order .................. .. 65
4.3.2 KCF at Navier-Stokes Order .................. .. 66
4.3.3 KIC at Euler Order .................. ..... .. 70

4.3.4 ICS at Navier-Stokes Order .................. .. 71
4.4 Summary of Results .................. ....... .. 72

5 TRANSPORT COEFFICIENTS .................. ..... 74

5.1 Homogeneous Order: The Cooling Rate ( . . 75
5.2 Euler Order Terms .................. ........ .. 76
5.2.1 Pressure .. .. ... .. .. .. . . .. .... 77
5.2.2 Euler Transport Coefficient (u ................ .. 80
5.3 N 1,'.i i-Stokes Transport Coefficients ................ 83
5.3.1 Shear Viscosity .................. ..... .. 84
5.3.2 Bulk Viscosity .................. ....... .. 86
5.3.3 Thermal Conductivity ................. .. .. 87
5.3.4 The p Coefficient .................. ... .. 90

6 KINETIC THEORY OF TIME CORRELATION FUNCTIONS ..... 95

6.1 General Formalism ....... . . ....... 96
6.2 Illustration: Enskog Kinetic Theory for Elastic Hard Spheres . 101
6.3 Deriving the Kinetic Theory for Inelastic Hard Spheres ....... 106
6.4 Structure of the Transport Coefficients . . . 112
6.5 Comparison with Ch('!i ip, -Enskog Results: Shear Viscosity .115

7 DISCUSSION AND OUTLOOK ................... .. 122

7.1 Summary of Primary Results ............... . 122
7.2 Context and Scope of this Work .............. .. 124

APPENDIX

A GENERATORS OF DYNAMICS FOR INELASTIC HARD SPHERES 128

B STATIONARY REPRESENTATION OF THE DYNAMICS ....... 136

C MICROSCOPIC CONSERVATION LAWS . . ..... 144

D SPECIAL SOLUTION TO THE LIOUVILLE EQUATION . ... 152

E CONSERVATION LAWS IN THE STATIONARY REPRESENTATION 155

F DETAILS IN THE k EXPANSION OF /C (k, s) . . .... 159

G DERIVATION OF THE FORMS OF TRANSPORT COEFFICIENTS .168

H ELASTIC HARD SPHERES: SOME DETAILS . . ..... 179

I ENSKOG KINETIC THEORY ................ .... 185

REFERENCES .................. ................ .. 207

BIOGRAPHICAL SKETCH .................. ......... .. 211

LIST OF TABLES
Table page

A-1 Hard Sphere Collision Operators ................... . 135

C 1 Microscopic Balance Equations ................ ... 150

C-2 Forms of the Forward Fluxes and Source ............. .. 151

C-3 Forms of the Backward Fluxes and Source ................ .. 151

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STATISTICAL MECHANICS AND LINEAR RESPONSE FOR A GRANULAR
FLUID

By

1i ,v 2006

C'!I ,i: James W. Dufty
Major Department: Physics

The objective of this study is to explore the application of methods from

non-equilibrium statistical mechanics to the study of granular fluids. This

exploration is carried out in the specific context of obtaining a hydrodynamic

description for these fluids that is based unambiguously on an underlying

microscopic theory. A particular model of a granular fluid amenable for theoretical

study, namely inelastic hard spheres, is considered. The isolated homogeneous

system is studied and the associated ensemble characterized. Then, small

perturbations away from this reference ensemble are considered with the aim of

isolating the hydrodynamic response of the system and hence obtaining microscopic

expressions for the various hydrodynamic transport coefficients. The primary

outcomes of this project are exact Green-Kubo and Helfand forms for all the

hydrodynamic transport coefficients. These expressions are a first of their kind

in that they do not have any a priori limitations on the domain of their validity,

unlike those obtained from kinetic theory that exist in the literature so far. These

exact forms are a well defined starting point for further analytic and numerical

analysis in order to obtain useful insight into the nature of transport in this

fluid. First steps in this direction are taken by comparing the exact forms to

their counterparts for normal fluids and by developing the kinetic theory of time

correlation functions that relate the results in the present work with those already

given in the literature.

CHAPTER 1
INTRODUCTION

1.1 Granular Materials as Objects of Theoretical Study

Granular materials are ubiquitous in both nature and industry. Naturally

occurring granular substances are the regoliths of planets (that is, the outer most

part of the planetary crust) and .,-I i. ,1 ,!--ical objects such as the rings of planets.

The regolith on earth for example ranges from fine dry sand to marshy and lumpy

clay. Industrially, most agricultural and pharmaceutical products are in granular

form in some stage of their production and transport. In several studies [1, 2], it

has been estimated that !i' of the potential capacity of industrial plants is wasted

due to difficulties in handling and transporting granular materials. These materials

have been widely studied in the engineering community for a long time [3, 4] and

are of active interest even todwi as any improvement in the understanding of the

dynamics of these systems translates into cost reduction in these industries.

Granular materials are studied in different regimes. Some of the interesting

phenomena exhibited by these systems are mediated by the interstitial fluid present

in them, either through mediating cohesion between grains or through a Stokesian

coupling of the grains to their flow. These systems are called wet granular media.

Granular systems in which the interstitial fluid does not p1 li an important role

in the phenomenology of the material are called dry granular media. The latter is

the class of systems of interest here. They are interesting from the point of view

of a theoretical physicist for a number of reasons. Dry granular materials exhibit

a wide v ,i i I ji of phenomena. A pile of sand is like a solid in the sense that it can

withstand application of normal stress (i.e., if you pour sand and let it come to

rest, it sits in a heap, bearing its own weight). But it is unlike a solid in that its

stress response is highly inhomogeneous with a few particles carrying the bulk of

the load [5]. When activated, it flows like a fluid, for example, in an hour glass or a

grain hopper. But the flow is characterized by different phenomena than in the case

of a normal fluid, such as j -,,,,ii,'.- novel instabilities and pattern formation [6, 7].

Further, model systems with simple prescriptions for inter-grain interactions exhibit

all of the phenomenology described above. Inelastic hard spheres or dissipative

Hertzian contact forces together with simple realizations of tangential friction

constitute very accurate models for capturing the physics of this system [11]. Also,

the numerical tool of Molecular Dynamics (M!1)) simulation can be brought to bear

on this system to supplement and elaborate experimental observations, and to

verify predictions and stimulate the formulation of theoretical analysis (for example

see [12, 13] for MD studies of various aspects of the physics of this system). Hence,

the rich phenomenology associated with this class of systems can be studied in a

theoretically tractable setting.

Theoretical interest in these systems is also motivated by the following

considerations. A granular material can be treated as a prototypical non-equilibrium

system in the following sense. Firstly, notice that most i,,i' r... 16pi models used

to describe granular materials have the constituent grains as their smallest entities.

But, these grains themselves are large particles and hence the thermal energy kBT

(kB being the Boltzmann constant) is very small compared to the gravitational

potential energy mgh of the grains at room temperature and under terrestrial

conditions. So conventional thermodynamics pl ,i,- no role in describing the

phenomena exhibited by granular systems. These are explained based on purely

dynamical considerations. For example, a mixture of two granular materials of

different grain sizes segregates when activated by vibration or rotation [8, 9, 10].

But for the above observation about the irrelevance of the thermodynamic

temperature in the dynamics of this system, the above phenomena would appear

to be a violation of the second law of thermodynamics. Secondly, the spectrum

of theoretical tools that can be fruitfully applied to study granular materials

in various regimes is very wide. For example, the well known phenomena of

compaction of granular materials under tapping can be modelled theoretically by

introducing a pseudo thermodynamics where the role of energy in conventional

thermodynamics is taken by the free volume in the system and the entropy is

now given a configurational interpretation [14, 15]. The broadly applied theory

of self organized criticality, used to explain the ubiquitous occurrence of "1/f

i. .-. in disordered non-equilibrium systems was first formulated in the context of

avalanches in a sand pile at the angle of maximum stability [16]. A granular system

in the jammed state can be described using the theoretical language applicable to

describe glassy dynamics [17, 18]. A sufficiently "fluidized" granular material can

be studied using the tools applicable to normal fluids in non-equilibrium states.

Thus dry granular materials provide the opportunity for the development and

applications of a wide variety of methods used in the study of non-equilibrium

phenomena.

1.2 Granular Fluids

The focus of this work will be on the fluidized regime. In order to better

characterize the regime considered here theoretically, the following points are

clarified.

When does a granular material behave like a fluid? A granular

material can be studied as a fluid when it is activated sufficiently in a certain way.

As stated earlier, the thermodynamic temperature pi'l ,l no role in these systems.

The only velocity that these grains possess comes from activation or induced

flow. If the activation is in such a way that this induced velocity on the scale of

grains fluctuates, then one can define a "granular temp. II ii associated with

this fluctuating velocity of the grain and expect that a statistical description in

terms of a course grained number density field, flow velocity field and temperature

field will be relevant in understanding the dynamics of the system. Hence the

activation must be in a way so as to induce a collisional flow in the medium. Such

an activated granular material is called a 5,i 1im1 ir fluid". Most granular flows of

interest ( for example vibration induced phenomena, gravitationally induced flow

down a bumpy incline, rapid shear induced flows) fall under this regime. Under

these circumstances, the standard tools used in the description of normal fluids can

be applied to study the granular system.

Internal degrees of freedom serve as a sink for the kinetic energy

of grains. Notice that even when the grains are assumed to be frictionless, the

collision between them is inelastic. These grains are macroscopic objects with

many internal degrees of freedom. A collision results in the loss of a fraction of the

kinetic energy of the grains to micro deformations on the surface of the grains and

to the excitation of other internal modes in the grain [19]. The energy lost to these

internal degrees of freedom does not contribute to transport in the fluid. So, in

most theoretical models of a granular fluid the grain is taken to be the microscopic

entity in the system, with the internal degrees of freedom of the grains acting as

a "black box" sink to the kinetic energy of the grains. Therefore, the total energy

of the system is no longer a conserved quantity in the theoretical models and this

serves as a technical complication in the analysis of the dynamics of the system.

But more importantly, the rate at which kinetic energy of the grains is lost to the

internal modes gives rise to an internal time scale to the fluid apart from the time

between collisions and this p1 i' an important role in the physics of these fluids.

A granular fluid exhibits two regimes of fluid like behavior. A well

known phenomenon in steady flows of granular fluids down inclined planes is that,

in such a state, the shear stress in the system goes as the square of the shear rate

5

[20]. This is called Bagnold scaling in the literature. Since the response of the

system under a shear strain is rate dependent, the fluid is said to be theological

or a complex fluid. Granular fluids exhibit theological behavior whenever the

external driving of the fluid is such that it probes the intrinsic time scale in the

fluid, namely the time scale set by the rate of loss of kinetic energy due to the

inelastic collisions between grains mentioned above. This behavior manifests itself

in shear flows and is an important property to fold into the theoretical description

of a granular fluid under shear. But, when the activation of the granular material

is in such a way that the external driving is well removed from this internal time

scale, for example in the case of activation through vibration at the boundary, the

granular fluid behaves like a Newtonian fluid. This latter regime will be the one for

which the results obtained in this work can be applied directly.

1.3 Theoretical Developments in the Study of Granular Fluids

Dry fluidized granular materials have been the focus of extensive study using

theoretical and numerical tools for the past twenty years in the statistical physics

community. A complete survey of avenues explored and results catalogued is

beyond the scope of the considerations at hand. A summary of results is presented

here, restricting attention to a particular context as follows. A one component

granular fluid is considered. Attention is focused on studies contributing to the

understanding of hydrodynamic transport processes in general and those giving

analytic expressions for the various hydrodynamic transport coefficients based on a

microscopic theory in particular. This summary paves the way to place context and

scope on the work presented here as part of this dissertation.

A theoretical model widely used as an idealized representation of a granular

material in its fluidized state is that of N smooth hard spheres that collide

pair-wise inelastically so that the collisions conserve momentum but there is a

fractional loss in the kinetic energy of the pair. The energy loss is characterized by

a single parameter a, the coefficient of restitution, with a = 1 corresponding to the

elastic limit of no energy loss. This is the generalized version of the Hard Sphere

model for normal fluids that has been extensively used in the theoretical study of

equilibrium fluids.

The theoretical development in the context of this model granular fluid

has followed in the same spirit as that of normal fluids a hundred years ago. In

the dilute regime, the dynamics of this granular fluid can be considered to be

governed by a Boltzmann kinetic equation [21, 22]. In this regime the granular

material is referred to as a granular gas. Extensive studies have been carried out

on the kinetic theory of this system [23]. These include, but are by no means

limited to characterization of the isolated homogeneous state associated with this

equation [24] and carrying out a systematic small gradients expansion using the

C'!i 'I'pi ,i-Enskog procedure, thereby determining the hydrodynamic transport

coefficients such as the viscosity and thermal conductivity from the underlying

kinetic theory (for example see [25, 26]). Also the effect of different driving forces

on the nature of the resulting reference steady states [27, 28] has been studied.

Further, numerical tools such as Molecular Dynamics simulations [29] and Direct

Simulation Monte Carlo method [30] for solving the Boltzmann equation have

been used to study states that are far from homogeneity, such as the Uniform

Shear Flow state, which is the prototype shear state for this system that manifests

its theological properties [31, 32]. These numerical tools have also been used to

evaluate the different transport coefficients obtained from kinetic theory. In the

process, potential stumbling blocks for carrying out numerical simulations of

these systems, such as inelastic collapse were identified and characterized [33].

The low density hydrodynamics that results from the above analysis has been

extensively studied analytically and numerically with specific applications to

many experimental scenarios [26, 34, 35]. The instabilities in these equations

have been ain i1v.. 1 Ito fruitfully predict pattern formations that occur under some

experimental conditions [36].

The next systematic step forward in understanding the hydrodynamic

description was taken in the form of trying to establish that the hydrodynamic

modes are indeed present in the spectrum of the Boltzmann collision operator and

that they are the slowest modes. A similar analysis carried out for normal fluids

in terms of the elastic hard sphere Boltzmann collision operator gave a positive

result, establishing that there alv-bi-, exists a length and time scale such that the

hydrodynamic description of the fluid is complete [37]. For the case of granular

fluids, it was established that the hydrodynamic modes are indeed part of the

spectrum of the collision operator [38]. But the issue of time scale separation

between these modes and the faster i11' n..- 'pi' modes was left unresolved due to

the complicated nature of the inelastic hard sphere Boltzmann collision operator.

Further progress in this direction was made possible by the use of kinetic models

for the collision operator that retained the essential features of the true kinetic

equation, but were analytically tractable enough to carry out exact calculations (for

example see [39]).

All of the above analysis is valid only in the limit of arbitrarily small densities.

But, most experimental realizations do not fall into this domain. The first step in

the direction of incorporating density effects in the hydrodynamic transport theory

for a granular fluid was taken in the form of studying a Revised Enskog Theory for

inelastic hard particles. This is a Boltzmann-like kinetic equation that incorporates

density effects phenomenologically in two v--i -. 1) It takes into account the finite

size of the hard particle (which is treated as a point particle in the Boltzmann

theory). 2) The collision frequency is now weighted by the pair correlation function

at contact that amounts to incorporating three particle effects in a mean field

like approximation, which further emphasizes the density effects. This kinetic

equation was analyzed for states close to homogeneity and hydrodynamic transport

coefficients identified using a C'! hni, ii Enskog procedure [40]. In the case of

normal fluids, these transport coefficients are found to be accurate up to relatively

high densities. A similar domain of validity was expected for these coefficients in

granular fluids as well. Further, the hydrodynamic equations thus obtained were

solved for a particular boundary driven scenario and the resulting hydrodynamic

profiles were compared with those obtained directly from experiment and very good

agreement was found [41].

All of the above mentioned theoretical developments have their basis in kinetic

theory and hence have limitations on the domain in density in which they can

be expected to be relevant. In the case of normal fluids, further development of

the theory of transport was stimulated by the application of the exact methods

of non-equilibrium statistical mechanics, namely, linear response and the time

correlation function method. This method yields exact expressions for the

transport coefficients that have no a priori limitations on their domain of validity.

The systematic study of granular fluid transport is at exactly this point. The

general non-equilibrium statistical mechanics formalism is in place [42]. First steps

in its application have been taken in the context of the prototypical transport

process in a fluid, namely diffusion [43, 44]. The work here aims to apply this

method extensively to energy and momentum transport processes in the granular

fluid as elaborated below.

1.4 Objective of this Work

The aim of this work can be stated in the broadest context as obtaining a

hydrodynamic description of a model granular fluid that has its basis unambiguously

tied to the microscopic dynamics of the system at the level of the grains that

constitute the fluid. This is done by using the methods of non-equilibrium

statistical mechanics in general and linear response in particular. A model

system consisting of smooth spherical hard particles is considered and a linear

response analysis is formulated and carried out to give exact expressions for the

hydrodynamic transport coefficients of this fluid. They have the form of Helfand

and Green-Kubo expressions that are time correlation functions over the reference

homogeneous ensemble. These results are compared with the analogous results

for normal fluids and the role pl i, d by the non-conservation of energy in the

transport processes of this system is explicated. Further, an approximate evaluation

of these exact expressions is carried out and compared to known results obtained

from kinetic theory.

For the purposes of clarification, consider a normal fluid that consists of atoms

that interact through a continuous potential. Then, a linear response analysis gives

the exact expression for the shear viscosity of this fluid in the form

1 ft
I = lim dt (HY (t) HY (0)),q (1-1)
too,yvoo VkBT Jo

where Hy, is the volume integrated microscopic momentum flux and ()qg denotes

the equilibrium ensemble average over the canonical ensemble. In other words,

the shear viscosity of a normal fluid is the time integral of the momentum flux -

momentum flux autocorrelation function calculated over the equilibrium ensemble.

This work presents a derivation of the analogous result for a granular fluid for the

shear viscosity and all the other hydrodynamic transport coefficients.

The important consequences of this work can be summarized as follows.

1. This is the first systematic and extensive application of the methods
of non-equilibrium statistical mechanics to granular fluids (except for
the prototype process of diffusion [44] and preliminary considerations of
Green-Kubo formula for the shear viscosity [45]).

2. The transport coefficients whose expressions are obtained here are the same
ones studied using kinetic theory (see [26] and others cited above) and used
in the hydrodynamic equations to explain and interpret experimental results.
But the expressions available in the literature up to this time are limited by

the fact that they are valid only at low or moderate densities. The expressions
obtained here are applicable for all densities and inelasticities.

3. In the process of obtaining the above expressions, notions such as the
Onsager regression hypothesis and the Fluctuation-Dissipation theorem
are revisited and a reinterpretation provided in the context of this inherently
non-equilibrium system.

4. For normal fluids, fruitful insight into the nature of energy and momentum
transport in the fluid were obtained by numerical evaluations using Molecular
Dynamics simulations [46, 47] and approximate analytical evaluations using
extensively developed methods such as mode coupling [48]. This work sets the
stage for a similar exploration into transport processes in granular fluids.

5. Experimental evidence indicates that a hydrodynamic description for the
dynamics of a granular fluid works well in the vibro-fluidized regime in
a considerable region of the parameter space of density and coefficient of
restitution. In the domain that it fails, it is unclear if the hydrodynamic
theory fails or the estimated values for the transport coefficients are no longer
valid. Numerical evaluation of the transport coefficients obtained here can
shed light on this problem, as the expressions are valid for all values of density
and coefficient of restitution.

1.5 Structure of the Presentation
The contents of this presentation are structured as follows.

1. In ('!i Ilter 2, the ingredients necessary to do statistical mechanics for a
granular fluid of N inelastic smooth hard spheres, namely, the generators of
the dynamics, the characteristic reference ensemble and the formulation of
time correlation functions over this ensemble are given.

2. In C'!i Ilter 3, starting from the macroscopic balance equations for the
hydrodynamic variables, using the fluid symmetry and the assumption
that the gradients in the hydrodynamic fields are small, a set of nonlinear
hydrodynamic equations describing a granular fluid are obtained phenomenologically.
Then, these are linearized about the homogeneous state and the linearized
equations are a' i1v. .1 to identify the hydrodynamic modes and the associated
eigenfunctions for this granular fluid. The linear equations obtained here are
parameterized by unknown coefficients such as the pressure and viscosity.
Thus, this serves as an identification of the precise targets of the microscopic
analysis that follows.

3. In C'!i ilter 4, the problem of linear response and its use in identifying the
hydrodynamic description of the fluid is formulated and solved to give a
formal expression for the hydrodynamic transport matrix as a matrix of time

correlation functions over the homogeneous reference state. These are the
primary results of this work.

4. In C'!h Ilpter 5, the details of this transport matrix are unfolded and the various
quantities that enter the hydrodynamic equations are identified explicitly.
These include expressions for the cooling rate, hydrostatic pressure, the shear
and bulk viscosities, the thermal conductivity and the p coefficient. Detailed
comments on the technical and physical content of these expressions are made.

5. In C'!i lpter 6, the formal aspects of a kinetic theory for inelastic hard sphere
time correlation functions are developed. Then, this is used to carry out
an approximate evaluation of some of the transport coefficients using an
Enskog-like approximation. The results thus obtained are compared to those
that exist in the literature.

6. C'!i lpter 7 is devoted to a discussion of the results obtained in this work and
the outlook for the future directions to be explored.

CHAPTER 2
STATISTICAL MECHANICS OF A GRANULAR FLUID

The focus of this chapter of the presentation is the setting up of the statistical

mechanics of a granular fluid. Statistical mechanics is the frame work that allows

the extraction of the macroscopic observables of a system as statistical averages

over the associated set of microscopic states that characterize the system. In

particular, non-equilibrium statistical mechanics is a many body tool that allows

the identification of the microscopic basis of transport phenomena in a system.

Also, as will be seen in the rest of this work, time correlation functions p1 i- the

same role in transport phenomena as ensemble averages do in thermodynamics.

These time correlation functions for a granular fluid are defined and characterized

in this chapter.

The layout of this chapter is as follows. The primary ingredient in the

microscopic prescription of the system are the interactions between its constituent

particles. This specification allows the calculation of the N particle trajectory for

a given initial condition that specifies the value of the position and velocities of

all the grains. Some of the models for grain-grain interaction are first discussed.

Next, the dynamics of the system is characterized in the case when the initial state

is a macrostate by identifying the generators associated with the trajectories and

phase space distributions. Then, the characteristic ensemble associated with a

homogeneous isolated granular fluid is identified and characterized. Further, the

essential machinery for studying perturbations and response in non-equilibrium

statistical mechanics, namely time correlation functions, are characterized.

Finally, some comments are made on the nature of fluctuation and response in

this inherently non-equilibrium system by considering some illustrative examples.

2.1 Microscopic Models for Granular Fluids

Most theoretical models for granular fluids consist of spherical particles that

interact upon contact, so as to lose a fraction of their kinetic energy as a result

of the interaction. In the present work, all particles are taken to be smooth (i.e.,

no tangential friction or rolling) and spherical. Then, one such realization for the

inter-grain interaction is the dissipative soft sphere model that gives the force

between a pair of particles i and j as

Fi (gij, qij) = f (dij) (kdijqij 7 (gij qj)Qj) (2-1)

where qi is the relative coordinate qi qj gi = vi vy is the relative velocity of

the two particles, qij is the unit normal vector and diy is the magnitude of normal

compression defined through

dij qij (2-2)

a being the diameter of the undeformed particles and k and 7 being determined by

the material properties of the particles. Notice that the force F consists of a spring

like conservative part parameterized by f and k, and a drag like dissipative part

parameterized by a friction constant 7. When f is chosen to be independent of dij,

the conservative part of the above interaction is precisely a linear spring of spring

constant k. Typically f is chosen to be of the form

f (d=) O(-(a- q) d (2 3)

which turns out to be the correct choice for the spherical geometry of the grains

[49], and this becomes the Hertzian contact force model for smooth spheres or

disks. The N body trajectory traced by this system of particles can be obtained

by using these force rules together with the additional assumption of pairwise

additivity of these forces.

A simpler model more amenable to theoretical many body analysis can be

obtained from the soft sphere models above in the limit that the springs governing

the contact force become infinitely stiff while the collision time for the event goes

to zero [49]. In this limit, instantaneous momentum transfer occurs and the model

becomes one of inelastic hard particles. In this case, the relative velocity after the

collision is related to its pre-collision value through a collision rule of the form

gj = (1 + a (g) (a gi) a. (2-4)

where the parameter a is called the coefficient of restitution and characterizes

the energy loss during the collision. Notice that the coefficient of restitution is

in general a function of the impact velocity. This can be understood from the

fact that the processes that lead to dissipation in these interactions (for example

micro-deformation on the surface of the grain) have a characteristic threshold

energy associated with them and hence cannot occur when particles collide

arbitrarily slowly. A reasonable model system would be one in which the coefficient

a is a constant independent of the impact velocity up to some threshold value

go, below which it decreases monotonically to one, which is the elastic limit [19].

Hence, for a sufficiently activated fluid the collisions do not sample the domain

of the impact velocity dependence of a and it may be treated as a constant

parameter. This simplified inelastic hard sphere model will be the inter grain

interaction considered in the rest of this presentation. The rest of the chapter

focuses on characterizing the N body trajectories that result from this interaction

and the statistical mechanics associated with them.

2.2 Generators of Dynamics for Inelastic Hard Spheres

Consider a system of N mono disperse smooth hard spheres of mass m

and diameter a. The system is assumed to have periodic boundary conditions,

that mimic an unbounded infinite system. Also, the system is initially in an

activated state with some finite energy E. A complete specification of the

initial state of the system involves knowing a point in 6N-dimensional phase

space F {qi, vi; i = 1,... N}, that gives the positions and velocities of

all the particles in the system. The state of the system at a later time t is

completely characterized by the positions and velocities of all particles at that

time Ft = {ql(t),... qN(t), VI(t),... VN(t)} The dynamics consists of free

streaming (straight line motion along the direction of the velocity) until contact

and at contact, binary collisions that leave the center of mass velocity of the pair

unchanged but instantaneously change the relative velocity gj = vi vj of the pair

at contact to its post collision value g'j given by a collision rule

g = g- (1 + a) ( g) (2-5)

Here & is a unit vector directed from the center of particle j to the center of

particle i through the point of contact. The parameter a (the coefficient of normal

restitution) is chosen a priori in the range 0 < < < 1 and remains fixed for a given

system. As noted above, the value a = 1 corresponds to elastic, energy conserving

collisions, while a < 1 describes an inelastic collision with an corresponding energy

loss for the pair given by

E, -( -a2) 2. (2-6)

The center of mass velocity (vi + vj) /2 is unchanged so that the total mass

and momentum of the pairs are conserved for all values of a. Subsequent to

the change in relative velocity for the pair i,j the free streaming of all particles

continues until another pair is at contact, and the corresponding instantaneous

change in their relative velocities is performed. The sequence of free streaming

and binary collisions determines a unique trajectory in phase space, Ft, for given

initial conditions. The collision rule is invertible so the trajectory can be reversed,

although with the inverted collision rule ( i. -I I i I 'g" collisions).

The statistical mechanics for this system [42] is comprised of the dynamics

just described, a macrostate specified in terms of a probability density p(F), and a

set of observables (measurables) denoted by {A(F)}. The expectation value for an

observable at time t > 0 for a state p(F) given at t = 0 is defined by

(A(t); 0) J dFp(P)A(FP), -- {qi(t),..., qN(t), vi(t),... N(t)} (2-7)

For theoretical analysis the dynamics described above can be represented in terms

of a generator L defined by

(A(t); 0)- dFp(P)etLA(P). (2-8)

Such a generator for the dynamics is identified in Appendix A, with only the

results quoted here. There are two components to the generator L, corresponding

to the two steps of free streaming and velocity changes at contact,
N 1 N N
L v- Vq, + iT(i ) (2-9)
i= 1 i= 1 ji

where the binary collision operator is given by

T(i, j) = 0(-gij, q ,)|gg qj 16(q oa)(b,- 1). (2-10)

Here qij is the relative position vector of the two particles, is the Heaviside step

function, and bij is a substitution operator

(2-11)

bijA(gij) = A(', .gj) = A(g'j),

which changes the relative velocity gij into its scattered value g'j, given by Eq.

(2-5). The theta function and delta function in (2-10) assure that a collision takes

place, i.e. the pair is at contact and directed toward each other.

The dynamics can be transferred from the observable A(F) to the state p(F)

by defining an adjoint generator L

dF p(F)etLA(F) dF (e-tLp(F)) A(F). (2 12)

The form generated by L is referred to as Liouville dynamics. Implicit in the

discussion above for the direct dynamics of A(Ft) for hard particles is the

restriction of the phase space to non-overlapping configurations. This is assured

when the generator L is used in the left side of (2-12) since all acceptable

probability densities p(F) must exclude the domain of any pair overlapping.

However, the right side of (2-12) does not have that restriction on the domain of

integration and consequently the generator for Liouville dynamics is not the same

as that for observables (as in the case of continuous potentials). Instead, direct

analysis of (2-12) leads to the result (see Appendix A)
N N N
vi qi-(i,), (2-13)
i= i=1 j(i

with the new binary collision operator

T(i, j) = 6(qij a) gij .ij |(O(gij q ij)a -2b (-g, ij qij)). (2-14)

Here bN is the inverse of the operator bij in (2 11)

1+a
b7lgi = gS- (a gij) -. (2-15)

In summary, the dynamics of phase functions is given by

(216)

(9t L) A(F, t) = 0,

and the Liouville equation associated with distribution functions is

(a + L) p(, t) 0. (2-17)

As an aside, note that the form of the generator L and corresponding binary
collision operator T(i,j) does not depend on the details of the collision rule given
by the operator bij, and therefore applies more generally to a class of hard collisions
[51]. In particular, it is formally the same for both elastic and inelastic collisions.
In contrast, the generator for Liouville dynamics is obtained by a change of
variables that introduces the Jacobian of the transformation between the variables

gij and 71 Hence it depends explicitly on the restitution coefficient a.
Note that from the Liouville Equation in (2-17), a set of equations analogous
to the BBGKY hierarchy can be obtained that would characterize the dynamics of
the reduced distribution functions in the form

(at + )f(1) (qi, pl, ...qi, p, t) dqi+ldpl+lT (k, + l) f (+) (qi, Pl,...-q+l, P+l, t)
k= 1
(2-18)

where the reduced distribution function is defined as

(l)(qp, ...qpt)- NN -)..(N -1) fdqi+ldp+l...dqNdPNp (, t) (2-19)

and L is the same Liouville operator as earlier, but now for a system of I particles.
In particular, the first equation of the above hierarchy is

(a + v V7q) f(1) (q, pi, t) dq2dp2T (1, 2) f (2)(qi, P, q2, P2, t) (2-20)

In this form it is easy to see that the connection to kinetic theory of a granular
fluid is to be made by exploring possible functional relationships of the two particle
distribution f(2) to the one particle distribution f(). If some such functional
relationship is identified or postulated, the above equation becomes a closed
equation for the dynamics of the one particle distribution function and hence

becomes a kinetic equation associated with the granular fluid at low densities. Also,

the BBGKY hierarchy is useful to reduce some N particle expressions to two and

three particle expressions exactly, rendering them more tractable to approximations

and evaluations. These ideas are elaborated further in subsequent chapters.

This completes the characterization of the dynamics of phase functions and

distribution functions. A pause is warranted at this point, to note a few things

about the dynamics of the system as formulated above. First of all it is observed

that the binary collisions used here to describe the evolution of the system is at the

trajectory level and is distinct from the "uncorrelated binary collision assumption"

that goes into a kinetic theory of such a system. It is the analogue of pair wise

additive forces in the case of particles that interact through continuous potentials

and hence does not imply the neglect of correlations of any kind.

Next, there are two points to be noted about the generator of dynamics

involving binary collisions. Suppose the initial macrostate p considered above is

characterized by some finite average energy E, then it is alv--, possible that it

contains F points which involve more than two particles at contact and hence the

generator of trajectories is ill defined on such configurations. This is a problem

with elastic hard spheres as well. But the resolution lies in the fact that these

states have zero measure in the space of all such activated states for systems with

large number N of particles. Hence it is valid to neglect trajectories originating

from such initial F points when calculating the macroscopic value of an observable

A as in Eq.(2-12) above. That is precisely what is done implicitly when use is

made of the hard sphere generators of the form Eq.(2-9). The second problem is

one that arises dynamically and is exclusively associated with the dissipative nature

of the collisions for these systems. As stated earlier, for sufficiently activated (i.e.,

finite initial temperature) systems, configurations with three or more particles in

contact have zero measure. For a system of elastic hard spheres, this is sufficient

to guarantee that all trajectories involving three particle interactions will also have

zero measure. But, for inelastic hard spheres, there is an additional dynamical

effect called "( !! '1-" [33]. It is possible for a group of particles to undergo

infinitely many binary collisions among themselves in a finite time, so as to loose

all their kinetic energy and come to rest. When such collapse happens, then

there exists a possibility of a three body interaction occurring subsequently. The

statistical weight associated with such events, that is, completion of collapse

that leaves two or more of the particles involved at relative rest and in contact,

is unknown. For the purposes at hand, it is assumed that such events have zero

weight in the sense that a collapsing pair is broken up before the completion of the

collapse and that binary collisions alone are sufficient to generate a well defined

trajectory for all later times and this issue will not be considered further in this

work.

2.3 Isolated Homogeneous System

In the previous section, the N body trajectory associated with this system of

inelastic hard spheres was characterized through its generator and the language

of general macrostate dynamics was defined. The next step in setting up the

statistical mechanics is to identify the "prototype" ensemble or macrostate p that

should be studied and to characterize this state. In order to do this, consider

an isolated homogeneous system. First, observe that this system is inherently

non-equilibrium. This can be seen by considering the phase function associated

with the total energy E = i 1mv of the system. The dynamical equation

governing the time evolution of its ensemble average is

a dp (F) a ( ()
at at
= m'n/drp(F}Y Y 6(qij-a}(l-a2)
i j4i
xO (qij gij) (gij qij)3, (2 21)

where the right hand side is manifestly negative. Define a granular temperature

through the equation

T(t) mt v(t) (2-22)
3 Ni= 2
Using the equation above, the temperature is found to be monotonically decreasing
with

a~ InT(t) -((t), (2-23)

where ((t) > 0 is the "cooling rate due to inelastic collisions given by

mN [
((t) = (1 ) 12T(t) dF( 12 3(g 12 2)(q q a)p(F,t). (2-24)

This shows that there is no i''1. .II to equilibrium" for a granular fluid since

there is no such stationary equilibrium state. However there exists evidence

on theoretical (at the level of kinetic theory, for an illustration in the context

of a kinetic model see [52]) and numerical grounds [53] that a wide class of

initial homogeneous states rapidly approach a universal state that is spatially

homogeneous (translationally invariant) and all of its time dependence occurs

through the average speed T(t). This is known as the Homogeneous Cooling

Solution (HCS). The equation governing its dynamics can be obtained by

eliminating the time derivative in the Liouville equation Eq.(2-17) using the

time evolution equation of the temperature Eq.(2-23) above to give

L (t) Ph (, T (t)) 0 (2-25)

with the definition
1 OX -
L(t) X= --(h (t) T (t) + LX (2-26)
2 OT (t)
Since the system at hand is hard spheres, the only energy scale in the problem

is the kinetic energy of the particles. Hence, the only way this temperature

dependence can occur is through a scaling of the velocities. This allows the

homogeneous cooling state to be written as a scaling state in the form

Ph (T (t)) ( t,. (t))-3Np, 3N*() V, (t) V2Th(t (2-27)

where ', (t) is the thermal velocity and is a microscopic length scale (for example

the mean free path) chosen to non-dimensionalize the distances in the problem.

First of all, using the scaling form of the HCS in the defining equation for the

cooling rate (2-24) above gives directly

h (tI) (2-28)

where (* is now a dimensionless constant that depends on the parameters of the

model and the reduced density of the homogeneous fluid. Next, using this result in

the temperature equation (2-23) above, gives
t + (0) -2 2m)2
Th (t) Th(0) ( f+ (0h) 22> (2-29)
%h} (A2 t2(

Also given is the limiting behavior at long times, showing that the temperature

becomes independent of the initial conditions. This form is known as the Haff's law

for a freely cooling granular fluid [54] and is the signature of a HCS that is readily

recognized in numerical simulations. Also, with this scaling form the L operator

defined above now takes the form

-1 N
(t)X (h() Vv, ((v, u) X) + LX (2-30)
i=
This equation must be solved self-consistently with (2-24) for (h (t) = (h (t ph)

which is a linear functional of ph. It is easily seen by direct calculation that none

of the equilibrium ensembles for normal fluids are solutions to this equation, even

with a generalization to the scaling form (2-25).

One immediate consequence of the scaling nature of the HCS ensemble

is described below. Let f(P) be an arbitrary differentiable function of the

momentum. By momentum conservation

0 J dFp (t) Lf (P) J dF (Lp, (t)) f (P)

S(/) dF Vv, ((v u) Ph (t)) f(P). (2-31)
i 1
Integration by parts then gives finally

SdFph (t) (P mNu) Vpf(P) 0. (2-32)

Since f(P) is arbitrary this implies that the HCS distribution is sharp with respect
to the total momentum
(P mNu (7qij vi-u
h (t) (t) ( (t) t) (233)

In summary, the characteristic ensemble of an isolated homogeneous granular
fluid is the HCS ensemble. It is a time dependent non-equilibrium ensemble that
has its entire time dependence through the cooling temperature. The characteristic
time scale associated with it is the cooling rate (h. Also, this ensemble is sharp with
respect to the total momentum, i.e., each trajectory in this ensemble originates
from initial F points that all have the same momentum P. This completes the
characterization of the homogeneous reference state of this system. This HCS
is going to pl i, the same role in the statistical mechanics of the granular fluid
that the equilibrium Gibbs state does for a normal fluid, as will be seen in the
subsequent chapters.
2.4 Stationary Representation and Time Correlation Functions
In this section, some technical simplifications that are a consequence of the
choice of microscopic collision model made above are unfolded. Since the HCS state
of this hard sphere system is a scaling state, a simple change of variables can be
made in order to absorb the homogeneous cooling dynamics in the definition of the

new time scale in such a way that the homogeneous cooling state is the stationary

state of this dynamics. To see this, define a set of dimensionless variables

q V u (t)
qi -, V ds = dt, (2-34)

And, for any distribution p, define an associated dimensionless form as

p* ({q*, V*}, s)= (' (t))3Np(, t). (2-35)

In terms of these variables the dimensionless Liouville equation becomes

8sp* + p* = 0, (2-36)

with the new generator for Liouville dynamics

,* 1 N
i 1

It is readily verified that L* is time independent (see Appendix B). The dimensionless

time scale is a measure of the average number of collisions in a time t and is

obtained using (2-29) and (2-34) to be

2 1, 0
s= In (t + (-)t ) (t) e- (0). (2-38)

This dimensionless form for the Liouville equation, (2-36), supports the HCS

as a stationary solution

hp = 0. (2-39)

Consequently, in the following it is referred to here as the stationary representation

of the Liouville equation. This representation essentially involves partitioning

the dynamics of any distribution into two parts, one due to the cooling Th (t)

and the other is everything else. The homogeneous cooling of the temperature is

generated by the scaling operator Sp* -= A >i V* (Vhp*). Since, in the

HCS this is the only dynamics, it is the stationary state of the dynamics in this

representation. The usefulness of this stationary representation lies in the definition

of time correlation functions over the HCS ensemble, which is considered below.

For observables A and B and a macrostate p, the two-time correlation function
is of the form

CAB(t) (A(t)B; 0) = dFp(r)6A(rF)B(F). (2-40)

Rewriting the above expression in terms of the generators identified earlier gives,

CAB(t) = Jdp(C) (etLA(C)) B(F), (2-41)

or equivalently, using the adjoint representation of the dynamics

CAB(t) = d A(P)e-L (p(F)B(f)). (2-42)

More generally, correlation functions at three times can be defined but the

representation in terms of generators can be ambiguous. For example, it is easily

seen that L(AB) / (LA)B + A(LB) and consequently

A(Ft)B(Ft) = eL"(AB) / (eLtA) (eLtB) (2-43)

In the case of elastic systems, when time correlation functions over the equilibrium

state are considered, time translation invariance and the stationarity of the Gibbs

state are used to map the three time problem onto one of calculating a two time
correlation function as

(A(t)B (t') ; 0)e, (A(t t')B (0) ; t')e, (A(t t')B (0); 0)e, (2-44)

It turns out that the stationary representation for the dynamics described above

allows the same manipulations to be carried out in the case of time correlation
functions over the HCS ensemble. The results are stated here and the details of

obtaining these results are given in Appendix B.

First consider two-time correlation functions over the HCS. These can be
written as

(A(t)B)h-s J dfpr (, 0)Ar(F)B = dFA()e -tL (ph()B(P))

B (v (0)) /dF*A({q*, v(t)v })e-"S (p*(F*)B*(F*))

S B (v (0)) CA (v (t)) d*A*({q~, v })

xe-_8 (p*(F*)B*(F*)) (2-45)

In the last equality a change of variables to the dimensionless form has been made

and CA,B are the appropriate functions of the scaling variables that are required

to make the observables A and B dimensionless. Aside from a possible explicit

dependence on time through v(t), depending on the specific choice of function

A, the time dependence is now given by the Liouville dynamics in the stationary

representation. Here, s is understood to be s (t, 0). For homogeneous functions of

velocity as will be considered in the rest of the presentation,

CB (V (0))~ Vb (0) ; CA (V (t)) V' (t) (2-46)

for some a and b and hence

CB (v (0)) CA (v (t)) ~ eC-a(*sV+b (0) (2-47)

which amounts to an additional simple time dependence associated with the

cooling. Also, there exists an equivalent representation for the stationary state

correlation function given in Eq.(2-45) above as

(A(t)B)h = CB (v (0)) cA (v (t)) dF* (es*A*(F*)) p*(F*)B*(F*) (2-48)

where
N
*= L*- + ( Vvt (2-49)
i=i

Further, three time correlation functions can be written as

(A (t) B ('); 0) C (v (t')) CA (v (t)) / dr* (c A*(r*)) p;(r*)B*(F*) (2-50)

where the s is now understood to be s (t, t') The derivation of these results is given

in Appendix B. It can be inferred from the results above that it is sufficient to

consider a class of dimensionless correlation functions of the form

CAB(s) (A*(s)B*(0)), J drPf*p(P*) {e"*A* (F*)}B*(F*) (2-51)

I dF*A*(F*)e-* (pP(F*)B*(P*)). (2-52)

Finally, a special property of hard sphere distributions can be used to to

extract the "backward dyir iir, generator for phase functions. It is shown in

Appendix A that for any hard sphere distribution p (i.e., one with an excluded

volume factor multiplying it), there exists an operator L_ such that, for any phase

function B,

L(pB) = (Lp)B + p(L_B), (2-53)

where the new operator L_ is for the form
N N N
L_ = vi V, T_(i,j) (2-54)
i= 1 i 1 ji

T_(i,j) 6(q, a)O(gj, q- ,) |g, qj (b1 1) (2 55)

In the dimensionless variables Eq.(2-53) becomes

Z (p*B*) = (Zp*)B* + p*(C B*), (2-56)

where
1 N
C = L* + 2 vZY v Y;L L_. (2-57)
i=1

In particular, when the distribution is the HCS p*, the first term in Eq.(2-56)

vanishes. Using this fact in the stationary representation for time correlation

functions gives

(A*(s)B*(O)) d A(*)e (p*(*)B*(*))

dP*A*(F*)p (F*)e-s B*(P*)

S (A*(O)B*(-s))c, (2-58)

This representation will prove useful when comparison with results obtained for

elastic hard spheres is sought in the subsequent sections.

In summary, in this section, a new representation for the dynamics of the

system has been identified such that the reference homogeneous ensemble, namely

the HCS is stationary in this representation. Further, this has been extended to

time correlation functions over the HCS so that they become stationary state

time correlation functions. This is a technical simplification that allows a time

dependent non-equilibrium problem to be mapped onto a stationary state problem.

This mapping allows for several formal manipulations of the time correlation

functions that would otherwise be complicated and also facilitates comparison with

known results for normal fluids. This concludes the setting up of the machinery

required to do non-equilibrium statistical mechanics for a granular fluid.

2.5 Fluctuation and Response in a Granular Fluid: Some Observations

A central idea in the theory of of non-equilibrium statistical mechanics

is the Onsager regression hypothesis [55] and its manifestation in the form of

fluctuation dissipation relationships between time correlation functions and

response functions. There has been considerable attention given to such fluctuation

dissipation relations in the context of granular fluids in the recent literature

[56, 57, 58, 59, 60]. But the conventional results associated with the fluctuation

dissipation theorem are inherently tied to the special properties associated with

the reference state, namely the equilibrium Gibbs state. As established in the

discussion earlier, granular fluids are inherently non-equilibrium. Even when there

is a driving force that gives rise to a steady state, this state is very different from

the Gibbs state and hence the fluctuation dissipation relationships take different

forms. These considerations are elaborated in what follows.

First consider the Onsager regression hypothesis. Briefly stated, this can be

formulated as follows. Suppose a small initial macroscopic perturbation of the

reference equilibrium state is considered, then the hypothesis states that regression

of this perturbation must be the same as that of a spontaneous fluctuation in

the system and therefore, the response of the system to this perturbation is

characterized by a time correlation function over the unperturbed reference state.

Secondly, the response function characterizing the linear response of the system

to an external field is found to be related simply to a time-correlation function.

This fact has been used to derive various fluctuation dissipation relations between

equilibrium time correlation functions and response functions.

In order to see how the above statements translate to a context where the

reference state is not the equilibrium Gibbs state, consider a granular fluid in

a non-equilibrium homogeneous stable steady state po that is the solution to a

Liouville equation of the form

Lpo = 0 (2-59)

where the operator L is now considered to include the boundary condition or the

external driving force that gives the steady state for the dissipative granular fluid.

The particular details of such driving are not important for the consideration

at hand and hence are not specified explicitly here. In order to understand the

response of the system to external perturbations, suppose at time t = 0 an initial

perturbation is set up through two sources, 1) an initial preparation of a given

weakly inhomogeneous density field and 2) a conservative one body external force

that couples into the microscopic density function, for example a gravitational
force.
The initial preparation mentioned above gives a macrostate of the form

p (F, 0) po (F) + dr J dr'g (r r') 6y (r') 6n (r, 0)] (2-60)

where y (r) is an as yet unspecified phase function and the 6 denotes its deviation
from its average value in the po state which guarantees that

SdFp(F, 0) =-1 (2-61)

i.e., the initial preparation is properly normalized and

gy (r r') dFpo (F) 6y (r') n (r) (2-62)

which ensures that
SdFn (r") [p (F, 0) po] 6n (r", 0) (2-63)

i.e., the initial preparation indeed has the prescribed number density field.
The external force enters as a correction to the Liouville operator in the form

Lpert L + L1, (2-64)

where
Lip JdrV(r) I{(r) ,p} (2-65)

with n (r) -= 6 (r qj) and {, } being the Poisson bracket

{AB A B A ). (266)
9qi Opi Opi Oqj

The perturbing part of the Liouville operator can be rewritten as

Lip = drV (r) a(r) B -p

J- drF(r). 6 (r qi) (2-67)
j ^ pi
Solving the Liouville equation

( + Lpt)p (F, t) 0, (2-68)

to first order in perturbation in the external force, with the initial condition given
by Eq.(2-60) above, results in

p (F, t) e-Ltp (, 0) + dt'e-L(t-t') (-Lpo (F, t'))

= po (F) + dr J dr'g971 (r r') (e-Lty (r') po (F)) 6n (r, 0)

+ dt'e-L(t-t') (-Lipo (F)) (2-69)

where the stationarity of the po state with respect to the unperturbed Liouville
operator has been used. Now consider the average of the number density in this
state,

6n (r", t) = dr (r") [p (F, t) po ()]

07
= dr J dr'g- (r r') C (r' r", t) 6n (r, 0)

+ dt' drF (r) R (r- r", t t') (2-70)

where
C (r r', t) = dF (r') (e-Ety (r) po (F) (2 71)
C (r- r',7

is the correlation function that characterizes the relaxation of the system due to an

initial perturbation and

R (r r', t) = dn (r') e (6 (r qi) (2-72)

is the response function that characterizes the response of the system to an

external force. This is the most general form that can be obtained for a general

homogeneous stationary state.

Notice that the first part of the Onsager regression hypothesis comes out

naturally from the linear response procedure if restated as the relaxation from

an initial perturbation relaxes in the same way as a spontaneous fluctuation in

the reference state, for the response to such a perturbation is given by a time

correlation function in the reference state. But, the relationship between correlation

functions and response function is not so straight forward. In order to see this,

first consider the above two expressions when the reference state is the equilibrium

state. First notice that if the phase function y is chosen to be n then using the

conservation law for the number density, then, g-1 just becomes the inverse

of static structure factor S in the reference state po (see Appendix H) and the

response in the density takes the form

6n (r", t) = dr dr'S- (r -r') C (r' -r", t) 6n (r, 0)

+ dt' drF (r) R (r r", t t') (2-73)

Or, equivalently, this can be given a Fourier representation as

sn (k, t)= -1 (k) C(k, t) 6n (k, 0) + dt'F (k) R (k, t t') (2-74)

Further, when po = peq then,

-peq q (), (2-75)
opi m

and one can identify the second term in the response function as

-3 6 (r qi) Peq (F) -= (r, t), (2-76)

where j, (r, t) is the number flux in the continuity equation

aOt + V j = 0. (2-77)

Hence the fluctuation dissipation relationship is recovered in the familiar form

tC (r r',t) = Vr, R (r r',t) (2-78)

The point of the above demonstration is that the familiar form of the

fluctuation dissipation relationship hinges on the fact that the Gibbs state is of

the form e-3H where H is the generator of the dynamics of the system. And

hence this particular form of the relationship does not in general apply for other

stationary states of a system. Further, other forms of the fluctuation dissipation

relationship known to hold for equilibrium systems, like that relating the diffusion

coefficient to the mobility coefficient are known to take different forms for granular

fluids [61]. Hence, even though a large body of the methods developed for the

study of normal fluids can be extended and fruitfully applied to granular fluids,

results obtained from these tools cannot be borrowed and used without critical

examination of the derivation associated with them.

In what follows, attention is restricted to the use of these tools established

above for the specific problem of deriving a closed hydrodynamic description for

a granular fluid and hence obtaining expressions for the transport coefficients

associated with such a description in terms of time correlation functions. These

expressions are exact and still have the N body problem in them, but at the same

time are amenable to controlled analytic approximations and specific numerical

evaluations.

CHAPTER 3
PHENOMENOLOGICAL HYDRODYNAMICS

In the preceding chapter, the tools necessary to use non-equilibrium statistical

mechanics in the study of granular fluids were established. The role of statistical

mechanics is to provide a bridge that links the macroscopic phenomena in a system

to its microscopic roots. The macroscopic description that is the objective of this

analysis is a hydrodynamic description of a granular fluid. In this chapter, such

a hydrodynamic description is obtained on phenomenological grounds in order to

identify the target quantities for the microscopic analysis that follows in subsequent

chapters.

Hydrodynamics in the most general context can be defined as a closed

description of the dynamics of a system in terms of its -! .v- vo ii ,i [62].

It is typically applicable on length scales long compared to the characteristic

microscopic length and time scales long compared to the characteristic microscopic

time (which in the case of fluids correspond respectively to the mean free path

and the mean time between collisions). On this scale, the slow variables are

conventionally the conserved quantities and the broken symmetry variables. This

can be seen by noting that the dynamical equation associated with a conserved

variable x is a "continuity equation" of the form

8x (r, t)
( + V f (r,t) 0, (3-1)
at

i.e., the time derivative is proportional to a gradient. Assuming that the system

approaches an equilibrium homogeneous state, it follows that the time scale

associated with the relaxation of this variable diverges as the system approaches

homogeneity and hence the dynamics persists after times long compared to the

mean free time, which characterizes the decay time of the other variables in the

system. A phenomenological route to obtain a hydrodynamic description of a

system is 1) identify the conserved quantities of the dynamics, 2) write down the

conservation laws for these variables, 3) propose a closure criterion that allows

the unknown fluxes in the above conservation laws to be expressed in terms of the

conserved densities, called the constitutive relations for the system. For example,

for a normal fluid, the locally conserved quantities are the number density, the flow

velocity and the energy density or the temperature and the constitutive relations

are the Newton's viscosity law and Fourier's law for heat transport. This gives the

Navier-Stokes order hydrodynamic equations for a normal fluid.

An analogous phenomenological derivation of the hydrodynamic equations

associated with a granular fluid is given in this chapter. The context of hydrodynamics

for a granular fluid has to be revisited carefully because this is an inherently

non-equilibrium system with no notion of approach to equilibrium associated with

it. As will be seen in the following, the -! v.--" variables in this system are not all

conserved quantities. Hence, care must be exercised to identify the precise sense

in which these hydrodynamic equations are complete in the long time limit. The

route taken to address these problems is the following. First, a set of hydrodynamic

equations are derived with the assumptions that go into the derivation stated

clearly along the way. Once the resulting description has been written down, it is

analyzed to give the context and domain of validity for the assumptions used to

obtain the description itself.

3.1 Nonlinear Navier-Stokes Hydrodynamic Equations

The macroscopic variables with respect to which a closed description for the

dynamics of a granular fluid is sought are a prior chosen to be the number density

n (r, t), the momentum density g (r, t) and the energy density e (r, t). Notice that

the first two fields are locally conserved quantities, but the energy density is not.

The starting point for deriving such a description are the macroscopic balance

equations associated with these fields given by

On(r, t) g(r,t)
+ Vr o (3-2)
at m
agi(r,t)
t +Vrhj(r,t) 0 (3-3)
Be(r, t)
e(r + V7r s(r,t) = w(r,t) (3-4)
at

where hij is the momentum flux, s is the energy flux and w (r, t) is the function

characterizing the local loss in energy due to the dissipative nature of the

collision between the granular particles. From a purely macroscopic point of

view, these equations are taken to define the fluxes and the source term. Hence

the specification of the fluxes is arbitrary up to terms of zero gradient and the

separation into source and flux in the energy equation is not unique. But if these

equations are viewed within the context of statistical mechanics as ensemble

averages of the corresponding microscopic conservation laws given in Table C-l in

Appendix C, then the fluxes and the source are unambiguously defined. This will

be the view point taken here.

As for normal fluids it is usual to replace the momentum density by the flow

velocity U (r, t) defined through the relation

g (r,t) mn (r,t) U (r,t). (3-5)

If a local rest frame is defined for the fluid at each point r through a Galilean

transformation r r U (r, t) t, then, a temperature T (r, t) is defined using the

energy density of the fluid in the local rest frame through the definition

1
e (r, t) -mn (r, t) U2 (r, t) e o (n (r, t) T (r, t)) (3-6)
2

Here eo (n, T) is some specified function of n and T. The two most common

choices are eo (n, T) = 3nT/2, or eo (n, T) = e, (n, T) the thermodynamic

function for the corresponding equilibrium fluid. The former is common in

applications of computer simulations, while the latter is the historical choice in

most formulations of hydrodynamics. For both normal and granular fluids, the

choice made constitutes a 1/. I,.:l/.: of temperature for non-equilibrium states and

has no a priori thermodynamic implications. With these definitions, the above

balance equations can be recast in the form

Dtn + nV U 0, (3-7)

DtUi + (mnn)- jPij = 0, (3-8)

o l(D+()T+ co- IT V -U+P j U + V.q 0, (3-9)
OT On
where Dt = Ot + U V is the material derivative that is the time derivative in

the local rest frame, Pij is the momentum flux in this reference frame, called the

pressure tensor

Pij = hij nUiUj, (3-10)

and q is the corresponding heat flux, identified as

qi si ( + o 2nmU )U PijU,. (3 11)

Further, in the rest of the presentation, the choice eo (r, t) = n (r, t) T (r, t) is

made. This gives the temperature equation in the form

(D + () T + (Pij j U + 7V q) = 0, (3-12)
3n

with the cooling rate defined through the relation

2
S(r, t) T (r, t) (r, t) (313)
3n (r, t)

These equations have the same form as those for a normal fluid, except for the

cooling rate that represents the dissipative nature of the interaction between the

particles of this fluid.

Hydrodynamics is a closed description in terms of the hydrodynamic fields.

But inspection of the microscopic analogues of these fluxes tells us that these local

rest frame fluxes are in general complicated functions. The closure of the above

balance equations with respect to the hydrodynamic fields is made by i-ii:

that the fluxes become normal functionals of the hydrodynamic fields after a short

aging time, i.e., the pressure tensor P takes the form

Pij (r, t) Pi [n (r, t) T (r, t) ,U (r, t) (3-14)

and it depends on space and time only through the hydrodynamic fields. Similar

forms are postulated for the heat flux and the cooling rate as well. This closure

assumption effectively converts the balance equation Eqs.(3-7)-(3-8) into

hydrodynamic equations for the five fields chosen to be the hydrodynamic variables

of interest. This assumption is the phenomenology involved in this derivation of

hydrodynamics.

One can give a physical interpretation for the basis of this assumption as

follows. Consider the granular fluid in an inhomogeneous state. Visualize the

system as being partitioned into cells such that each cell has enough particles so

that multiple collisions can occur in times short compared to the time taken by

a typical particle to traverse the size of the cell, but the size of the cell is small

compared to the whole system. Now, notice that the hydrodynamic variables

here are the course grained versions of precisely those microscopic quantities that

prescribe the collision rule between grains. Hence a statistical average of these

quantities over only the particles within the cell will not change except through

boundary effects that include collisions with particles across the defining walls of

the cell. With this picture, it is easy to see that, at times t such that t is large

enough that several collisions have taken place within the particles of the cell, but

small enough so that only a small number of collisions have occurred across the

walls, the only quantities in each cell that would not have relaxed yet would be

the hydrodynamic variables and any other observable will acquire its space and

time dependence through these fields only. This is the regime of normal dynamics

in which the closure assumption above is a reasonable one and the hydrodynamic

equations are expected to be the relevant dynamical equations of the system.

Further, when the gradients in the system are small, the functional form of the

fluxes can be expanded in powers of the gradients of the hydrodynamic fields. The

Navier-Stokes hydrodynamics entails retaining terms up through second order in

the gradients to give the constitutive relations for the fluxes as

Pi p(n, T)6 I(n, T) + + OU d u 1(nT)6jV. u (3-15)

q- -A(n, T)VT-p(n, T)V (3 16)

S- 0 (n, T) + ((n, T)V U+T(n, T)V2T

+("(n, T)V2n + (1 (VT)2 + (2 (wn)2 + 3 (VT) (Vn)

+4 (ViUj) (ViUj) + (5 (ViU) (VjU) (3 17)

Note that the cooling rate is required to second order in the gradients, while

the pressure tensor and heat flux are required only to first order. The pressure

tensor has the same form as Newton's viscosity law for a normal fluid, while

the expression for the heat flux is a generalization of Fourier's law. These

expressions include the unspecified functions p(n, T) and (o (n, T), as well as

unknown transport coefficients (U(n, T), (T(n, T), ("(n, T) and so on, the shear

viscosity qr(n, T), the bulk viscosity K(n, T) the thermal conductivity A(n, T),

and the new coefficient associated with heat transport in a granular fluid, p(n, T).

All of these must be provided by experiment or the theoretical justification of the

above phenomenology.

This completes the phenomenological identification of a set of hydrodynamic

equations for a granular fluid. The rest of the chapter will focus on the linear

hydrodynamics about a homogeneous state. But, in closing this section, some

remarks are made. Although the Navier-Stokes equations are based on the small

gradient forms for the constitutive equations, it does not mean that they are

limited to systems close to a homogeneous state. They are applicable locally

over domains larger than the mean free path even when the hydrodynamic fields

still vary significantly throughout the system. Consequently, a wide range of

experimental and simulation conditions for granular fluids have been well-described

by the Navier-Stokes equations (for example see [63]). Therefore, getting reliable

estimates of these hydrodynamic transport coefficients provides a tractable

theoretical handle on a large body of experiments done on fluidized granular

materials.

3.2 Linearized Hydrodynamics

In this section, the hydrodynamic equations above are considered in the

context of weak spatial inhomogeneities. As a first step, the homogeneous state

associated with these hydrodynamic equations is characterized. Then, the linear

about the nature of response predicted by these equations and the stability of

the homogeneous state. As a first step, consider the homogeneous limit of the

Eqs.(3-7)-(3-9) above. These are of the form,

aOnh 0, atUh 0, (O + Co (n, Th (t))) Th(t) 0, (3-18)

that is, the homogeneous state has a constant density, a constant flow velocity and

a homogeneous temperature that is cooling with time with a characteristic cooling

rate (o. Consider the cooling equation in particular. Notice that the underlying

fluid is one of hard spheres. This is a system with no characteristic internal energy

scale. So, the only energy scale in the problem is the temperature. This, together

with the fact that the cooling rate is a normal functions of time, leads to the fact

that, on dimensional grounds,

S(o nT(t)) =(( h) T/2 (t) (3-19)

Eq.(3-19) is the macroscopic analogue of Eq.(2-28) in the previous chapter. Hence

the solution to Eq.(3-18) above is indeed the Homogeneous Cooling Solution

considered in the previous chapter in the context of statistical mechanics. This

homogeneous dynamics of the temperature is the same as identified in Eq.(2-29) in

the previous chapter

Th (t) T(0) ( ( (3-20)

Thus the homogeneous cooling state is the solution to the homogeneous hydrodynamic

equations. This is the first distinction from the hydrodynamics of normal fluids in

that the homogeneous reference state that emerges is inherently time dependent.

Next, the equations (3-7) through (3-9) are linearized about this homogeneous

state. Without loss of generality, pick the constant homogeneous flow velocity Uh

above to be zero. For this purpose, introduce dimensionless variables of the form

n nh T Th (t) U (321)
6n* 6= T* = rU* = (3-21)
nhTh (t) I ,(t

The dimensionless space and time scales are the same as those introduced in the

previous chapter through Eq.(2-34),

r* =- ds ) dt. (3-22)

In this form the coefficients in the linearized N ,',i. r-Stokes equations are constants,

independent of space and time. The subsequent analysis is most conveniently done

in terms of a Fourier representation with respect to space

(k, s) J dr*eikr* y*(r*, s) (3-23)

where the set {yI} are chosen to be

y (Sn*, T*, k-* U*, 6 U*, e2 6 (3-24)

Here 81 and e2 are unit vectors orthogonal to each other and to k, defining the

transverse flow field components. The linearized N ,1.-. i-Stokes equations are then

identified as

(63a + /)3 (k, s) 0, (3-25)

where the transport matrix Chyd is found to be block diagonal with a "longitudinal"

part corresponding to a, f = 1, 2, 3, given by

0 0 -1k

-hp* di k -ikV lik* + (4 +n*) k*2
alnnh l_ 2 ] *
(3-26)
The I1 in-., -," components decouple from the longitudinal degrees of freedom in

the above equation and are given by

h + T*k*2 0
Sd(k*) l=2 a, ,= 4, 5 (3-27)
0 + r*k*2

The dimensionless transport coefficients are defined by

h = h, C =(u, T= b n (3-28)

2 2 1 1
A* = A, p* = 2* = K* = PK. (3-29)
dvhnhf dvhTh/ r,,I,, (I mnhj,
This completes the derivation of the linearized N ,. i.. -Stokes equations for small

deviations from the homogeneous state. Notice that the transport coefficients

(1 through (5 have dropped out in the linearized equations. These equations

contain the information about the response of a homogeneous granular fluid to long

wavelength perturbations. The unknown parameters in these equations will be the

target of the microscopic linear response cin iiJ-; that is carried out in subsequent

chapters. In the last section of this chapter, the above equations are analyzed with

the purpose of establishing the notions associated with approach to hydrodynamics

for a granular fluid. Also, the stability of the underlying homogeneous state

scrutinized.

3.3 Hydrodynamic Modes and Stability

The hydrodynamic equations given above were derived from the macroscopic

balance equations for the variables number density, flow velocity and temperature.

As noted earlier, in the case of normal fluids, such a description is expected to be

"complete" on long time scales, for these were locally conserved variables and hence

the time scale for relaxation of long wavelength perturbations in these variables

was divergent. This rationale has to be modified in the context of granular fluids

because the energy density is no longer a locally conserved variable.

For the purpose of understanding the choice of these variables, consider the

linear hydrodynamic equations above. The transport matrix at k = 0 is

0 0 0 0 0
s aIn (h (-* 0 0 0
9nnh 2
Kchyd (0) 0 0 0 0 (3-30)
2

0 0 0 ;h 0
2
0 0 0 0 -

Recall that the homogeneous reference state about which these equations are

linearized is time dependent and cooling. This time dependence was accounted for

by making a nonlinear change of variable in time so that the cooling is incorporated

in this choice of time scale and the homogeneous state is stationary. The /Chyd (0)

identified above is the dynamics associated with homogeneous perturbations
of the reference state. This is a trivial dynamics in the sense that it is just the
cooling of the new homogeneous state as described in the variables in which
the original reference homogeneous state is stationary. Further notice that in
the case of the hard sphere granular fluid considered here, the analogue of the
property Ot oc Vr that characterizes the hydrodynamic variables for normal fluids
is (981 + /Chy (0)) oc Vr, i.e., the hydrodynamic variables here are those whose
characteristic time of approach to this residual dynamics diverges in the long
wavelength limit. It is in this sense that the temperature can be treated on the
same footing as the flow velocity and the number density.
Next, further understanding of the content of the hydrodynamic description
and the fact that the hydrodynamic response of a granular fluid is very different
from that of an elastic fluid can be obtained by considering the hydrodynamic
modes for this system. The eigenvalues and eigenvectors of the generator for this
dynamics, /Chy, defines the five Navier-Stokes order hydrodynamic modes

/Chyd (k);() 7 (k)( (k), i = 1..5. (3-31)

The eigenvalues of the matrix /Chd are determined by the cubic equation

73 72 ( + ) k2 + (A* (*) k2

( 422 2 1 (2,* 2 1 9,lnp]
( + -* + K) k + ( k p
S2 2 3 2 3 U 2 alnnj
1 ln (h 1 l1np]
+kp (h nnh + = 0 (3 32)
k'P 2 8 Innh 49Inn

and the decoupled shear modes are solutions to the equation

+ *k*2 0 (3-33)
27

If the limit a -- 1 is taken for this equation, then, its solution to order k2 give

the familiar hydrodynamic modes associated with normal fluids, namely the two

propagating sound modes, the heat mode and the two transverse shear mode [37].
But, when the solution to the above equation is considered for a / 1, to order k2
they are

A1) (k) (- In h p k2, (3 34)
h(i*OIn nh u
A (2)* 2p* 8* In (h 2, p
A(k) ( A*-(-2 nh ))kk (335)

A 3(k) + 2 h + ( 2 k2 (336)
2( 2(Q 3 AInnh In nA

+ + k2,

1 1
X(2')(k) --(h 2 (3 37)
2 2
Notice that all the eigenvalues are real and hence there are no propagating modes

in the system, and the a -- 1 limit of these modes do not correspond to the

familiar hydrodynamic modes of a normal fluid. The drastic difference in the

nature of the hydrodynamic modes obtained as the elastic limit of the above

eigenvalues is due to the non-analyticity of the eigenvalues and eigenvectors about

the point a = 1 and k = 0. Close to the elastic limit, (* which goes as (1 a2)

(see Eq.(2-24) in the previous chapter) and k are small parameters and the type
of modes obtained depends on how these parameters approach zero [26]. This
is an indication of the fact that the inelasticity, even when small, gives rise to

drastically different transport in the fluid. But, for the purposes at hand, attention

is restricted to the a / 1 forms of these modes.

Next observe that there exists a critical wavelength k' defined by

k, (3-38)

such that for k < k'g the shear mode becomes unstable. Similarly there exist

threshold wavelengths associated with some of the other modes such that

these modes become unstable as well. This implies that the homogeneous state

characterized by these hydrodynamic equations is unstable to sufficiently long

wavelength perturbations that excite these modes. This instability of the HCS to

these long wavelength modes has been well established in the literature [64, 65].

The above statement is the mathematical content of the linearized hydrodynamic

equations. What this implies physically is that the response due to the unstable

modes grows until such time as the linear theory breaks down and further analysis

of the dynamics has to be carried out using the full nonlinear theory in the

previous section [66].

Finally, further insight into the nature of the hydrodynamic response of this

fluid can be obtained by looking at the eigenvectors of the transport matrix /ICh

that excite the above hydrodynamic modes. To lowest order in k these are found to

be

W6T + 2 J% I2inh < k U*, *e2 e- J* (3-39)

The first of these modes is excited when the condition

0 In nh

This can be interpreted as follows. The cooling rate (h(Th, nh) has the form

(h(Th, nh) = T 2(h( fh). It then follows that this condition for exciting the first
mode corresponds to variations in the temperature and density that leave the

cooling rate constant. This is the first manifestation in this presentation of the

novel coupling between the density and temperature fields in the granular fluid

through the cooling rate. The second mode in Eq.(3-39) is due to a temperature

perturbation at constant density, while the third is due to a longitudinal velocity

perturbation at constant temperature and density. The last two are the response to

a transverse velocity perturbation, again at constant temperature and density.

This concludes our analysis of the response of the homogeneous state to small

spatial perturbations. Recall that the unknown quantities in the hydrodynamic

equations above are the pressure p, the cooling rate (, the transport coefficients

shear viscosity r1, the bulk viscosity K, the thermal conductivity A, the p coefficient,

and the source transport coefficients (u, (T and (,. The subsequent chapters

focus on identifying these transport coefficients exactly from the underlying

micro-dynamics in the form of time correlation functions over the HCS ensemble.

Further, note that although this is done in the context of linear hydrodynamics,

these transport coefficients are the same functions of the density and temperature

in the nonlinear equations and hence the results can be used in the non linear

equations as well.

CHAPTER 4
LINEAR RESPONSE

Hydrodynamic transport coefficients are those quantities that characterize

the response of the fluid to gradients in the hydrodynamic fields internal to the

fluid. For normal fluids, the method of linear response has been successfully

used to get exact expressions for these transport coefficients starting from

the microscopic theory [37] in the form of time correlation functions over the

homogeneous reference state. Thus, non-equilibrium statistical mechanics allows

the identification of transport coefficients in terms of the time dependence of

fluctuations in the equilibrium ensemble just as equilibrium statistical mechanics

allows the identification of thermodynamic quantities such as the specific heat

in terms of the ionp!Ilude" of static fluctuations in the equilibrium ensemble.

For example, the viscosity of a fluid is determined by the time integral of the

autocorrelation function of the momentum flux at a time t with the momentum

flux at the time t = 0 (see Eq.(1 1) earlier). The key idea that makes such

identifications possible is Onsager's hypothesis that perturbations relax in the same

way as a spontaneous fluctuation in the system.

As was shown in C'! lpter 2 earlier, the Onsager regression hypothesis, when

suitably reformulated, works for granular fluids as well. Hence, it is expected that

there exist representations for the hydrodynamic transport coefficients of a granular

fluid in terms of time correlation functions of fluxes over the reference state.

Identifying the general method for obtaining such an expression for the transport

coefficients is the objective of this chapter.

The structure of the presentation is as follows. First, the formal procedure of

using linear response to extract the hydrodynamic response of a fluid and hence

obtain a microscopic representation for the transport coefficients is outlined.

Then, some technical simplifications that render this formal procedure analytically

tractable are pointed out, in the form of identifying a special initial preparation

that excites only the hydrodynamic modes in the long wavelength limit. Finally,

a formal expression for the hydrodynamic transport matrix that was identified on

phenomenological grounds in the previous chapter is given in the form of a matrix

of time correlation functions. Further unfolding of this matrix to identify the

detailed forms of each of the transport coefficients is reserved until a later chapter

in the presentation.

4.1 General Procedure of Linear Response

In this section, the basic procedure of the linear response analysis associated

with the response of the fluid to internal gradients is formulated. For this purpose

notice that, as identified in the previous chapter, the homogeneous hydrodynamic

state is the one with a constant density and flow velocity, and a temperature that

cools such that

Th (t) Th(0) (i ) (41)

This is the Homogeneous Cooling State. The N particle statistical ensemble that

corresponds to this macrostate is the HCS ensemble identified and characterized

in C'! Ipter 2 in Eqs.(2-25)-(2-33). Next, note that the linearized hydrodynamic

equations given by Eq.(3-25) characterize the response of the fluid to weak

inhomogeneities in the hydrodynamic fields with respect to the HCS. This

response can be captured at the level of statistical mechanics by the following

procedure. Let the initial ensemble be a weakly inhomogeneous ensemble, with its

inhomogeneities parameterized by the hydrodynamic fields, that is

P =Ph + drb,(F; r)6y, (r), (4-2)

with

y, (r) {6n (r) T (r) 6U (r)} (4-3)

where

6x = x Xh (4-4)

is the deviation of quantity x from its value in the HCS. Thus the y,'s are

precisely the variables whose dynamics was given in Eq.(3-25). Further, the
functions b,'s are such that the initial distribution p is normalized and does indeed

have the prescribed hydrodynamic fields. That is,

SdFb(F; r) 0 ; J dFa (F; T, r') b(F; r) 6 (r r') 6, (4-5)

The first condition above makes p properly normalized. In the second condition
above, the a,'s are given by

a, (F; T, r) h ((; r), (F; r) eo,n (F; r)) g (F; r) (4-6)
eO,T mnh

where
S(F; r) ( 1

(F; r) =pj + E (qij) 6 (r -q), (4-7)
i= 1
g (; r) Pi
are respectively the microscopic number density, momentum density and energy

density, and
0eo aeo
eo,n = IT, eO,T = n (4-8)
On OT
with eo being the chosen function that defines the temperature for the system

(see Eq.(3-6) in C'i plter 3 earlier). These are precisely the phase functions whose

ensemble average are the linearized hydrodynamic fields 6y,'s, that is,

6y, (r) dFa, (F; T, r) 6p (F) ; 6p (F) p (F) (F). (4-9)

So the second condition in Eq.(4-5) guarantees the requirement that the initial
ensemble p does indeed have the prescribed values for the hydrodynamic fields.

Further, as with the hydrodynamic equations earlier, the choice eo = nT is made

to give

a, (F; T, r) h (F; r), (F; r) 3T (F; r) (; r) (4-10)
31, 2 mMnh

The specific choice for {b,(r)} is left arbitrary at this point.
At a later time t the ensemble is obtained as a formal solution to the Liouville

equation Eq.(2-17), in the form

p (F, t) h (F, t) + dr e-b(F;r)) ya (r). (4 11)

Now taking averages with the set of functions {a,} over the above ensemble a

response equation is obtained in the form

6ya (r, t) = dr'C,3 (r, t; r', 0) y3 (r', 0) (4-12)

with the response function C being given by

Ca (r, t; r', 0) J- dFa (r') (e-tb (r')). (4 13)

The response function is a time correlation function over the HCS ensemble, like
the ones defined and characterized in C'i lpter 2.

Before further analysis, it is useful rewrite the above equation in the stationary

representation introduced earlier in Eq.(2-34) by transforming to the dimensionless
variables

q i v U ,(t)t
qs = V ds dt. (4-14)

The hydrodynamic fields are non-dimensionalized with their values in the HCS
ensemble

n*, =7T* = 5U* (4-15)
n n T Th(t) U ,, (415)

The phase functions {ao(r)} that are the observables of interest in the dimensionless
form are
az (n(* jn ),g*) (4-16)

where the superscript denotes the non-dimensional quantity with

( i*,,g*) -= e(r) ,-- g ). (4-17)
(nh nhTh nhmvh

Also, it is useful to take advantage of the translational invariance inherited from
the homogeneity of the HCS to introduce a Fourier representation through the
definition
(k, s) dreikx(r*,s). (4 18)

Using these, Eq.(4-12) above can be rewritten as

5y (k, s) C p (k; s) 5by (k, 0), (4-19)

with

C~p (k; s) = dF*a (k) (e _(-k)) (4-20)

where explicit use has been made of the fact that the time correlation function C is
only be a function of r r', due to the homogeneity of the HCS. This dimensionless
response equation (4-19) will be the focus of study in the rest of this chapter.
To identify the macroscopic hydrodynamic equations from this expression, it
is useful first to rewrite the response equation (4-19) in the form of a transport

equation similar to (3-25) in the previous chapter,

(69,3 + KCt (k, s)) y (k, s) = 0, a,= 1...5. (4-21)

A formal expression for ICa (k, s) is readily identified from (4-19) as

K, (k, s) = 0,C0, (k; s)) C' (k; s). (4-22)

As argued in the previous chapter, one expects that the hydrodynamic description

identified in Eq.(3-25) is the complete description of the dynamics of the system

in the limit of wavelengths long compared to the mean free path and at times

long compared to the mean free time of the particles in the fluid. Hence, the

hydrodynamic matrix given in Eqs.(3-26) and (3-27), when it exists, follows from

this formal result for small k (long wavelengths) and long times,

hId (k) lim kp (k, s) (4-23)
s->oo,k<<1
lim (c/3 (0, s) + ik (k.VkK/C (k, s)) +..)). (4-24)
s 00 \ \ \ / k=0 / /

Comparison of this expression with the forms (3-26) and (3-27) not only provides a

"derivation" of the linear hydrodynamic equations, but also gives the coefficients of

those equations in terms of the response functions.

That completes the formal prescription of extracting the linear hydrodynamic

transport matrix from the full microscopic dynamics. Note that the above

derivation of hydrodynamics has been accomplished without any constraints on the

functions ba that characterize the perturbation chosen in the initial state, beyond

normalization and the moment conditions associated with the hydrodynamic

variables. It turns out that the tractability to analytical and numerical analysis

of the results obtained in Eq.(4-24) above rests on using the degree of freedom

afforded by the flexibility in the choice of ba to simplify the process of the long

wavelength expansion described earlier. In the following section, a special choice

of bay's is made and the simplifying properties of this choice in the analysis of the

transport matrix pointed out.

4.2 Special Initial Preparation

When the linear response analysis is carried out for normal fluids, it is found

that analytically tractable expressions are obtained for the different transport

coefficients when the initial preparation is chosen to be the linearized local

equilibrium ensemble [37]. This section identifies the analogous simplifying initial

preparation for a granular fluid. This is done in the following way. First, a special

homogeneous solution to the Liouville equation is identified that is related to the

HCS in a simple manner. Then, this homogeneous solution is generalized for weak

inhomogeneities and the special initial preparation required for the linear response

procedure to be applied to a granular fluid is identified. Also, the particular

simplifying features of such a choice are elucidated.

4.2.1 Special Homogeneous Solution to the Liouville Equation

Recall that the HCS ensemble is a solution to an equation of the form

S*p (f*) 0. (4-25)

where

Sh 2 (Vv V +L (4-26)

This equation is used in Appendix D to identify a set of functions T, (F) such that

they have the property

: IC-h (0) q (4 27)

where K T (0) is the transpose of the generator of the hydrodynamic equations in

the homogeneous limit identified in Eq.(3-30) and 'T's are the functions

ayah (t) (4 28)

that is, they are derivatives of the HCS ensemble with respect to its parameters,

which are the homogeneous hydrodynamic fields.

Now suppose an initial homogeneous perturbation to the HCS ensemble of the

form

p (F, 0) Ph (F, 0) + T, (F) 6y, (4-29)

is considered, with the perturbing hydrodynamic fields 6ya are understood to be

homogeneous. Then, this ensemble at a later time, in dimensionless form is

p* (F*, s) p* (F*) + (e-ihydT (o0) (*, 0) y. (4-30)

where use has been made of the property of the ''s given in Eq.(4-27) above.

Calculating the ensemble average of the volume integrated forms of the functions

aa's given in Eq.(4-16), that is,

aa J drt*a (F*; r*), (4-31)

gives the result

6y* (s) 6(e (O)s (e- hyd(o0)s) J:. (4-32)

which is precisely the solution to the homogeneous hydrodynamic equations. In

getting the above result, use has been made of the readily verifiable fact that

SdF*a*= (P*)= 6. (4-33)

This implies that a perturbation of the form given in Eq.(4-29) gives rise to a

dynamics that is the microscopic precursor to the macroscopic hydrodynamics of

the fluid. In other words, such a perturbation gives rise to a purely hydrodynamic

response in the fluid at all times.

4.2.2 Local Homogeneous Cooling State Preparation
In this subsection, the form of the initial preparation in Eq.(4-2) is chosen so
that in the homogeneous limit, it reduces to the special solution to the Liouville
equation characterized in Eq.(4-30) above. Then, the simplifying properties of
such a choice are enumerated. For this purpose, first define an initial local HCS
distribution. This distribution is similar to the local equilibrium distribution for
elastic collisions and represents a system that has the HCS form locally but with a
point wise varying temperature, density, and flow velocity fields. Formally, the local
HCS is obtained from the HCS of Eq.(2-27) in C!i lpter 2 by the transformation

( -3N V -)U vi U ) 2Th(qi)
i,1 (qD) m
(4-34)
so
S(0) ( (q 0)) 3 P ( U (q, 0) } ( 0) (4-35)
I h 0(q,, 0)
where the dependence on an inhomogeneous density field n (qi, 0) has now been
made explicit. Clearly, the initial HCS is recovered for spatially homogeneous
temperature, density, and flow fields. Then, if the choice for the b,'s in Eq.(4-2) is
Jplh (f, 0)
b, (F; r) Ph (F, 0) (; r) (436)
6y, (r, 0)r) (4 36)
then, in the homogeneous limit this will correspond to the choice that gives the
special solution to the Liouville equation characterized above, that is

S1' (r) T, (F). (4-37)

Suppose this choice is made, then the transport matrix defined in Eq.(4-22) above
becomes,
ICa (k, s) = (8,C (k; s)) C (k; s) (4-38)

with

Co, (k; s)= V* d*a (k) e- (-k). (4-39)

This is the form of the transport matrix and response functions that will be

considered in the rest of the presentation.

4.2.3 Simplifying Properties of the LHCS Preparation

In this subsection the particular properties of the above choice of initial

preparation that simplify the analysis of the transport matrix are identified. As a

first step notice that it follows from Eq.(4-30) and Eq.(4-37) above that at k = 0

the transport matrix in Eq.(4-38) reduces to

IC, (0, s)=- ( (0; s)) C- (0; s) = K/C (0), (4-40)

that is, the transport matrix is purely hydrodynamic at all times in the homogeneous

limit.

Next observe that, in order to go from the full transport matrix to the

hydrodynamic transport matrix, two limits have to be carried out, one that of

long wavelengths (i.e., k -i 0) and that of long times (i.e., s -- oc) as stated in

Eq.(4-23) earlier. The special initial preparation is such that the time derivative

of both the functions ?7 (k) and ,, (-k) have an intrinsic k ordering that can be

used to k order the transport matrix so that the long time limit can be taken in a

simple manner. This can be seen as follows. First, it is shown in Appendix D that

the phase functions 2,7 (k) obey a dynamical equation of the form

atd (ks)
S + (0) a* (k, s) ik f (k,s) 6p* (k, s), (4-41)

with

ff (r, s)- {*,( s* *) h (4-42)
3 \ /

where g is the momentum density and hence the number flux, s is the heat flux ,
hg is the microscopic momentum flux and I is given by

~2 3 2* ( 2 9In6 (h
(k, s) (k, s) + -e (k, s) n (k,s) + (h 1 n (k,s)
(4-43)
In the above equation, w* is the dimensionless form of the source in the energy
balance equation given in Table C-2 in Appendix C. As shown in Appendix D, the
homogeneous part of the source term has the property

1(O, s) (1 P) 2 (0). (4-44)
3

The operator P in the above equation is a projection onto the set of functions

{(4's given by

PX (F*) = (0) f d*X (r*) ( (0)

(0) dF*X (F*) T3 (F*) (4-45)

Thus, the source term is orthogonal to the homogeneous part of the initial
preparation. So, if this balance equation is used to obtain a dynamical equation for
the correlation function C then,

9,C, (k; s) + Cay (0) C, (k; s) ikiD, (k; s) = S (k; s) (4-46)

with

Dn (k;s) 1=V dF*f (k, s) (k) (447)

and

So (k; s) = JdF*T (k, s) (k). (4-48)
San~ ~ (kJ)6a-*7

As a consequence of the orthogonal projection identified in the homogeneous limit

of *, it is clear that S,0 (0; s) = 0 and therefore Eq.(4-46) implies

(a896, cy (0)) C, (k; s) oc ik (4-49)

Therefore, the special initial preparation allows the time derivative to be used to

expose the intrinsic k ordering of the correlation function.

Similarly, it is shown in Appendix D that the adjoint functions ', obey a

dynamical equation of the form

'(k, s) + K (0) (k, s) + ik 7 (k, s) 0 (4-50)

with the definition

ik 73 (k, s)= (Z6 ChyT (0) (k, s) (4-51)

Thus, the fact that the initial preparation gives a purely hydrodynamic response in

the homogeneous limit translates into the fact the dynamical equations associated

with the direct and adjoint functions can be used to k order the transport matrix

formally before the long wavelength or long time limit is taken. In what follows,

Eq.(4-41) will be referred to as the direct conservation law and Eq.(4-50) will be

referred to as the adjoint conservation law.

Lastly, it is claimed that the initial preparation guarantees that all the time

dependent quantities in the k expanded transport matrix are orthogonal to the

invariants of the generator of the dynamics and hence the long time limit turns out

to be patently well defined. This property is elaborated upon in later sections of

this chapter.

Summarizing, a special initial preparation has been characterized above such

that it satisfies the properties 1) in the long wavelength limit, the transport matrix

KC is purely hydrodynamic at all times, 2) the operator (a8I + /CKh (0)) acting

on both the direct and adjoint functions turns out to be proportional to ik and

hence can be used to expose the natural k ordering present in the transport matrix

prior to the hydrodynamic limit being taken and 3) this initial preparation makes

the long time limit of the k expanded transport matrix patently well defined for

all the time dependent quantities turn out to be orthogonal to the generator of

the dynamics. This section is concluded by remarking that these are precisely the

properties that are associated with the local equilibrium preparation of normal

fluids that lends the transport matrix in the hydrodynamic limit to be expressed in

a tractable form.

4.3 k-Expansion of the Transport Matrix

In the preceding section, the correct initial preparation needed to extract

the hydrodynamic response was identified and a formal expression for the full

transport matrix in terms of the response function was given. Further, several

simplifying properties associated with this choice of initial conditions were noted.

In this section, advantage is taken of the above mentioned properties to carry out

a long wavelength expansion of Eq.(4-38) above. Here, three representations for

the various hydrodynamic parameters are given. Each representation is formally

equivalent to the other two. Each in turn is suitable for either interpretation of

structure and content, developing different analytical approximation schemes or

numerical evaluation schemes. In order to simplify the notation and expose the

relevant structure without distractions, all the Greek indices associated with the

different hydrodynamic fields are suppressed in the following.

As a first step, the most direct method of carrying out the k expansion is

given. For this purpose, define

C(k, s) = () (s) + ik]C1) (s) + (ik)2C(2)(s) + ..

(4-52)

and

/C(k, s) = C() + ikC(1) + (ik)2C(2) +.. (4-53)

where C 1), K1) and so on depend on the unit vector in the longitudinal direction

k, but this dependence is left implicit here in order to simplify the notation. As has
been established above /C() Chyd (0) at all times. Then the transport matrix, to

Euler and Navier-Stokes order turns out to be, respectively,

K (s) lim [-( ((, + hd (0))C(1) (s)) ekhyd(O)s (454)

K,(2) (s) lim (-(a + K" (0))C (2) (_) (1s) C (1s)) Chd()s (455)
s>>O(

where explicit use has been made of the order k = 0 results. The matrix elements

in Eq.(4-54) contain the expression for the hydrodynamic parameters that include

the pressure p and the transport coefficient at Euler order, namely (u. The

matrix elements in Eq.(4-55) contain the expressions for the shear viscosity Tr,

bulk viscosity K, and the transport coefficients associated with heat transport,

namely A, p, (, and (T. These are formal expressions defined in turn through terms

in the k expansion of the elements in the bi-orthogonal set that consists of the

functions {aa}'s and {' }'s defined earlier. The above route is the most direct

way to obtain explicit forms for the elements of the transport matrix up through

the Navier-Stokes order and result in Helfand forms for the various transport

coefficients in a form most suitable for numerical evaluation. But, these expressions

are not transparent with respect to the formal structure of the result. In order to

illustrate the content of these expressions and interpret the structure, an alternate

route that makes explicit use of the conservation laws and balance equations

associated with these variables is described below.

The key idea in carrying out a k expansion of the transport matrix in such

a way that the results are amenable to interpretation and theoretical analysis in

terms of approximate evaluations is the following recognition. The hydrodynamic

variables in the system are those which have the property that (89, + K/Ch (0)) oc
ik. The microscopic precursors of the hydrodynamic fields, namely the phase
functions {aa}'s and the adjoint functions ( s also have this property within
the correlation functions associated with the special initial preparation detailed
above. Notice that the expression for the transport matrix in Eq.(4-38) has one
time derivative. Hence, the procedure used to k order the transport matrix would
be to introduce a second time derivative using an identity of the form

X (s)= ds'sX (s') + X (0) (4-56)
Jo

on the time dependent quantities in IC, and eliminate each of the time derivatives
in favor of a gradient using the special properties associated with the bi-orthogonal
set {a3}'s and {,' }'s that occur in the response function C. Details of carrying out
this prescription are outlined below.
As a first step, one can use Eq.(4-46) above to rewrite the transport matrix in
Eq.(4-38) in the form

KC (k, s) ""C (0) i (k; s) + () k; s)] (k; s) (4-57)

which essentially involves using the direct conservation laws to evaluate the first
time derivative. Note that the k = 0 form of the transport matrix has been
extracted. Before proceeding further, it turns out to be useful to introduce an
integrating factor exp (Kh""dT (0)) in the adjoint conservation equation that
absorbs the k = 0 dynamics of that is, define

i (k, s) = exp (Chyd (0) s) (k, s) (4-58)

Then, Eq.(4-50) above becomes

9.' (k, s) + ik. (k, s) 0

(4-59)

with

k 73 (k, s) e exp (J (0) s) ( b* (C" (0)) (k, s) (4-60)

Next, Eq.(4-39) above is re expressed in terms of correlation functions involving
I's to give

C (k, s) /d (0) [ikD (k; s) + S (k; s)] 1 (k; s) (4-61)

where

C3(k) d (k)e(k; ) (khyd()s)

V* dF*; (k) ; (k,s). (4-62)

Similarly,
V ) 1 f -
Dn, (k; dF (k) (s) ) d (k) ( k (4-63)

and
-S (k; s) =62 V* dFr* (k) (-k, s). (4-64)

Further, using form of the adjoint conservation laws in Eq.(4-59) above for the
dynamics of /, the following equations for the response functions above can be
obtained.
a,C3 (k; s) ikiE, (k; s)= 0 (4-65)

,D3 (k; s) ikFj (k; s)= 0 (4-66)

s9,q (k; s) kikN (k; s) 0 (4-67)

where
a, (k; s) = dF*a (k) 7 (-k, s) (4-68)

F (k; s) f dF* (k) 7* (-k, s) (4-69)

(k; s) = -62 d*l (k) r* (-k, s) (4-70)

Now consider the transport matrix as given in Eq.(4-61). Use an identity of
the form in Eq.(4-56) to write

D (k; s) j ds',,D (k; s') + D1 (k; 0) (4-71)

and similarly for S (k; s) and C (k; s). Then use Eqs.(4-65)-(4-67) to eliminate
these time derivatives to obtain the transport matrix as

/C (k, s) /hYd (0) {kkiD (k; 0) k2kjk ds'T (k; s')

+S (k; 0) + ik ds'N (k; s')}

x I iki ( ds' (k; s')) -1 (k; s) (4-72)

This completes the formal implementation of the prescription described earlier
which exposes the inherent k order present in the full transport matrix. Notice that
any perturbative expansion is yet to be done, and all the manipulations done so far
are exact. This intrinsic k ordering is present in KC due to the special nature of the
dynamics of the bi-orthogonal set {ap}'s and {(~p}'s as manifested by the direct
and adjoint conservation laws in Eq.(4-41) and Eq.(4-59) above.
In the remainder of this chapter, the expression (4-72) is considered explicitly
at Euler order and at N ',i. r-Stokes order to identify the various parameters in
the hydrodynamic matrix and to elucidate the structure of the result. Also, in the
process two new forms for the hydrodynamic parameters are identified that are
equivalent to the forms in Eq.(4-54) and Eq.(4-55) earlier.
In order to simplify the presentation of the results that follow, introduce the
notation

/C (k, s) = F (k, s) + Cs (k, s)

(4-73)

where the superscript F denotes the terms coming from the fluxes in the direct

conservation laws and hence correspond to the parameters in the constitutive

relations for the fluxes given in Eqs.(3-15) and (3-16) in C'! plter 3, namely

the pressure, the shear and bulk viscosity, the thermal conductivity and the p

coefficient in the heat flux term. The superscript S denotes the terms coming from

the source in the direct conservation equations (4-41) and hence correspond to

the parameters in the constitutive relation in Eq.(3-17), namely the transport

coefficients (u, (, and (T. Each of the above parts of the transport matrix are

considered in turn, first at Euler order and then at N .',1 i-Stokes order, and the

structure of the results obtained is discussed.

4.3.1 CF at Euler Order

This is the part of the transport matrix that contains the hydrostatic pressure

and the density and temperature derivatives of the pressure. By direct examination

of Eq.(4-72), it can be recognized that the flux part of the transport matrix /CF at

Euler order is

1CF(1) A
1i) D-kD, (0; 0)

V* dF*fa* (0) (-0, 0) (4-74)

In order to better interpret the content of the above expression, it is useful to

rewrite the expression in laboratory variables. The details of doing this are given in

Appendix G. But for the purposes here, it is sufficient to recognize that it has the

form

K(1) -da, (Tb (t)) Vk, dF (0) )' (0; T: (t)) (4-75)

where da, is a pre-factor determined by the dimensions of the quantities fo and ,

depending on time through the temperature. The temperature dependence of' ,

has been made explicit in this notation. Also, recall that

h (Th (t)) (476)
(4-7b)
0h, (t)

This allows Eq.(4-75) to be rewritten as

S 1) = -d (Th (t)) k f (0); t (4-77)

Thus, the Euler order terms in the constitutive relations (3-15) and (3-16) are
derivatives with respect to the homogeneous hydrodynamic fields of the ensemble

averages of the volume integrated microscopic fluxes over the HCS ensemble
at a time t. For the case of normal fluids, these terms are determined by the

derivatives of the ensemble average of the fluxes over the equilibrium state [37].

Hence the same structure is retained here, with the HCS state taking the place of
the equilibrium state. The only time dependence is that of normal time dependence

through the cooling homogeneous temperature, which is precisely as was found in
the case of the linearized hydrodynamic equations. Note that this is true for all

s, that is no long time limit needed to be taken to make this Euler term purely

hydrodynamic. Explicit phase functions involved and further interpretation of these
results for specific terms such as the pressure are given in the next chapter.
4.3.2 /IC at Navier-Stokes Order

This part of the transport matrix contains the transport coefficients shear
viscosity, bulk viscosity, thermal conductivity and the p coefficient. Terms at order
k2 that arise from the correlation functions associated with the direct fluxes are

grouped together to get

F(2) k1) (0; 0)

+ ds' {1 (0; s') VD (0; 0) (0; s')}} (4 78)
Jo /S

The content of this expressions is as follows. Putting in the various correlation
functions defined above through Eqs.(4-63)-(4-70) gives the result (see Appendix F
for details)
1 i f d- ,i ,(1)
F(2) V* df (0) (0)

-kik ds' J dPf*f* (0) (1 PT)

x e_ h 7*d (0) (4-79)

where 1) is the first order term in a k expansion of (k) and pT is adjoint of
the projection operator defined in Eq.(4-45), given as

PTX -) (0) dF *f (0) (*), (4-80)

and the generator of time dependence of the adjoint flux 7 has been made explicit.
Notice that & (0) are the invariants of the dynamics generated by ( I lChT)
and the (1 pT) operator projects orthogonal to these invariants. Next, attention
is drawn to two features of the above expression. First, the (1 PT) in the second
line Eq.(4-79), results in the following identity (See Appendix F for the details),

(1- pT) e_*-khydT )r 7* (0)= (1 eT) *f1hyd' )} (1- pT) 7* (0)
(4-81)
Hence the time generator (e *IhdT) ) acts on a quantity that is orthogonal to
its invariants and hence the time dependent quantity in the correlation function can
have a well defined long time limit as claimed at the end of the previous section.
Summarizing, the transport coefficients in Eqs.(3-15) and (3-16) are identified
through the relation
1-" {- ~ W- (
S liml{ V ikj df*fP* (0)F() (0)

-kik, f ds' { drF** (0) ( T) (0s')}} (4-82)

where the lim above denotes the thermodynamic limit, namely V -- 00 and

N -- oo such that n = N/V is a constant and the long time limit of s -- oo.

Thus, the expression for the transport coefficients that occur in the constitutive

relations associated with the hydrodynamic equations consists of a flux-flux

correlation function. Notice that the property in Eq.(4-81 allows the introduction

of a projection operator that subtracts out the components of the flux along the

invariants of the dynamics, ensuring that a long time limit exists. This is known as

the Green-Kubo form of the transport coefficients.

The time independent first term in the above expression remains to be

interpreted For this purpose, observe that the dynamical equation associated with

* (k) Eq.(4-59) yields the identity

8 *Wlj(0, s)
O~ ) (0, s) (4-83)
as

Hence, Eq.(4-79) can be written in the alternate form

1 -t ~ )
CF(2) V kikj d*f* (0) () 0)

+kk ds' JdPf* (0) (1 pT) 8 (0 (4-84)

This form allows the recognition of the first term as the value of the integral in

the second term at the lower limit. In the case of normal fluids where the particles

interact through continuous potentials, this term is zero and the Green-Kubo

expression for the transport coefficient reduces to just the flux-flux time correlation

function characterized earlier. The reason that such a term is present here is

technical. It is associated with the fact that the form of the correlation functions

for s ~ 0+ and s ~ 0- are different. There are two causes for this difference. One is

that the system consists of hard particles and the dynamics is discontinuous about

s = 0 because of instantaneous momentum transfer present in the collision model.

An analogous term exists for elastic hard sphere transport coefficients as well. The

second cause is again a discontinuity in the dynamics of the system about s = 0,

but this time due to the fact that the collisions dissipate energy and hence the

direction of time in the reference state is no longer arbitrary. This implies that such

an instantaneous piece is alv--x present in Green-Kubo expressions for granular

fluids, even when the underlying collision model involves a continuous potential

(like the soft sphere models defined in Eq.(2-1)).

Finally, as shown in Appendix F

PT*( (0) = 0, (4-85)

and hence, without loss of generality Eq.(4-84) can be written as

/CF(2) lkk df **(o)( p (0) PT)_l) (0)

fs t (0, s')
+kik, ds' df ** (0) (1 T) as'

And therefore the time integral can be carried out to obtain

K/C''"" lim -kiik, 1 dPf* (0) (1 pT) J) (0, s). (4-86)
V* J

This latter expression is also called the Helfand form for the transport coefficients.

This equivalent form turns out to be the more convenient quantity for numerical

evaluation of these transport coefficients using Molecular Dynamics simulations and

for developing approximate analytic evaluation schemes using kinetic theory (see

C'! lpter 6 later in this presentation).

In summary, it has been shown here that the Green-Kubo form for the

Navier-Stokes order transport coefficients associated with the flux terms has the

properties that 1) it has a part that is a direct flux adjoint flux time correlation

function, 2) the generator of dynamics in this correlation function acts on a

quantity orthogonal to its invariants and hence has a well defined long time limit

and 3) it has an instantaneous part that partly arises due to the instantaneous

momentum transfer associated with hard sphere models and partly due to
the inelasticity of the grain collisions. Finally, a Helfand form of the transport
coefficients has been identified in Eq.(4-86) above.
4.3.3 ICs at Euler Order
This is the part of the transport matrix that contains the source transport
coefficient that occurs at Euler order. From the expression in Eq.(4-72), grouping
terms that arise due to the source in the direct conservation laws, up through Euler
order, one gets
S-k ds'Ni (0; s') + S (0; 0) (4 87)

Substituting for the forms of these correlation functions gives (Appendix F has
some intermediate details)

4/C ki- ds ds f d ) (-0, s') + df T (0) ^1' (0) (4-88)
v j Jo J J74

Recall that the term 1(0) has the property that

S(0) = (1 -) W* (0) (4-89)

Hence, the above expression can be written in the equivalent form

K^(1) 1 ds dF (0) (t1 pT) 7 (0, s')

+ (0) ( PT) )i* (0)} (4-90)

Comparison of Eq.(4-90) with Eq.(4-82) shows that the long time limit of this
correlation function is well defined as the time dependent quantity is orthogonal to
the invariants of the generator of this dynamics. Further, as shown in Appendix G,
spherical symmetry of the HCS gives that the only nonzero entry of the type above
is for l = 3. Examining the phenomenological transport matrix in Eq.(3-26) in the

previous chapter, it can be concluded that

S= lim{- kt ds' dF* w* (0) (1 pT) (-0, ')
V Jo J o
d+k- (0) (1 T) (T) (0)} (4-91)

This is the Green Kubo expression for the transport coefficient occurring at Euler

order, again the limit being taken as described earlier.
Further, as earlier, the time integral can be carried out to get

5 limk3- dP*~* (0) (1 p) 3 (0,s) (4-92)

This is the Helfand form for this transport coefficient. Note that in this case a long

time limit has to be taken in order to obtain the hydrodynamic form, unlike the
Euler terms that come from the flux part of the transport matrix. In the case of

normal fluids, the Euler level hydrodynamics is entirely non-dissipative, that is,

involves no entropy producing processes. The collisional loss in energy in the case
of granular fluids gives rise to this transport process at Euler order.
4.3.4 /Cs at Navier-Stokes Order

This is the part of the transport matrix that contains the representations of
the transport coefficients (, and (T in terms of time correlation functions. Again,

starting from Eq.(4-72) this part of the transport matrix can be identified as

,/CS(2) k i { (1) (0; s') Kl () (s) (0; s') + (2) 0). (4-93)
-0 l 1

When the forms of the different correlation functions in the expressions above,
which are unfolded in Appendix F, the transport matrix is identified as

K/2)C kikj ds' dF*l) (0)* (-0, s') + d *i)i (0) (0

+k kj sds'- dFi(0) 7 ) (-o, s') + dF* (0) (2)ij (0)

As there are several terms here, it turns out that the structure of the result is
easier to see when the time integrals are carried out to get the Helfand form of this
part of the transport matrix. This gives

S(2) ksk { d *l (l) o (l(0, s) + dF* (0) (2)i (0,
-( (0, s) +s)}
( / j
-k-ky (* J dF*F (0) 1 (0, s) dF*3 (0) ((0, s)

The terms in the first line above the the direct part of the transport coefficients
while the terms in the second line above is the subtracted part. Notice that th th
subtracted part of the transport coefficient now is a product of two time correlation
functions. This kind of structure occurs for normal fluids for transport coefficients
at Burnett order in the k-expansion. In the case of granular fluids, this occurs at
Navier-Stokes order because of the transport coefficient that is present at Euler
order in the k-expansion.
The structure of the Green-Kubo forms for Burnett transport coefficients
is not well understood even for normal fluids. Hence the source part of the
Navier-Stokes transport matrix will not be considered further in the present
work.
4.4 Summary of Results
In this chapter the formal identification of the elements in the hydrodynamic
transport matrix in the form of Green-Kubo and Helfand expressions that are time

correlation functions over the HCS ensemble has been carried out. These are the

primary results of this work. Therefore this chapter is concluded by enumerating

the key results in it. The general method to extract hydrodynamic response of a

fluid from the solution to an appropriate initial value problem was described.

1. A special homogeneous solution to the Liouville equation was identified and
characterized. This solution has the property that its dynamics is completely
given by the linearized hydrodynamic equations in the homogeneous limit.
Equivalently, it can be stated that the dynamics of this state is completely
specified by the hydrodynamic modes of the fluid.

2. A special initial state was chosen such that, in the homogeneous limit, it
reduced to the special solution above. This was the local homogeneous cooling
state. Then, the properties of this initial state that result in the simplification
of the subsequent linear response procedure and the process of formally
identifying the hydrodynamic limit of the transport matrix were identified.

3. The formal result for the hydrodynamic transport matrix was derived at the
Euler and Navier-Stokes order and it was shown that the hydrodynamic limit
is well defined in each case, that is, the long time limit existed.

In the next chapter of the presentation, the specific phase functions that enter

into the formal results above for each of the transport coefficients is identified

and the resulting expression compared with the known results for normal fluids

to shed light on their physical content and hence the implications to transport in

these systems. Also, in ('! Ilpter 6, a kinetic theory of time correlation functions

is developed that makes contact with the various expressions for these transport

coefficients that have been obtained from Boltzmann and Enskog kinetic theories in

the literature.

CHAPTER 5
TRANSPORT COEFFICIENTS

In the previous chapter, the primary results of this work were derived. They

are exact expressions for all the parameters in the phenomenological hydrodynamic

equations that were given in C(i lpter 3. For the transport coefficients, these exact

expressions are of the Helfand and Green-Kubo forms, time correlation functions

over the HCS ensemble. The technical content of these expressions were partially

discussed and the general structure of the result was elucidated.

In this chapter, further unfolding of these exact results is given by making

explicit the different phase functions involved and by comparing the results with

their corresponding analogs in the elastic limit, where applicable. Recall that the

linear hydrodynamic transport matrix was identified in Eq.(3-26) in C'!i Ipter 3 and

0 0 -ik
(k*) ln ((p* (*) -*2 + (A* (*) *2 (p*
Qha 4 +nnh++
I,*l V *i* + (4T]* + v*) k*2
aP In nh 2
(5-1)
for the longitudinal degrees of freedom corresponding to a, = 1, 2, 3, and the

decoupled transverse part was

Chyd (k*) (

+ + rl*k*2

0

0
S+ T*k*2
2

The unknown parameters in this equation are the cooling rate (h, the pressure p,

the Euler order transport coefficient (u, the shear and bulk viscosity rl and K, the

thermal conductivity and diffusivity A and p, and the two transport coefficients

a, = 4,5

(5-2)

from the source at Navier-Stokes order, (, and (T. Each of these quantities is

identified from the corresponding formal expression in the previous C'! ipter and

discussed in turn.

5.1 Homogeneous Order: The Cooling Rate (h

The homogeneous dynamics of this fluid is parameterized by the cooling rate

in the Homogeneous Cooling State. As a first step note that the cooling rate in any

macrostate p was first identified in Eq.(2-24) as

((t) (1 a212T(t) dP(g1 12 )3(g12 2)(q12 )p(F,t). (5-3)
12T(t)

In particular, when the ensemble in question is the HCS, this can be rewritten in

the stationary representation defined in Eq.(2-34) as

S- c (t) (1 a 22) d (gq)3gT q a2)( 2 o*)p*(F*) (5-4)
h- (t) ( 6 dr* *9A, ( ),

Further, since the cooling rate is determined by only a two particle function, use

can be made of the hierarchy of distribution functions associated with the HCS

ensemble, as given through Eq.(2-18) earlier and written as

h = (l-a2) (nh3)2 d12(g29122)3 (g12*q2)(2-j2 )f' (q2 2,v v) (5-5)

where

(nh (2 ) 2, v, v) N (N 1) dqdv ...dq* dv p() (5-6)

and the notation giving f(2) as a function of q,2 makes use of the fact that the

HCS must be invariant under translations. Lastly the integral over the two particle

phase space can be recast into integrals over the relative and center of mass

coordinates. Using the 6 function in the above expression that puts the two

particles at contact to do part of the relative coordinate integral, one obtains

(h 1 d( ) dg ,e12(g2.- )T3 (g,2-*) dP,12 f2) (-*, v, v) (5-7)

Notice that the cooling rate is generated by a function of only the relative

velocity of pairs at contact and is obtained as an average over the two body

distribution function at contact. In particular, it is the average of the third

moment of the normal component of the relative velocity of colliding pairs. This

is the microscopic expression associated with the parameter characterizing the

homogeneous cooling dynamics of the unperturbed and homogeneously perturbed

hydrodynamic states of this system.

5.2 Euler Order Terms

In this section, the transport matrix at Euler order is examined to identify

the parameters to this order in the hydrodynamic equations. First, recall that

the constitutive equations (3-15) and (3-16) used to obtain the hydrodynamic

equations were written down taking into account the underlying symmetries of

the fluid, namely homogeneity and isotropy. These same symmetry properties

characterize the homogeneous reference ensemble as well. These symmetry

properties can be used to conclude that the various terms at Euler order that

are zero (like the 21 matrix element which would correspond to a term proportional

to the density gradient in the dynamical equation of the temperature fields) are

indeed vanishing when calculated from the statistical mechanical prescription as

well. As an illustration consider the 21 matrix element at Euler order arising from

the flux part of the transport matrix. This has been identified in Eq.(4-74) as

C21 -ki21 (0; 0) (5-8)

where

D21 (0; 0) (j- df ** (0) T (0). (5-9)

As was stated earlier, it is useful to consider these correlation functions in the
laboratory variables. Putting in the form of the flux f7* (0) and restoring the
dimensions to the above equation gives

1 / 1 a2 (
j721 (0; 0) = 1 dF s g Ph (t). (5 10)

Now notice the homogeneous cooling state is taken to have zero flow velocity when
the density derivative is being carried out. Hence the ensemble average in the
above expression is that of a vector over a homogeneous and isotropic distribution.
Therefore, this integral must vanish. Similar arguments show that each of the
zero terms at Euler order follow from the statistical mechanical derivation as well,
including the vanishing of Euler order terms in the transverse components and the
decoupling of the transverse modes from the longitudinal modes. The details of
these arguments are given in Appendix G.
5.2.1 Pressure

In what follows attention is restricted to the case of the non vanishing elements
only. Notice that for the flux part of the transport matrix, they are all related
to the pressure of the fluid. Consider first the case when the observable is the
temperature, namely / = 2 in Eq.(5-8) above. The only non zero term in this row
in the D matrix at this order is the 23 matrix element.

D23 (0; 0) V* dP*f* (0) 4 (0)

V* 3 (o) (0) ( ) (511)

As stated in the previous chapter, a physically interpretable form of this result is
accessible if it is transformed back to the real variables. Restoring the dimensions

to the various quantities gives

(1) i VT71 (t fd( ( (t)N
K3F(1) -k D3 (0; 0) kk (t ) nh gUj g p (t) 3
(5-12)

Now, the homogeneous cooling state in the above equation is one with a finite flow

velocity U. It is recognized that the flow velocity enters into the homogeneous

cooling state through the velocity of the particles, i.e.,

Ph (U) h ({v U})) (5 13)

Hence, one can change variables in the F integration as vl vl U, and use

the transformation properties of the heat flux and the momentum under Galilean

transformations to get

(1) 21 1 1
K23(1) 3 V T (t) h df (0) h (T (t))(5-14)
3 3 V Th (t) nh 3

The details of this transformation can be found in Appendix G. Comparison with

the hydrodynamic matrix in Eq.(5 1) above gives the identification

nTh (t) p* p j= / dfhi (0) ph (Th (t)). (5-15)
V 3

This gives us the definition of the hydrostatic pressure as the average of the trace

of the volume integrated microscopic stress tensor over the HCS ensemble at

a time t. Further, as was done for the cooling rate above, putting in the form

of the flux h given in Table C-l, transforming the expression to the stationary

representation and carrying out the F integrals to map this onto an average over

reduced distribution functions over the HCS gives the result that the pressure is the

sum of two terms in the form

p* =p + p,

(5-16)

where

P (h 3 f dv* (vI) v (5-17)

and

1 (nh,)2 1 (2)
PV -3 4- (1 a) ldqdvtdqidv{f 2) (q12, V, v )

x5 (q42 a*) 0 (-q 2 g2) (q2" g12)2}. (5-18)

The first term above in Eq.(5-17) is the kinetic part of the pressure that arises

purely from the transport of momentum associated with the free streaming of the

particles. It can easily be seen that the kinetic part of the pressure pK gives

PK = 1 (5-19)

and hence

pK = nhTh (t) (5-20)

which is the full pressure in the limit of low densities when the collisions become

infrequent and is indeed the full result for ideal gases. Next, the second term above,

determined by the two particle distribution function at contact, is the "potential"

part of the pressure that arises due to the interaction between the particles, which

in the case of hard spheres is generated collisionally. Now focus on the collisional

part of the pressure pc. As earlier, this can be rewritten by transforming to relative

and center of mass variables to give

(nh f3)2 1 + a) 3 2 g2)(a 2
PC' 3 4 dg2(9 91)R-9*

which identifies the collisional part of the pressure as generated by the average

of the second power of the normal component of the relative velocity of colliding

pairs over the two particle distribution function. Also notice that the center of

mass velocity of the pair P12 l' 1 no role in the collisional part of the pressure, as

should be expected.

As an illustration for the structure of this quantity, let us consider its

evaluation in the elastic limit. When a -- 1,

f (2) f (2)*
2) -,V,) V (a,, V ,V ) g (a )r-3 -1/2(v2+v2)) (5 -22)

where g (a) is the pair distribution function at contact and it depends only on the

magnitude of a. Evaluating Eq.(5-21) above, in this limit gives

(rh 3)2 *3
PC =--i a 3g (a) 327 (5-23)
3

Hence, the collisional part of the pressure is proportional to the density, and

at high densities, dominates over the kinetic part of the pressure. Further, it

is reasonable to expect that pc for the inelastic system has a similar behavior

as well, in the sense that the pair correlation function at contact would still be

isotropic, being a largely geometric quantity. The coefficients of the density will

be modified by the inelasticity, through the coefficient of restitution a, and the

velocity correlations that will be present in the two body distribution function.

This completes the characterization of the microscopic definition of pressure

obtained from this analysis. Also, it follows easily that the terms in the matrix

elements associated with the longitudinal flow velocity field, namely the 31 and

the 32 element, which are the density and temperature derivative of the pressure,

come out consistent with the above definition of the pressure. The details of this

are given in Appendix G.

5.2.2 Euler Transport Coefficient (u

The last parameter in the hydrodynamic equations at Euler order that remains

to be characterized is the transport coefficient (u that occurs in the dynamical

equation of the temperature field. This coefficient is identified in Appendix G

starting from the formal expression given in Eq.(4-91) earlier to be

(* = lim{ ds' J dF*W* (*) e(se- (* A3 Mii
11
1 JdF*W*A/Mi} (5-24)
V* 3

where

W* (*) (0) + ( E* N* + (, + 1 N* (5-25)
3 2 3 n nn

with

1 1(1_ a2) Y-- 9-(5 -26)
(0) n- (1 a2) (q~ (* g (g .q)3 (q o*) (5-26)
1 m4l

being the source in the energy balance equation given in Table C-l and E is the
total energy of the system given by

E* v2,' (5-27)

and also
N
N* (5-28)

Also, the adjoint density in the above expression is

MA -E q;* (V7*p), (5-29)

a space moment of the velocity derivative of the HCS. This is the Green-Kubo form
of the Euler order transport coefficient associated with granular fluids.

As a first step in unfolding the content of this expression, it is observed that
the phase function W was obtained by unfolding the action of (1 P) on the

energy source w. And recall that

PX (F*) (0) f *X (*) i (0), (5-30)
7) X

i.e., 1 P projects out the homogeneous dynamics. This allows the function W

to be interpreted as the function characterizing that part of the rate of change of

energy that is not due to the homogeneous dynamics of the underlying state, i.e.,

not due to the cooling reference homogeneous temperature. This is what would be

expected given that (u characterizes the relaxation of spatial perturbations to this

homogeneous dynamical state. Next, in order to characterize the adjoint function,

consider M in the elastic limit, i.e., when the coefficient of restitution a goes to 1.

In this limit, the homogeneous cooling state goes over to the equilibrium ensemble,

which gives the result

Mil a qliv P:q (5-31)

which is a space moment of the momentum density. It follows that the flux 7

becomes the longitudinal part of the momentum flux h given in Table C 3 in the

appendix times the equilibrium distribution function. Now, in the case of inelastic

systems, it will be a different function, reflecting the non equilibrium nature of
the reference ensemble. But note that it must still be a measure of momentum

transport, albeit in the non equilibrium ensemble, because this transport coefficient

is the measure of the contribution of a divergence in the flow velocity to the rate of

change of temperature at a point r.

Also, as a further illustration of the content of the expression in Eq.(5-24), the

instantaneous part is simplified to give

(* nst 1 d*W*Mi
S3 V* d*W*M

x V* dP2 2* (-,) d (( v g(5g32)
x [dPe12 f' (,7, v*, v*) (5-32)
J1

Compare this result with that obtained for the collisional part of the pressure in

the fluid in Eq.(5-21) above to see that

(1- a) p (5-33)

Hence, it can be said that (u is the contribution of the source to what would

physically constitute the hydrostatic pressure of the fluid. If a small volume

element of the fluid is considered, then the amount of pressure that the fluid

element can exert on its boundaries is decreased by the energy lost locally due to

collisions. Part of the effect of this transport coefficient is to decrease the effective

pressure in the system, as can also be seen from the fact that the transport

coefficient enters the hydrodynamic equations in the form p* + (*. At the level

of linear hydrodynamics, the two coefficients are indistinguishable in their physical

consequence.

Further, the time integral in Eq.(5-24) can be carried out in the above

expression and hence can be rewritten as

(* = lim{- dF*W (*) e-(*-3)M (5-34)
V V*3

This would be the Helfand form of this transport coefficient. As will be shown in

the subsequent chapter, this form of the transport coefficient is most convenient

to make connection with results obtained from kinetic theory. This completes the

analysis associated with (u.

5.3 Navier-Stokes Transport Coefficients

In this section, the transport coefficients occurring at Navier-Stokes order are

analyzed along the same lines as done in the previous section.

5.3.1 Shear Viscosity

The shear viscosity is represented by the 44 and 55 matrix elements of IC. This

element is identified in Appendix G starting from the expression in Eq.(4-78) as

T]* = lim kikjeiieij { dF*h*,Mj

0 V*
ds'- Jfd ,(e-4- ([* -\) M4 j'} (5-35)

where

1
= Y, IvI
~3nh{I

( t + a) o*6 (q *i a*)
Im
x (-qm g,7) (qLm 2 P qLLj (536)

and AMij is the same function defined in Eq.(5-29) above. This is the Green-Kubo

form of the shear viscosity. Further, using the fact that k = 0, Eq.(5-35) can be

written as

lim dF*r M, fds'- dF*he-(Z*- (* 4)
7 1 l r 1 0* -- A 4 h
o V* J Jo V* J
(5-37)

where AM/ denotes the traceless part of the tensor AM/ given in Eq.(5-29). As

noted above, in the elastic limit

M a-1 q- I Vipeq (5-38)

and hence

S-A4) ij a-,l ij Peq (5-39)

the backward momentum flux in the elastic limit, which has the same form as

the h- in Table C-3, with a 1= Thus, the above expression reduces to a time

correlation function over the equilibrium ensemble of the volume integrated forward

momentum flux hij with the backward momentum flux h, together with an

instantaneous part [67]. The different fluxes in the forward and backward time

direction and the instantaneous part are artifacts of the hard sphere nature of the

interaction of the particles as explained in Appendix C. In the case of inelastic

particles, the adjoint flux contains explicit information about the nature of the non

- equilibrium state and is a measure of momentum transport in this state.

Further, the time integral in Eq.(5-35) can be carried out to give

1* lim 1 T dF* e- 4)S, (5-40)

which is the Helfand form for the shear viscosity of a granular fluid. Lastly,

the instantaneous part of the Green-Kubo form is written in terms of reduced

distribution functions in order to illustrate the structure of the formal result. The

reduction is carried out in the same way as for the pressure earlier, with the result

*Inst 1 1 *
io V* 7a 1 7"v

(1 + a) (3nh) / dad 12 ( (- g) ( g))
60
x JdP12f h 2) v*, v*) (5-41)

i.e., it is the average of the normal component of the velocity of colliding

pairs averaged over the two body reduced HCS distribution. Notice that the

instantaneous part is purely collisional, reflecting the fact that it comes about

because of the boundary condition associated with hard sphere dynamics about

the point of contact. Further, using the first equation of the BBGKY hierarchy

associated with the HCS, the above expression can be rewritten as

1
*Inst (1 + a) (f, ) a 20 (5 42)
60

where Va, is the average collision frequency as determined by the loss part of the

right hand side of the hierarchy (see for example [37]).

5.3.2 Bulk Viscosity

Starting from the formal expression Eq.(4-78) in the previous chapter, it is

shown in Appendix G that the Green-Kubo expression for the bulk viscosity can be

identified as

9 V* j Jo Vit*
(5-43)
where hkj and Mai are the same functions identified in Eq.(5-36) and Eq.(5-29)

respectively. The quantity hS) is the -,l,1i i 11 l" momentum flux defined as

s2) (1 P)a

SOlnPh N p* E* N) (5-44)
2 8 In ln 2 3

These additional terms are necessary for the existence of the long time limit of

the correlation function, as established in the previous chapter. Without loss

of generality, the subtracted pieces can be added to the instantaneous term in

Eq.(5-43) as well, as they give zero contribution under the ensemble average.

Further, if the instantaneous part of the Green-Kubo expression in Eq.(5-43) above

is evaluated, it is found to be the same as that for the shear viscosity, but for an

over all numerical factor reflecting the different tensor contractions involved in each

case.

*Inst *inst. (5-45)
3
Finally, the time integral can be carried out to give

lim dF*le(f )e-(-)4 s4 (5-46)
9 V* J 1

This is the Helfand form for the bulk viscosity of a granular fluid.

Thus the physical content of the expressions for the bulk viscosity are the same

as those for that of the shear viscosity, except that the tensor contractions here

measure the isotropic part of the momentum transport.

5.3.3 Thermal Conductivity

As with the earlier Navier-Stokes coefficients considered above, starting from

the expression in Eq.(4-78), the dimensionless thermal conductivity A* is identified

in Appendix G as
A* lim AIst + ds'GTerm (') (5-47)

where

AInst -1 dF*S(Af, (5-48)
V* 3
and

GThem (s') = dF*S(s)-(-)' (* A) /i. (5-49)

In the above, S' is two thirds of the volume integrated heat flux,

Si 2 1 { 2v* *( -*)(l + a)
3n I
x (-q g ) (g qL)2 (PL*m ql) qL } (5-50)

and S(s)i* is the -dl'i I i heat flux" with the invariant total momentum P

subtracted from it,
S(S)i* S- + 1 Pi. (5 51)

Lastly

a =c -i AV* (v~p* (F*)) q] (5-52)

is a function generated by the non equilibrium reference state associated with this

system. Eq.(5-47) is the Green-Kubo expression for the thermal conductivity of a

granular fluid. It is a time correlation function of the subtracted heat flux with a

flux that characterizes heat transport in this non equilibrium reference state.

In order to illustrate the nature of the adjoint flux, consider the elastic limit of

Mi above,

13 [2 (1) (1) ] *
|l- = 2 3e I (0) (0) pe
S-1)*, (5-53)
2 a2 Peq

and hence A2) M in the elastic limit becomes the equilibrium distribution

function times the backward heat flux given in Table C-3 and the Green-Kubo

expression becomes correlation function of the forward heat flux with the backward

heat flux. As noted earlier, the fact that the forms of the two fluxes are different

is an artifact of the hard sphere nature of the interactions. Now notice that the

equilibrium ensemble is the information entropy ensemble in the sense that it

consists of all the accessible states given the constraints on the system. Hence it

is physically intuitive to think that the amount of heat transport in the system

for a given temperature gradient will be proportional to how correlated the heat

flux stays with itself over time. In other words, if a heat flux is set up at a time

t = 0, the extent to which it persists in the same direction at a later time t

would be a measure of heat transport in the system. If a significant fraction of the

trajectories in the macrostate allow this persistence, then the correlation function

in Eq.(5-47), which is a measure of this persistence averaged over all trajectories

will be large and hence the system will have a large thermal conductivity. But in

the case at hand, the reference ensemble is a non equilibrium ensemble that is a

complicated function of its constraining variables as the collection of trajectories in

this macrostate is more severely constrained than in the equilibrium case. Hence

the extent of heat transport in this system depends strongly on the collection of

trajectories present in the macrostate as reflected by the fact that the adjoint flux

in the correlation function for the thermal conductivity is not simply the heat flux

but is generated by the non equilibrium macrostate itself.

As a last step in illustrating the content of the Green-Kubo expression above,

the instantaneous part of the thermal conductivity is expressed in terms of reduced

distribution function in the hierarchy associated with the HCS to give

A*Inst 18 ) 4(1 + a) ddg AdP 2{2(g2 ) 2 (P .a)2
t s t th (2)it e p
+.(g 2 )32(, g 2),* (*,,v v ). (5-54)

Notice that as with the earlier transport coefficients, the instantaneous part of

the transport coefficient is purely collisional reflecting its origin which is the

discontinuity in time for colliding configurations at contact. Also, this is the first

transport coefficient where the center of mass momentum of the pair is part of

the phase function being averaged, reflecting the fact that this is the first term so

far that is associated with heat transport, unlike all the previous terms that were

essentially associated with momentum transport.

Finally, as demonstrated in Eq.(4-86) in C'!i pter 4, the subtracted part of the

heat flux can be introduced in the instantaneous term without loss of generality

and the time integral in Eq.(5-47) can be carried out to give

A* lim 1 dF *S(S)i*- ( 2- g (5-55)
V* 3

This is the Helfand form for the thermal conductivity of a granular fluid, with

S(s)'* and ,, are as defined in Eq.(5-51) and Eq.(5-52) respectively. This

completes the characterization of the exact expression for the thermal conductivity.

5.3.4 The p Coefficient

This transport coefficient is identified starting from the formal expression given
in the previous chapter to be

11
8* = li1m{ Jd dF*S'*Ri
V* 3

(2 In h ds 1 d*S(s)
2= ds'-- dP*se- s)T
SIn nh V*
xe A2)' (* -A2) A} (5-56)

where S'*, S(s)i*, and ,, are as defined in Eqs.(5-50), (5-51) and (5-52) respectively.
Also,
i t -r n (r, 0)'
K ^, ((t))3Ne Ldrk.rfl (5-57)

and
8 In (h
Q R 2 0n(M. (5-58)
8 In nh
The expression in Eq.(5-56) is the Green-Kubo expression for the p coefficient
which is a new transport coefficient associated with the transport of heat in
this system that is not present for normal fluids. First notice that the second
time correlation function is the same as that occurring in the expression for
the thermal conductivity given in Eq.(5-47) above. Also, unlike the cases of

2. the adjoint density 1i is given implicitly in terms of the local HCS state.
This is because, the reference state as has been characterized in this work is a
homogeneous state. The formalism as developed here does not make explicit the
density dependence of this state and hence is unknown up to this point. A formal
way to extract the density dependence would be to introduce an external potential
that couples into the density, in the Liouville operator governing the dynamics of
this system and solving the inhomogeneous problem. Then a formal procedure of
inversion can be used to eliminate the external potential in favor of the density.

A more physical way to get density dependence of the macrostate would be to

generalize the state to open systems, i.e., define a "grand canonical" homogeneous

Cooling State. This treatment will be part of future analysis of this problem.

For the moment, the result is considered at the formal level above and the

content of the expression elucidated below. As a first step, it is established that

this transport coefficient is uniquely related to the dissipative nature of the

interactions in the system by showing that in the elastic limit, this coefficient

vanishes. It is apparent from the form of Eq.(5-56) that this unfolded expression

does not have a transparent elastic limit. Hence, for the moment, the more formal

result in Eq.(4-86) from which the above explicit expression was obtained, is

considered.

P = kk{Vj dF*S'*

ds' dF*S(s) 1 (O,)}. (5-59)
3 0 V*

First, observe that, in the elastic limit

b* (k) |, ~* (k) ? (k) p (5-60)

where t* (k) is the Fourier transform of the equilibrium direct correlation function

of the fluid. The details of obtaining this form are given in Appendix F. It follows

from the above that

*(1)i (0) a1 i (0) p/,, (5-61)

for, normalization and fluid symmetry imply that ?* (0) = 1 and *(1) (0) = 0.

Therefore,

7i (0, s) = = e-L (Pi*peq) PL*peq, (5-62)

a time independent quantity. This identifies all the quantities in Eq.(5-59) in the

elastic limit. First notice that the instantaneous part of the correlation function