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# Statistical Mechanics and Linear Response for a Granular Fluid

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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

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................... 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

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

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

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

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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

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

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

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

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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

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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

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

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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

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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

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

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

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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

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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)

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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

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

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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

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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

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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.

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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

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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.

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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)

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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

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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)

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

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

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

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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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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)

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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)

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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

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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

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

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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

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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

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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.

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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.

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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

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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.

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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

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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.

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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.

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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.

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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.

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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

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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

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

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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

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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),

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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

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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.

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

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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

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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

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

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

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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)

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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

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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.

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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.

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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

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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

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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

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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

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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

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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

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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.

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

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

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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

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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

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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,

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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

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

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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)

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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

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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.

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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

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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

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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

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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.

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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

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

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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~! `

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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)

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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.

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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

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

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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)

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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):

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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

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

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

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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

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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

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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

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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.

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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

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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)

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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

PAGE 168

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

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Thisequationcanbeintegrateduptogive Also and Hence,thetransportmatrixinthehomogeneouslimitbecomes

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

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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

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(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:

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

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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.

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

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Therefore1P>e ThesearetheresultsusedinthetextinChapter4.

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

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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.

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

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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.

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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)

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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 ).

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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

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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)

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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)

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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

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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.

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

PAGE 189

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

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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

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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

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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 ).

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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)

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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

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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

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

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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

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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.

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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)

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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)

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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

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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)

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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):

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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.

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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,

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

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

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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()

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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)

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ThatconcludesthedetailsoftheEnskogevaluationofthedierenttransportcoecients.

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## 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
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
System ID: UFE0013684:00001

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

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.

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