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Nonequilibrium fluctuations and transport in sheared fluids

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Nonequilibrium fluctuations and transport in sheared fluids
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Lutsko, James Francis, 1964-
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vii, 150 leaves : ill. ; 28 cm.

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Approximation ( jstor )
Correlations ( jstor )
Distribution functions ( jstor )
Fluid shear ( jstor )
Hydrodynamics ( jstor )
Mathematical variables ( jstor )
Simulations ( jstor )
Transport phenomena ( jstor )
Viscosity ( jstor )
Wavelengths ( jstor )
Dissertations, Academic -- Physics -- UF
Fluid dynamics ( lcsh )
Hydrodynamics ( lcsh )
Physics thesis Ph.D
Shear flow ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph.D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 146-149.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by James Francis Lutsko.

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Full Text

















NONEQUILIBRIUM FLUCTUATIONS AND TRANSPORT
IN SHEARED FLUIDS






BY






JAMES FRANCIS LUTSKO
















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 1986



































To my parents.

















ACKNOWLEDGMENTS


I would like to thank Jim Dufty for his guidance, encouragement and patience throughout the time I have worked with him. His supervision and friendship have proven invaluable in the course of my work.

In addition, my gratitude goes to the other faculty who have guided me through my studies as well as to my fellow travelers in graduate school. In random order, they are Dave, Brad, Pradeep, Gary, Jim, Simon, Tom, Bob and Marti. Thanks also go to Bob Caldwell for assistance with the numerical analysis. Finally, special thanks are extended to Stephanie for her support, encouragement and friendship.





























iii
















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS..................................................iii

ABSTRACT........................................................... vi

CHAPTERS

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

2 PHENOMENOLOGY AND SIMULATION .............................12

Phenomenology...........................................12
Experiments ...............................................13
Simulations ...............................................15

3 FLUCTUATION MODELS......................................22

Generalized Langevin Equation ............................22
Long Wavelength Fluctuations .............................29
Small Length Scale Fluctuations ..........................32

4 LINEARIZED HYDRODYNAMIC MODEL.............................. 34

Linearized Model........................................... 34
Green's Function............................ .............37
Correlation Functions.................................... 47
Physical Properties......................................52
Probability Densities................ .................57

5 NONLINEAR FLUCTUATING HYDRODYNAMICS........................63

Naive Renormalization ........................................63
Formal Renormalization........................... ........69
Application of Formal Renormalization ..................... 74

6 SHORT TIME MODEL........................................80

General Properties of the Reduced Distribution
Functions ...............................................80
Explicit Form of Matrix Elements..........................83




iv










Approximations ............. .............................86
Hydrodynamic Equations..................................90

7 ANALYSIS OF THE SHORT TIME MODEL..........................96

Equilibrium Generalized Hydrodynamics.....................96
Nonequilibrium Generalized Hydrodynamics..................97

8 CONCLUSIONS........................................... 121

APPENDICES

A LINEARIZATION OF THE LANGEVIN EQUATION ..................125

B GENERALIZED EIGENVALUE PROBLEM ..........................133

C DYNAMIC STRUCTURE FACTOR ................................139

D ELIMINATION OF THE THREE POINT FUNCTION .................142

BIBLIOGRAPHY......... ....... ......................................46

BIOGRAPHICAL SKETCH..............................................150

































V















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

NONEQUILIBRIUM FLUCTUATIONS AND TRANSPORT IN SHEARED FLUIDS

BY

JAMES FRANCIS LUTSKO

December 1986

Chairman: James W. Dufty
Major Department: Physics


Hydrodynamic fluctuations in a sheared fluid far from equilibrium are described by means of nonlinear stochastic differential equations. Two models resulting from two different limits of the exact Fokker-Planck equation describing hydrodynamic fluctuations in an arbitrary fluid are presented. The first model results from restricting attention to fluctuations which take place on length scales large with respect to the mean free path. The resulting Navier-Stokes-Langevin equations are specialized to shear flow and the associated linear hydrodynamic modes are determined. These are used to calculate the correlation functions of fluctuations around the steady state with the full nonlinear dependence on shear rate necessary for large shear rates. It is then shown how the coupling of the linear modes through the nonlinearities in the stochastic equations allows one to calculate the effect of the nonequilibrium state on the macroscopic transport properties.


vi










The second model results from the short time limit of the exact Fokker-Planck equation and describes fluctuations even on atomic length scales provided the mean free path is small enough. The linear modes of this model are obtained and it is found that they differ considerably from the long wavelength modes. In particular, it is found that at large shear there are two sets of propagating or sound modes. A large shear instability is also found in this model and is tentatively identified with a large shear dynamic phase transition observed in computer simulations of sheared fluids.







































vii















CHAPTER 1
INTRODUCTION


Statistical mechanics is a methodology for determining the

physical properties of many body systems from the microscopic dynamics of its constituents. It is applicable when the properties one wishes to determine arise as a result of the interaction of a large number of the elementary constituents or when the time scales of interest are large with respect to the typical interaction time of the constituents. When this is the case, attention is shifted from the dynamics of individual constituents to the density in phase space (i.e., the space defined by the degrees of freedom of the system) of the system as a whole which is characterized by the distribution function of the system. The dynamics of the distribution function is given by the Liouville equation, which expresses the conservation of phase space density in a conservative system, together with appropriate boundary conditions, initial conditions and source terms representing the action of nonconservative external interactions.

When the distribution function is constant in time, the system is said to be in a steady state. Equilibrium is the particular steady state obtained by conservative systems and, in general, is unique in that the associated distribution function is known. For this reason, the statistical mechanics of equilibrium systems is well developed and





1





2



quite successful at predicting macroscopic properties such as transport equations and thermodynamic susceptibilities.

The statistical mechanics of systems not in equilibrium is, in contrast, poorly understood. This difficulty arises from the fact that nonequilibrium states are difficult to characterize; the distribution function is not known. For systems close to equilibrium linear response theory provides a means of studying nonequilibrium systems by relating their properties to those of equilibrium systems (e.g., fluctuation-dissipation theorems relate transport properties to equilibrium fluctuations). However, far from equilibrium one has only the Liouville equation.

To get a handle on systems far from equilibrium, one typically restricts attention to some subset of the degrees of freedom which obeys an approximately closed dynamics on the length and time scales of interest. Closed, in this context,.means that couplings to the degrees of freedom not in the subset being studied are negligible on the average. For example, the macroscopic equations of continuum mechanics for the local conserved densities (e.g., hydrodynamics) often provide an adequate characterization of the nonequilibrium state. The problem then becomes one of determining the average dynamics of these variables (i.e., macroscopic transport equations) and their fluctuations. At present, almost nothing is known about the transport properties of systems far from equilibrium.

This semi-phenomenological approach was first proposed some fifty years ago by Onsager and has essentially been verified for systems near equilibrium.2,3 Recently, it has been extended to apply to










systems far from equilibrium,4,5 but it is difficult to evaluate the success of such an extension since it has been applied to very few problems. The reason it has not been more thoroughly tested is that its application is straightforward only if the nonequilibrium state is characterized by a few simple macroscopic properties.

One such nonequilibrium system is a simple fluid in a state of uniform shear flow. In uniform shear flow, the pressure and temperature are spatially constant while the flow velocity varies linearly with distance in a direction perpendicular to the flow velocity (Figure 1.1). In a rectangular coordinate system (denoted as x-y-z), the flow field, uo(r), may be written



u (r) = a*r (1.1)
-0


where the shear tensor, a, is



a = axy (1.2)



and "a" is the shear rate. The deviation of this system from equilibrium is controlled by a single parameter, the shear rate, giving a simple characterization of the nonequilibrium state.

A sheared fluid is one of the simplest nontrivial nonequilibrium systems imaginable and, indeed, may appear so simple as to be uninteresting. However, even in this simple system drastic effects due to deviations from equilibrium are present, for example in its transport properties, giving the system an intrinsic as well as












Y=L U L U oL








XI
I



Flg. 1.1 Uniform shear flow.






5



pedagogical interest. In the remainder of this introduction, we will first describe three anomalies in the behavior of sheared fluids, and then we will describe the methods and physical ideas which will be used to attempt an understanding of the system. The introduction concludes with an outline of the work to be presented.

Sheared fluids are of value as model nonequilibrium systems not only because of the simplicity of the macroscopic state, but also because a great deal of effort has gone into developing methods of simulating such systems on a computer.6-13 This is of prime importance because, as will be discussed in detail later, laboratory experiments involving simple atomic fluids are at present limited to the small shear rate regime (i.e., close to equilibrium). As will be discussed in the next chapter, experiments are possible involving more complex rheological fluids, such as polymeric fluids and glasses,14 and colloidal suspensions15 which exhibit many of the properties to be discussed here for simple fluids. We concentrate on simple fluids, however, to simplify the theoretical description. Thus, the only "empirical" data which exist for simple fluids at large shear rates are from simulations. In fact, it is this body of "empirical" results which has spurred most of the recent theoretical interest.4,16-25

The first important anomaly observed in these simulations was

associated with the phenomena of shear thinning. As some fluids are sheared, their viscosity drops; they become "thinner" or less viscous. The dependence of viscosity on shear rate had been calculated in 1969 by Yamada and Kawasakil6 as






6



n(a) = n0 - lal/2 + O(a). (1.3)



The nonanalytic nature of the dependence on the shear rate was previously unsuspected and indicates the breakdown of nonlinear response based on a formal expansion of the Liouville equation in the shear rate. In addition, the result of the simulations was that n1 was measured to be two orders of magnitude larger than Yamada and Kawasaki predicted. Several other authors16 calculated n1 and obtained different numbers, although all are two orders of magnitude too small. A recent proposal by van Beijeren26 for the resolution of this problem will be discussed below.

In 1982, a second anomaly observed in simulations was reported by Erpenbeck.12 He found that, at very high shear rates, the system passed from the amorphous fluid phase into an ordered phase in which the particles line up one behind another in hexagonally packed "strings" (Figure 1.2). There has recently27 been some debate about the actual nature of the string phase but it is generally agreed that a high shear rate instability, or dynamical phase transition, exists. We shall present below a model exhibiting an instability on atomic scales which matches up qualitatively quite well with that seen in the simulations (note that macroscopic hydrodynamics is believed to be stable for planar shear flow28).

A third anomaly has been reported by Evans7 as occurring in twodimensional fluids. His simulations show the shear viscosity remaining constant as the shear rate increases until at a critical










cr 0
O 0

O
O






00 0 0 a >a*: O O

Fig. 1.2 Large shear rate order-disorder transition where ac is the
critical shear rate.






8



shear rate the fluid begins to thin in the expected manner. Evans attributes this to a small shear rate instability for which no theoretical understanding exists as yet. However, the transport properties of two dimensional fluids are anomolous even in equilibrium (two is the critical dimension) leading one to believe that Evans' observations may be related to this fact.

Most of the work to be presented here will consist of an attempt to construct and analyze models of fluctuations around the nonequilibrium state in dense sheared fluids. Once the fluctuations are statistically characterized, one is able to calculate time dependent correlation functions which in turn can be related to observable physical properties of the system. The phenomenological ideas behind this approach are very similar to those in the EinsteinSmolochowski model of Brownian motion.29 One considers the dynamics of the hydrodynamic variables (analogous to the position of the Brownian particle) around the steady state. The hydrodynaniic variables are appropriate to describe properties on a scale which is large compared to the microscopic correlation length. Because these variables are the local conserved densities, it is assumed that they decay according to the macroscopic conservation laws (e.g., their time evolution is governed by the Navier-Stokes equations) with the exception that the equations include stochastic sources which represent the thermal noise responsible for the creation of the fluctuations.4'30 Finally, the stochastic sources are generally taken to be Gaussian white noise with correlations fixed by a fluctuationdissipation theorem.









Having these equations, one in principle solves them for the

hydrodynamic variables as functionals of the stochastic sources and thus may calculate time dependent correlation functions using the known statistical properties of the sources. As alluded to above, these correlation functions can in turn be used to calculate various physical quantities such as the effect of the steady state on transport coefficients and thermodynamic susceptibilities.

While providing a closed and intuitively appealing model of fluctuations, this description is entirely phenomenological and as such represents an uncontrolled approximation. Several authors5,31-33 have addressed the question of the formal derivation and validity of fluctuating hydrodynamics and we shall outline their ideas in a later chapter. The result of these studies is that the foundations and validity of this approach are much better understood.

Thus, one approach to the study of nonequilibrium systems is to replace the microscopic description of the system, as characterized by the distribution function and Liouville's equation, with a hydrodynamic description. A hydrodynamic description of a system is obtained as a result of two approximations. The first is that attention is restricted to the local densities of the conserved variables (e.g., mass, momentum and energy in a simple fluid). On length scales which are large compared to the microscopic correlation length (i.e., the mean free path, �mfp), these variables are expected to decay slowly since their integrals over the whole system do not change at all with time. Thus, on these length scales, these variables are assumed to approximately decouple from the remaining,






10



quickly fluctuating, microscopic degrees of freedom which give rise to dissipation and thermal fluctuation of the slow variables. Concurrent with this, if k is the wavevector (inverse wavelength or gradient in real space) of the conserved densities, it is required that (kimfp) be small. The equations governing the dynamics of the conserved densities may then be expanded in this small parameter with truncation at second order yielding, for a fluid, the usual Navier-Stokes equations.

In a sufficiently dense fluid, the mean free path is actually smaller than the typical size, a, of the atoms making up the fluid. In this case, the hydrodynamic description will include terms of all order in (ko) while still expanding in (kimfp). The result is a hydrodynamic description of the (dense) fluid which contains information about the structure of the fluid and is valid on atomic (i.e., ka - 1) length scales although still restricted to lengths which are large relative to the mean free path (kimfp << 1). One might think, and indeed it was long thought, that atomic scale fluctuations require knowledge of all small scale fluctuations; after all, the reason for considering only conserved densities at long wavelengths is that they decay on macroscopic length and time scales while all other fluctuations decay on much shorter, microscopic, scales. However, while studying models of small length scale fluctuations, de Schepper and Cohen34 discovered that even in this regime the conserved densities decayed much slower than other variables and gave the dominant contribution, on these scales, to the density-density correlation function. This indicated that restricting










attention to the hydrodynamic subspace was a valid means of studying fluctuations even on very small length scales. Thus, hydrodynamic models can be used to study even atomic length scale fluctuations provided the mean free path is small enough. In particular, van Beijeren's explanation26 of the discrepancy between theory and simulation results for the size of the shear rate dependent renormalization of the shear viscosity is based on this generalization of fluctuating hydrodynamics to include atomic scale fluctuations in dense fluids.

The work to be presented below consists of modeling hydrodynamic fluctuations as a means of understanding the nonequilibrium state. In Chapter 2, we will discuss experimental and simulational means of studying steady state shear flow and the connection between the two. The third chapter discusses the formal ideas of statistical mechanics used to construct models of nonequilibrium fluctuations and, in particular, to construct fluctuating hydrodynamics at long wavelengths as well as generalized fluctuating hydrodynamics. This is followed by Chapters 4 and 5 analyzing the long wavelength model, the linearized and nonlinear versions respectively, with the aim of deriving various physically observable consequences of the nonequilibrium state. The sixth and seventh chapters contain, respectively, the development and analysis of the small wavelength model and include a discussion of the hydrodynamics and large shear instability occurring on these length scales. The dissertation concludes with a discussion of the results obtained, the completeness of the present description and future directions.















CHAPTER 2
PHENOMENOLOGY AND SIMULATION


In this chapter, we discuss the basic phenomenology, such as length and time scales and strength of shear, which affects the observability of deviations from equilibrium behavior. We then discuss the possibility of performing laboratory experiments which measure nonequilibrium properties. Finally, we discuss the methods of simulating uniform shear flow on a computer.



Phenomenology

In macroscopic hydrodynamics, one typically has three types of

parameters: thermodynamic derivatives such as the sound velocity, co, transport coefficients such as the kinematic viscosity, v0, and the wavevector, k, the magnitude of which is inversely proportional to the length scale of the phenomena one is interested in. The time scale of hydrodynamic dissipation 0s (vk 2) (in equilibrium, momentum fluctu-vok t
ations dissipate as e ). In hydrodynamics, one usually uses perturbation theory treating the wavevector as a small parameter. The dimensionless parameter which actually occurs is (v k/c ) thus requiring that v0k << c0 or v0k2 << c k. Physically, this means that the rate of hydrodynamic dissipation is small compared to the rate of sound propagation. This is consistent with the Navier-Stoke's





12






13


equations governing hydrodynamics since they are themselves the result of an expansion to second order in the parameter (k�mfp), where 2mfp is the mean free path, and in fact c0 /0 z- mfp. Typical values of these constants for water at STP35 are ~ 1.8 * 10-2 cm 0 a
c0 - 1.5 * 105 cm/s so zmfp = 10 cm. Thus, the applicability of
6 -1 -1
hydrodynamics requires that k << 8.3 * 10 cm , or ka << 10
-8
where a - 10 cm is a typical atomic radius.

When a fluid is sheared, a new time scale, the inverse of the

shear rate (denoted here as "a"), is introduced. When the shear rate is as large as the rate of dissipation, we expect that largd deviations from equilibrium behavior will occur. For this reason, we refer to a - v0k2 as the large shear regime. When studying the long wavelength hydrodynamic model, we will require that a < v0k2 thus including the large shear regime while allowing us to use perturbation theory (since, given this bound, a << c0k).



Experiments

Various methods exist for measuring the static and transport

properties of fluids. For example, neutron scattering can be used to measure the static and dynamic structure factors (related to the pair correlation function and transport properties, respectively) on atomic length'scales.36 Light scattering samples the dynamic structure factor on hydrodynamic length and time scales.37 These methods and others are available to measure the corresponding properties for nonequilibrium fluids. However, the present ability to test theories of sheared systems is limited by the shear rates which can be obtained






14



in the laboratory. At present, the largest shear rates reported are less than 104 HZ.38 This shear rate is in the large shear regime
1/2 2 -1
when k - (a/vo) ~ 7.45 10 cm . This means that until we look at length scales as large as about 10-2 cm the effects of shear will be small compared to the usual equilibrium properties. Thus, light scattering, which measures properties on a length scale of the incident beam's wavelength - 10-5 cm, and neutron scattering, which is not practical for wavelengths above about 10-8 cm, require shear rates about two orders of magnitude larger than those attainable in the laboratory.

There are systems, other than simple fluids, in which similar nonequilibrium effects occur and which have the practical advantage that they have long relaxation times. Because the large shear regime occurs for a - inverse relaxation time (- v0k2 in a simple fluid) it thus becomes accessible in these systems. One example of such a system is a colloidal suspension in which the relaxation time t - d2/D for a typical interparticle spacing, d, and diffusion constant of the spheres, D. In this system, the density of spheres determines the relaxation time and the nature of the probe is not restricted as in a simple fluid. Experimental studies of sheared colloidal suspensions have been performedl5 and numerous nonequilibrium phenomena are observed including a string phase similar to that seen in simulations of simple fluids.

Another class of systems exhibiting easily accessible

nonequilibrium phenomena is macromolecular (e.g., polymeric) fluids





15



and glasses.14 The polymeric systems are not simple fluids because they are composed of anisotropic molecules. However, Simons et al. have shown that strong similarities exist between the phenomena seen in simple and complex fluids. These fluids demonstrate a wide variety of non-Newtonian behaviors including shear thinning. The accessibility of these phenomena again arise due to the fact that the constituents, the macromolecules, have long relaxation times lowering the lower bound of the large shear regime.

In this work, we will restrict our attention to simple fluids.

As the preceding discussion indicated, we will not have direct experimental tests of our results. The reason we do not consider colloidal suspensions, polymeric fluids or some other such system is that the results from computer simulations of simple fluids have shown that even our understanding of the simplest system is incomplete. However, because effects,.such as shear thinning and the string phase, seen in the simulations are observed in physically related laboratory systems, we have some confidence in the simulations. Thus, we propose to attempt to understand the simplest sheared system and hope, in the process, to gain some understanding by analogy of the more complicated systems in which the nonequilibrium state has important physical consequences.



Simulations

Most of the data for simple fluids have been generated by

nonequilibrium molecular dynamics6-13 rather than in the laboratory.






16



Such simulations require both the establishment of the linear shear profile as well as the maintenance of constant temperature.

In equilibrium molecular dynamics, one considers a box with

periodic boundary conditions containing N particles. At each time step in the simulation, one solves the equations of motion (e.g., Newton's equations) for all of the particles and moves them accordingly. The periodic boundary conditions have the effect of decreasing the size (number) dependence of the results allowing one to achieve the bulk, or thermodynamic, limit with on the order of 103 particles. In this way, one may study static properties, such as the pair correlation function, and dynamic properties like the decay of correlations. Clearly, simulations have the advantage over experiments in that one can obtain much more data about the system since one has the microscopic state at one's disposal., They are, however, limited by the number of particles, N, and the length of time which can be simulated. In particular, the size restriction limits the smallest wavevectors which can be studied.

The linear shear profile is generated by modifying the simple

periodic boundary conditions of equilibrium simulations. The system remains periodic in the x and z directions but when a particle with momentum p passes through the y = L plane at time t and with coordinates (x,L,z), it is reentered at y = 0 with coordinates (x-aLt, y, z) and momentum 2 - maLx. These boundary conditions are known as LeeEdwards' boundary conditions. They become more transparent when expressed in the local rest frame coordinates, q and p, defined as






17



. = 2 - a t



. = 2 - ma . (2.1)



In these coordinates, the local flow velocity is zero and it is seen that the Lee-Edwards' conditions correspond to simple periodic boundary conditions in the rest frame.

The temperature, T, is a measure of energy fluctuations around the steady state and is defined by



3 1 N
NkBT = < (i - ma. * q)2> i=1

N
<1 (Pi)2>, (2.2)



where kB is Boltzmann's constant. Substitution of the conditions of spatially uniform shear flow into the Navier-Stokes equations shows that a sheared fluid will undergo viscous heating. Specifically, one finds that


_E 2
a= na , (2.3)



where e is the internal energy density, leading to



E(t) = E(O) + na2t. (2.4)






18



For water sheared at 104 Hz at STP this leads to a heating rate of 0.024 K/s. For temporally uniform shear flow, particularly for the very high shears applied in computer simulations, an external, nonconservative force is necessary to hold the temperature constant. This heat extracting interaction is called a thermostat. This is equivalent to adding a drag force to the equations of motion:



d



d
dt = i x(r) i

-2

(r) = J (F. - a ) / 2. (2.5)
J J


where the form of A(M) has been chosen to hold the temperature as defined in eq. (2.2) constant.

The equation of motion form for the thermostat is important

because, with the Lee-Edwards' boundary conditions for imposing the flow field, the simulations can be mapped onto a well defined mathematical problem: the solution of Liouville's equation, modified to account for the thermostat, with the imposed boundary conditions. The modified Liouville equation is obtained from the general form of Liouville's theorem,39

a 1 a a a a + + F * ]p(r,t) ai [A(r)ip(r;t)]= 0
Sq, a(2.6)
(2.6)





19



where the repeated index, i, is to be summed and runs from 1 through N and the dot indicates an ordinary vector scalar product. So, (2.6) may be rewritten as



- p + Ep = 0, (2.7) with

A A
� = + fint + zext (2.8) where the free streaming and interaction terms are, respectively,



0 i 3q,


� = F i (2.9) int -i Dp
-i


while the external term, due to the nonconservative thermostat, acts on the distribution function as



ext p(r;t) = [(r)i1p(r,t)]. (2.10)



Because this operator includes the nonconservation of volume in phase space due to the nonconservative force (Liouville's theorem states that d (p(r)dr) = 0 and with nonconservative forces - dr * 0), it has a different form when acting on any other phase function.






20



Specifically, if A(r) is an arbitrary phase function, we have that



� A(r) = - ) * A(r). (2.11)
exti � i


The bar indicates the difference between these operators and we note that �0 = �0. It is easy to see that � and � are adjoints with respect to phase space integration.

Finally, we should mention that when working with hard spheres, the potential is singular and the interparticle force, Fi, appearing in these equations ill defined. This difficulty is eliminated by replacing the interaction term with the so-called T-operators which generate the singular hard sphere dynamics. Henceforth, for hard spheres, we will write


^+
�int - T�(,pi'j;qij) (2.12)



where � refers to forward and backward time equations. The form of these operators can be found in the literature40 and, for example,



T(x1,x2)A(1 ,x2) = - a2fd; ((- o * ~12 12 -12



* [b12 - 1] A(x,X 2)} (2.13)



where x, = a Iq, etc.; a is the hard sphere diameter, 12 = - 2






21



and


b AA AA
bl2A( l' 1; 2'2) A(ql' 1 - o ~ ;l2 '2 + " 12). (2.14)





We note here that the T-operators are not self adjoint, as is the interaction term for continuous potentials, eq. (2.9), so their adjoints are denoted with bars as well.















CHAPTER 3
FLUCTUATION MODELS


In this chapter, we discuss the statistical mechanical basis of the models to be analyzed. Specifically, the models we will study will be seen to be two different limits of a general Fokker-Planck equation. The detailed form of the models will also be given.



Generalized Langevin Equation

Our basic approach, as discussed in the introduction, to studying nonequilibrium fluctuations around shear flow is to choose a restricted set of variables whose dynamics we wish to study. At a fundamental level, any classical system of N structureless particles, or atoms, is described by the 6N coordinates, collectively called r, which are the positions, qi, and momenta P, of the atoms (where i ranges from 1 to N). The distribution function of the system, p(r;t), which describes the density of states in phase space, satisfies the Liouville equation,



t p(r;t) + � p(r;t) = 0; (3.1)


the explicit form of which was discussed in a Chapter 2. The set of restricted variables we choose to study is the set of conserved





22






23


A A
densities. These are mass density, p(r), energy density, e(r), and momentum density, j (r). A caret over a function indicates that it is a phase function. Their definitions are


S N
p(r) = Z m A(r-_)
n=1

^ N N
e(r) = E - _ A(r-qn) + F Z V( I-,j I) A(r- _)
n=1 nnn'

A N
ja(r) - Z pn A([-n) (3.2)
n=1


where A(r) is some localized function of r (for instance, a Dirac delta function). These variables will be denoted collectively as y(r). Our reasons for choosing this particular set were alluded to in the introduction. At length scales large with respect to the mean free path, we expect that the conserved densities are the only variables with long relaxation times (i.e., long with respect to the collison frequency). Therefore, we expect that the statistical properties of the system at these scales can be adequately described in terms of these variables with the other degrees of freedom being modelled as rapidly varying sources. In dense fluids, the mean free path can be smaller than the atomic dimensions and it is found, as one might guess, that the hydrodynamic variables, or conserved densities, still have long relaxation times and so still, on the average, obey an approximately closed dynamics. Because the other degrees of freedom are more rapidly varying, they may again be modelled as stochastic sources.






24



In the analysis to follow, it will prove convenient to work with the Fourier transform of the variables in eq. (3.2). This is defined to be


^ =ikfr
-1-
y(k) = V-I dr e y(r). (3.3)



We also note that the vector y(k) is explicitly,



y(k) = (p(k), e(k), (k)). (3.4)



For notational convenience, we will write



ya + y (k) (3.5) where the Latin index includes both the vector index a and the wavevector k. Repeated Latin indices are to be summed. The average of ya, or y (k), over the nonequilibrium ensemble, p(r;t), will be denoted as where




f J dr p(r;t)ya (3.6) and are the macroscopic variables one would measure in the laboratory.

In general, one also defines time dependent phase functions, y(k;t), as





25


-�t
ya (k;t) = e y (k) Ya(t) (3.7) where



= fdr Ya e- p(r;0) = fdr [e-t ya] p(r;O) = fdr Ya (t) p(r;0) = . (3.8) So, the macroscopic variables are the average over an initial ensemble, p(r;O), of a (t). The fluctuations of a (t), denoted as 6ya(t), are defined to be 6Ya(t) = Ya(t) - . (3.9) Because we are restricting attention to the set of variables y(t), it is useful to introduce the distribution function of these variables, p(y;t), defined to be





26



p(y;t) = fdr' 6(y-y') p(F';t), (3.10) where y is some possible value of the phase function y. So, for any function of the y's, say A[y], it is clear that


= fdr A[y] p(r;t)



= fdy A[y] p(y;t). (3.11) Finally, for a steady state such as uniform shear flow, we have that pss(r;t) = pss(r;0) - Pss(r). So, (y a;t> = ss



= Yo0a (3.12)


Grabert et al.41 show, using the projection operator technique introduced by Zwanzig,42 that the distribution p(y;t) satisfies the following exact generalized Fokker-Planck equation, a p(y;t) + -a v [Y] P(Y;t) - fvads fdy' Daby,y';s] ps(Y') aya
aay

p(y';t-s) } = 0 (3.13) SYb' P S(y')





27




where pss(y) is the steady state distribution, Va[y] is called the drift vector and D aby,y';s] is called the diffusion matrix. Repeated Latin indices are again to be summed in (3.13). The functionals v and D are expressed in terms of steady state averages and their detailed form is given in Grabert et al.41 Here, we only note that



- {Va [y] p (y)} = 0 (3.14) ayaa

so p s(y) explicitly satisfies (3.13).

Associated with (3.13) is an exact generalized Langevin equation for the variables Ya(t),



t Ya(t) a - aby(t)] + ds DabY(t-s);s] Fb[Y(t-s)]


a A
- A Dab[y(t-s);s]}
ayb(t-s)


= R (t) (3.15)


where the thermodynamic forces, Fb[y(t-s)] are defined in terms of the steady state distribution as


SIn p s(Y)
Fb[y(t-s)] = - = y(t-s (3.16)



which also appears in (3.13). The functional va is the same as that appearing in (3.13) while






28



Daby;s] = fdy' D [y,y';s] l (3.17)
ab ab y=y.


The sources Ra(t) have the following constrained averages,



0 = fdr' 6(y -y ') R '(t) p (r ')



D b[Y;t] = dP' 6(y -y ') Ra'(t) Rb'() p (r') (3.18)



where the hat on Ra(t) indicates it to be a phase function and where a primed function has argument r'. The first of equations (3.18) indicates that Ra,(t) is orthogonal, under the steady state average,
A
to any function of y and indicates that the constrained average of R (t) at fixed y is zero. The second of equations (3.18) has the form of a fluctuation-dissipation theorem relating the correlation of the "fluctuating force" to the diffusion matrix.

Now, eq. (3.15) together with properties (3.18) may be modelled as a stochastic differential equation if the sources R (t) are idealized as stochastic processes, denoted as Ra(t), which have the form,


Ra(t) - Mmnly] Sn(t) (3.19) where


=


= (27) 6(k+k') 6(t-t') A


A aa 6(t-t') (3.20)





29



where A,' is a constant matrix and the amplitudes Mmn[y] are fixed by the fluctuation-dissipation theorem, eq. (3.18). Finally, the variables Ya(t) are replaced by stochastic variables denoted as a (t) = y (k,t) and the average over the distribution function replaced with that over the stochastic sources,



ss a R(t) (3.21)


since the variables Ya(t) may be considered to be functionals of the stochastic sources.



Long Wavelength Fluctuations

If we restrict our attention to long wavelength fluctuations, then the preceding Langevin model can be explicitly evaluated. Specifically, we only consider wavevectors such that



kimfp << 1. (3.22)



In this case, the various functionals appearing in eq. (3.15) may be evaluated in terms of a power series in ktmfp. If we truncate the expansions at second order, then the model may be given explicitly,5 in terms of the usual microscopic conservation laws,



atp + V 0






30



at


Sj + t = 0 (3.23)



where sa and taB are the heat flux vector and stress tensor, respectively. These in turn each have a convective part, a dissipative part and a stochastic part, where the latter represents the stochastic source,


8 R
s = eu + t uB + A T+s a = a a o 3r a


t = u ( u ) - V + t (3.24) ta =uJ + P - aB Dar o aB a


where we have introduced the fluctuating velocity, ua = p j , the fluctuating pressure and temperature, p and T, and transport coefficients, n , Ko and A . These fluctuating thermodynamic functions and transport coefficients are the same functions of y as one has in equilibrium; i.e.,



p[y] = pey]



= pe[p,'] (3.25)



where pe is the equilibrium pressure functions and the last line






31



indicates that pe depends only on the (local) density and internal energy density, E, where E is given by


1 -1 .2
E = e - .p (3.26)



The dissipative parts of the heat flux vector and stress tensor are seen to be given by a fluctuating version of Fourier's law and Newton's viscosity law, respectively, where


2
S6 6 + 6 6 - 6 6 (3.27) aBPv ap av av Sp 3 aB Pv


Finally, as previously mentioned, the stochastic sources are modelled as Gaussian white noise with covariances determined by the fluctuation-dissipation theorem, eq. (3.18),


R 2 1/2
S = (k A T ) s (r,t)
a Bo a


R 1/2 1/2
tB = (kBTno0) s1s(rt) + (kBTio) /s2s (r,t) (3.28)



= = = 0



= 2 6 (r -r 2) (tl-t2)
a 1't)s (r2't2)> = 2 6a 1-2 1 2


= 2 A 6(r -r2) (t -t2)



= 2 6 6 6(r -r2) 6(tl-t2) (3.29)
-- 1 2 p -26B






32



with all other cumulants equal to zero. Equations (3.23) - (3.29) completely specify the fluctuating hydrodynamics model which may be summarized as a Langevin equation for the conserved densities, Ya(t), with a deterministic part given by the Navier-Stokes equations, local equilibrium transport coefficients and susceptibilities supplemented by the stochastic sources, eqs. (3.28) and (3.29), which model the excluded degrees of freedom. Finally, we note that this model is given in the language of eq. (3.15) in Appendix A where it should be noted that



D aby;t] = 0 (3.30) ayb


as is shown by Zubarev and Morozov.5



Small Length Scale Fluctuations

In a dense fluid, the mean free path may actually be smaller than the typical atomic length scale, to be denoted here as a (for a gas of hard spheres, a would be the hard sphere diameter). In this case, one may study atomic length scale fluctuations with the present methods. However, the evaluation of eq. (3.15) requires care for while we still expand in (k mfp) we must keep all orders in (ka). Quantities dependent on (ka) carry information about the small scale structure of the fluid. We can evaluate (3.15) while retaining the desired information regarding the small scale structure by taking its short time limit to obtain






33






at ya(t) - Va[Y(t)] = Ra(t). (3.31)


The drift vector is defined to be



Va[Y(t)] = - {<6(y-y) � Ya>ss/<6(y-y)> ss}i . (3.32)



In general, this is a complicated nonlinear functional of y(t). However, as is explained in Chapter 5, we will only be interested in this functional expanded to second order in the deviations from the stationary state. Thus, we expand as



v [y(t)] = v[y (k)] + 6Yb(t) * [ A v [y(t)]] + etc.
6yb(t) 0
(3.33)



The first order term is



6 -1
S v a[y(t)] = (<6y6y> ) b<6y y > ss (3.34)
y b(t)


This model is explicitly evaluated in Chapter 6 and analyzed in Chapter 7. While it may seem strange to study the short time limit of the generalized Langevin equation, it should be noted that similar short time equations have been found to provide qualitatively accurate descriptions of dense fluids even out of the short time regime (an excellent example is the Enskog equation43).















CHAPTER 4
LINEARIZED HYDRODYNAMIC MODEL


In this chapter, we linearize the Navier-Stokes-Langevin

equations described in Chapter 3 and use them to calculate various statistical properties of the system. We will do this in such a manner that the results apply for shear rates up to and including the large shear regime as previously defined. Of particular interest will be the hydrodynamic propagators and the two time correlation functions. We will discuss the applications of our results to light scattering experiments and will make contact with the small shear results of other authors. The chapter concludes with a description of various formal statistical properties of the system highlighting the difference of this model from local equilibrium.



Linearized Model

The complete hydrodynamic model was given in Chapter 3, eqs. (3.23) - (3.29), and will not be repeated here. The variables we choose to work with are the mass density, total energy density and momentum. So,



y -+ (p,e,j) (4.1)







34





35



and we denote the steady state averages of these quantities with subscript naught,



y (P0,'e'0,) (4.2)



and the deviations from the steady state, the fluctuations, are denoted by Za,



Z (r,t) - y a(r,t) - y (r,o) (4.3)



where we also use internal energy density and velocity in place of total energy density and momentum density. Finally, we scale and Fourier transform Z (r,t), to obtain the variable Z(k,t) in terms of which we wish to work,



Za(k,t) = fd3r k*r Z (r,t) (4.4)


Z a(k,t) + c 6p(k,t), 6E(k,t), e (k)u(kt)] (4.5)



2 -2 O 2 o 0
c1 = p-( ) ; c = ( - ) (4.6)
1 0 8p0 0 3C0 0


where p0 = PLE POO] is the steady state, macroscopic pressure; h0 = O + P is the enthalpy density and (e (k)} are a set of ^(1) ^
pairwise orthonormal vectors with e (k) = k/k = k and E is the internal energy density and u = p-j is the velocity. The scaling, internal energy density and u = p j is the velocity. The scaling,





36



(4.6), will simplify the form of the equations below and we emphasize that cl and c2 are equilibrium thermodynamic susceptibilities.

In the course of Fourier transforming and linearizing the

equations, we find that we must transform the convective terms which '-1
include the operator p0O * V which on an arbitrary function f(r) looks like


-1
P00 * V f(r) = a irj - f(r). (4.7)


The transform of which is


3 ik*r 3 ik-r r
fdr ik a..r f(r) = a (-i -)fd3re f(r)
- ajr la k. r


= -a k f(k). (4.8) ij 8k i


Thus, it turns out that the form of the linearized equations is


a t
-- Z (k,t) - a k Z (k,t) + H (k;a) Z (k,t) = R (k,t) (4.9) Bt a 3 j a - a - a


where Ra represents the stochastic sources which, at linear order, do not depend on the fluctuations, Za. The details of the linearization including the explicit forms of H a and Ra are given in Appendix A. Equation (4.9) is the primary result of this section. In the following sections, we will analyze the hydrodynamic (i.e., normal) modes described by this equation. For zero shear, H is the usual






37



matrix of linearized hydrodynamics the eigenvalues of which describe sound propagation, heat diffusion and momentum diffusion.

An important feature of eq. (4.9) is the coupling of variables of different wavevectors implicit in the gradient term. This unusual feature has important consequences on the evolution of fluctuations and is known as linear mode coupling.



Green's Function

To solve equation (4.9), we introduce the Green's function GaB(k,k';t) defined to satisfy



"- G (k,k';t) - a ki G (k,k';t) + H o(k;a)G B(k,k';t) = 0
a i k i -a(kaku;t)


with Ga(k,k';O) = (2w)3 6(k - k') 6a. (4.10)



In equilibrium, i.e., a = 0, the solution of (4.10) requires nothing more than the diagonalization of the hydrodynamic matrix H . Before solving (4.10) for nonzero shear, we recall the equilibrium solution.39

Let the eigenvectors of HaB(k,a=0) be denoted as

((i)(k)}ji=1,5 and the corresponding eigenvalues be {A(i)(k) i=1,5 so that



H a (k,O) o (k) = i) (k) i)(k). (4.11) a - a






38



Furthermore, as Ha is in general nonhermitian, we introduce the biorthogonal set of vectors, {In(k)} i=1,5' defined to satisfy


(i) (j)
n (k) . ( (k) = 6... (4.12)



In terms of these functions, the Green's function can be immediately determined to be



G (k,k;t) (k) )(k')(2) (k-k')exp - [Ai(k)(t-t0)] i=1
(4.13)



and tO is a constant which can be fixed by an additional boundary condition. The solution to eq. (4.9) is then given by


dk'
Z(k,t) = f 3 G (k,k';t) Z (k',t )
(2r)

dk'
+ dt d f ) G (k,k';t-T) R (k',-) (4.14) t 3(2n)3 as 8


where the wavevector integrations extend to infinity.

Because the hydrodynamic matrix H.0 is the result of a truncation at second order of an expansion in (kimfp), the eigenvalues can be determined by perturbation theory in this same small parameter. In equilibrium, one obtains two sound (propagating) modes, a heat mode and two degenerate shear modes,





39




1 2
A (k) = -ic k + 2 rOk


1 2
A2(k) = ic k + 2 r k



A (k) = DOT k



A4(k) = VOk2



A 5() = 0 k2 (4.15)



where explicit forms for the speed of sound, co, the sound damping constant, r0, and the thermal diffusivity, DOT, will be given later. For the present, we note only that the modes all decay like exp-(constant)k2t and that these results can be used to calculate the intensity of scattered light from an equilibrium fluid (as described below). Upon doing so, one obtains the classic Landau-Placzek formula44 which is perhaps the most successful application of fluctuating hydrodynamics. In fact, comparison of this result and experiment provides one of the best known means of measuring the equilibrium transport coefficients.

Returning to the nonequilibrium problem, eq. (4.10), we introduce a shear rate dependent generalization of eqs. (4.11) - (4.12) as follows:





40



-a k - ( (k;a) + H (k;a)( (k;a) = M (k;a)( (k;a) ijk ak a - a a


(i) (j)
n (k;a) * (k;a) 6 . (4.16) 1j


The presence of the gradient operator in (4.16) greatly complicates the problem. However, as discussed in Chapter 2, we restrict the size of the shear rate by requiring that the rate of convection be no larger than the rate of dissipation yielding the condition,



a < vk2. (4.17)



This choice is what allowed us to neglect heating in these equations,
2 14
since the rate of heating goes like a - k (see eq. 2.3), and makes possible a perturbative solution of eq. (4.16). The details of this calculation are given in Appendix B. The eigenvalues so obtained are


1 2 2 A = -ic k + - (r k + ak k /k )
1 0 2 0 x y


1 2 2
A = ic k + - ( k + ak k /k 2)
2 0 2 0 x xy










2
A3 = DOT k



4 = vgk2 - akxk /k2



A5 =~vOk2. (4.18)






41



We see that the degeneracy of the shear modes has been lifted while the sound damping constant has been shifted and the heat mode remains unaffected at this order.

If we now write the Green's function as



G (k,k';t) = 1 n (k;a)( (k';a)G )(k,k';t), (4.19) i= a


then we find that the function G(1) satisfies the following equation:



- a jk -k + A (k,a)] G (k,k';t) = 0
Bt ij i ak.
i


G (k,k';0) = (2w) 6(k-k'). (4.20) This scalar equation is easily solved and if we introduce a time dependent wavevector as



k.(t) = k. - kjaji t = k. - 6iy k xat, (4.21) then the solution of (4.20) is



G i)(k,k';t) (27)3 6(k-k'(t))E(i)(k,t)



where E (i)(k,t) = exp - t d A(k(-T)). (4.22) The explicit form of the propagators in (4.22) is





42




E(1)(k,t) = [k/k(-t)]1/2 exp{-ic ka(t) - 1 2 )
S(1 ) 1/2 .k/kt 2 0 t



E (2)(k,t) = E (k,t)



E(3)(k,t) = exp{-DOT k 2(t)



E (k,t) - [k(-t)/k] exp-k(-t)



E(5)(k,t) = expf-vk 2B(-t)} (4.23) where a(t) = (2ak xk)-1(k y(-t)k(-t) - k yk) k(-t) + k (-t) sgn(ak ) k + ky sgn(akx) y x


8(t) = (akk2 )-1 {k2(k (-t) - k) + 1 (k3(-t) - k)


2 2 2
k = k + k . (4.24)
x z


These functions behave for long times as a(t) + (ak /2k)t2


1 23
8(t) + 1 (ak /k) t (4.25)
3 x





43



Because lim B(t) > 0, the modes are all stable as expected (see Drazin and Reid and Case28 for further discussion of the stability of

-macroscopic shear flow).

In order to illustrate the physical significance of some of the shear rate dependent effects described by these equations, consider the evolution of a purely hydrodynamic sound excitation along the direction of flow. Specifically, we take Z (k,) () 6(k-k with k = k0x. The evolution of hydrodynamic excitations is given by eq. (4.14) when conditionally averaged over the stochastic force with fixed initial condition Z (k,0). Denoting this average by a double bracket we have that


dk'
<> = f 3 Ga (k,k';t) Z (k',O)
(21)


= 6(k - k (t)) (1)(k) E ()(k,t). (4.26)



Inverting the Fourier transforms yields,


(1) 2 1/4 Ok8(t,-a)/2 <> = E (k0(t))(1 + a2t2)/4 e-r0B(t-a)/2



x cos{k [x(t) - 0 a(t,-a)]} (4.27)



where a(t,a) and B(t,a) were defined in (4.25) and x(t) = x + ayt. The time dependent coordinate x(t) is due to the expected Doppler shift in the sound wave which travels in a moving medium. This kinematic effect may be suppressed by restricting attention to the





44



y = 0 plane and without loss of generality, we may chose x = 0 as well. Thus, the remaining shear effects are due to gradients of the velocity field and not the magnitude of the velocity. Figure (4.1) shows <>/<> for the choice of parameters (ck0/Okr2) = 10- , which is typical, and the two cases (a/r kO) = 0 and 1. Note that in the former case,


-rOk2t/2
<>/<> = e cos(c0 k 0t) (4.28)



which is the expression for ordinary sound propagation. Two effects are easily seen. First, the effective sound velocity increases with time relative to that for zero shear rate on this time scale. Specifically,


Ck a(-t) 1 21/2
c k {( + T2)/2+ ln[(1 + T ) + ]} (4.29)
C00kot 2


where T R r 0k t. Second is that the attenuation is enhanced, relative to that for zero shear, as time increases,


r0k08(t,-a) 1 2 2
= 1 + a . (4.30)
0 0

These relative magnitudes of convection and dissipation would be very difficult to reach with simple fluids but are possible for laboratory colloidal suspensions.


























Fig. 4.1 Propagatiqn and damping of sound wave in shear flow for a/rFkO = 1 (-) and
for a/rk0 = 0 (---).






46



















-








.




















e- -un
qllm tl ' � n l
milIII a ilirmt lb l! ii 411o Iiib
r'al ~l






47



Correlation Functions

The correlation functions of the deviations of the fluctuations from the steady state are useful for a number of reasons. From them one may calculate the dynamic structure factor, which describes the intensity of light scattered from the system and transport properties such as the shear rate dependent renormalized viscosities and they characterize the deviation of the system from local equilibrium. In this section, we calculate the two point correlation functions.

The two point correlation functions are defined to be



C a (k,t'k',t';a) = (4.31)



where the brackets denote the nonequilibrium average which in the present case corresponds to an average over the stochastic forces. Time reversal invariance and stationarity require that



C s(k,t;k',t';a) = Cas(k,t-t';k,O;a)



C a(k,t;k',t';a) = P P C (k',t;k,t'-a) (4.32)



where Pa = �1 is the parity of Za under time reversal. Therefore, we may consider t > 0 without loss of generality.

If we define



(k,t) = n W(k;a) R (k,t), (4.33) a a-






48



then, using eq. (4.14), we find that Ca (k,t;k',O;a) Ca (k,k';t) = fO dTfodT{ -(i) (k)5 )(k')E (k,T)E ()(k',T')
i,j


x (4.34) where we have taken tO + - in eq. (4.14) and used the boundary condition that Z (k,--) is finite. It is clear that



= (2) 36(k+k')6(t-t')F (i)(k) (4.35) where the form of F(i) (k) is easily determined. Thus,


3 (1) (i) (i) C (k,t;k') = (21) 6(k+k'(t)) (i) (k)n (-k')E (k,t)C (-k';a).

(4.36)



The functions C S(k;a) are related to the equal time condition functions because, in particular,



CaB(k,t=O;k';a) = (2)36(k+k')C a(k;a). (4.37) The explicit form of these functions is






49



C (k;a) = I-r ( { (i) E(j (-k)E() (k,)E() (-k,T)
ae 0 a 8
i,j


x F(ij)(k(-T))}. (4.38)



The nonzero elements of C a(k;a) are given in Tables 4.1 and 4.2.

Some immediate information may be gained from these

expressions. In particular, eq. (4.36) shows that the two body correlations decay in the same way, i.e., with the same propagator E(i)(k,t), as the macroscopic modes. This represents a generalization of Onsager's regression hypothesis to nonequilibrium states. Furthermore, eq. (4.38) is a fluctuation-dissipation theorem for the nonequilibrium system. These statements are more obvious when cast in the form of differential equations:



[ a k, -]C (k,M';t) + H (k;a) C (k,k',t) = 0 (4.39)
at ii a1j k- - o aS


[-6 a.k. -- + H (k;a)] C (k;a) + [H (-k;a)] C (k,a) = R (k) aa ij i 3k. a aS - aa ao a
3


(4.40)



where RaB(k) is the amplitude of the fluctuating forces,



s 6(k+k') 6(t-t') R (k). (4.41)



Equation (4.39) shows more clearly the linear regression law while (4.40) has the usual form of a fluctuation-dissipation relation.





50




Table 4.1 Equal time correlations.



2
C (k;a) = kB x [1 + A (k,a)] pp B 0 0 T 1C p(k;a) = kBTOhOOx T[1 h- T T + A1 (k;a)]




-1 T2 2 C (k;a) = k T x Ex T oC + [h - T -] + h A (k;a)] CE B 0 T T 0 v 0 0 xT 01





-1





C44(k;a) = kBToP0 [1 - A2(k;a)] C45(k;a) = k BTOPO A3(k;a)




-(
C55(k;a) = k T0P0 [1 - A (k;a)] 55 B 0 0 4





51



Table 4.2 Nonequilibrium contributions to equal time correlations.




kk k (-t) -r0 k (-t) A(k;a) = aY fdt e k3 (-t)


k k (-t) -2 k(-t) A2(k;a) = 2a odt X - e
k


k k (-t) k -2v0k2 (-t) A (k;a) = aodt[2xky F(kt) +




A4(k;a) = 2aodt F(k,t)[ x y- t F(k,t) + -le



k(-t)
F(k,t) = A(k(-t)) A(k)



kk k
A(k) = z tan -1 Y)
k k k
x


k(t) = k - k * a t



2 2 2 k2 . k + k
x z






52



Physical Properties

We now discuss two physical properties which may be determined

from the correlation functions. They are the long range nature of the correlations and the light scattering spectrum.

The density-density equal time correlation function in real space is easily seen to have the form


dk
- ik*r
C (r;a) = f ) e - - C(k;a)
pp (2)


= 3kB TO PXT[6(r/t) + /r F(r/i)] (4.42)



where the isothermal compressibility, XT = p ( )T' and where I = (r /a)1/2 is a typical length scale associated with the shear rate. The limit F(r/t)=O yields the usual equilibrium result where the 6-function is an idealization of the atomic scale correlations which actually exist in the fluid. The part porportional to r/k has been obtained before via an expansion to first order in the shear rate.20 Because of the form of F, this actually requires that r/k << 1 and yields



F(0) = -a rirj /8rYar2 (4.43)



where Y = cp/cv is the ratio of specific heats. This does not, as has been claimed,20 however, allow us to conclude that the correlations are long-ranged because it is the product of a short distance expansion. To determine the true nature of the correlations, at large






53



distances, we consider a somewhat simpler problem than the large-r expansion of F(r,t). Specifically, we consider



fdyjdz C (r) T kBTOPOXT[C(x/) + YH(x/t)]. (4.44)



The part of the correlation function due to the deviation from equilibrium, H(x/t), is given by



H(x/1) - (2w)-1 fO dt{(1 + t-3/2 -/2 (1 + t2)
0 3



x exp[-(x/i) (4t(1 + t2)) ] (4.45)



and for large (x/) behaves as



H(x/i) 4 0.828(x/)-5/3. (4.46)



At short distances, this function is Gaussian and the complete function is shown in Fig. 4.2. It is thus clear that the correlations are in fact long ranged, decaying as a power law.

The second physical property we wish to study with the

correlation functions is the intensity of light scattered from the fluid. It can be shown39 that the intensity of light at the point r with frequency w due to the initial scattering of a coherent beam of wavevector kO and frequency w0 is given by
















0.6
I

H(X/1)



0.4






0.2







0 2.0 4.0 6.0 80 10.0 X/i

Fig. 4.2 Nonequilibrium contribution, H(X/1), to density-density correlation function
along direction of flow for a/rOkO = 1 (-) and asymptotic form (---).
0 0o





55





1 2 2
I(r,w) = j 10 w (a/2 C0r)sin 0 S(k,_) (4.47)



where I0 is the spectral intensity of the incoming beam, a is the polarizability of the particles composing the medium, r * EO a EO cos, ED is the amplitude of the coherent beam, k = kj - (W/CO)r and Q = w - O0 and where the structure factor, S(k,Q), is defined as


dk dk
f'*dt d-1 -2 int S(k,Q) = fdt (k-k )8(k+k )C (k t;k ,0;a)e . (4.48)
6 1 2 pp -1
-* (21)


Thus, we see that the density-density correlation function determines the intensity of scattered light. The functions, j(k), are Fourier transforms of form factors which limit the volume of integration to that irradiated. In Appendix C, it is shown that S(k,0) can be written in terms of nonequilibrium Brillouin and Rayleigh peaks:



S(k;Q) = S R(k;Q) + S B(k;Q) + S B(k;-) (4.49)



where


1 a -1 -2 (1) (1)
S (k;a) = 2Re{[-it + X (k;a) - - ajk ] C n (C (k;a)l B 1 2 ij k 1 1 a al


1 a -1 -2 (3) (3)
SR(k;Q) = 2Re{[-it + X3(k;a) - 1 ai k - C1 na C (k;a)}.



(4.50)





56





In equilibrium, the gradients are not present and these

expressions reduce to simple Lorentzians. Expansion of eqs. (4.19) to first order in the shear rate yields previously obtained results; however, the general evaluation of (4.19) is difficult. A quantity which can be calculated directly is the Landau-Placzek ratio, R, defined as



R(k;a) i dw SR(kw)/fdw[SB(k,w) + SB(k,-w) S[Cpp (k;a) - Polh0 Ccp(k;a)]/a2C pp(k;a) + p0/h0 C p(k;a)]



(4.51)



where a = p0c1 /ho02. In particular, the integrated Rayleigh intensity is unchanged from equilibrium, fdw S (k,w;a) = (Y - 1)/Y (4.52) where Y = ratio of specific heats. An explicit form of (4.51), obtained using the expressions for the equal time correlation functions given in Appendix C, is



R(k,a) = (Y - 1)/[1 + YA (k;a)] (4.53)





57



where A = [C (k;a) - C (k;0)]/C (k,0). In Fig. 4.3, we plot
1 pp - pp - pp
AR - [R(k;a) -_R(k;O)]/R(k;O) as a function of e for
2
k = k (cosex + siney) and a/r Ok = 1. The small shear rate expansion of these results is in agreement with earlier work by Kirkpatrick et al.20 The result, eq. (4.53), is unexpected in that it states that the integrated intensity of the Rayleigh peak is unaffected by the shear.



Probability Densities

In this section, we recast our previous results in the language of the probability distributions of the fluctuations. In particular, the probability distributions are more useful than the description in terms of fluctuations for the calculation of higher order correlation functions or for averages of more general functions.

The probability and joint probability density are defined by



P(Z,t) = <6(Z(t) - Z)>



P(Z,t;Z',t') = <6(Z(t) - Z) 6(Z'(t') - Z')> (4.54)



where the averages are taken over the stochastic forces, {Ral, and over a specified ensemble of initial values, {Z(0)}. The 6-functions are short for



6(Z(t) - Z) H 6(Z (k,t) - Z (k)). (4.55) a,k

















0.30





0.10


U /4 I 34 n














Fig. 4.3 Deviation of the Landan-Placzek ratio, eq. (4.52), as a function of angle
for a/nk 2= 1 and k = 0. The value at e = /2 corresponds to k = 0 for which the
0 z x
equilibrium result holds.

CO





59


We begin the evaluation of eq. (4.54) by calculating a related quantity--the conditional probability density defined by



W(Z,t;Z',t') = P(Z,t;Z',t')/P(Z',t'). (4.56) Because the system is being modelled by a Markovian process, the conditional distribution is independent of the initial values, {Za ()} and because of the neglect of heating, the system is stationary so W(Z,t;Z',t') = W(Z,t-t';Z',O). So, without loss of generality, we consider



W(Z,t';Z',O) = <6(Z(tIZ') - Z>R (4.57) where, as indicated, the average is only over the stochastic forces and Z(OIZ') . Z'. From eq. (4.14),



(tdZ') = G(t)Z' + dT G(t-T) R(T) (4.58) where we have suppressed internal summations. An integral representation of the 6-function in (4.57) and use of (4.58) yields W(Z,t;Z',O) = fdA eiA[Z-G(t)Z']



- fdA exp{iX[Z - G(t)Z'] - 12M}





60





= Idet[M(t)]-1 i2exp{- [Z-G(t)Z'] * M-1(t) * [Z-G(t)Z']} (4.59)



where the second line follows from the first upon use of the Gaussian statistics of the source term. The matrix M(t) is given by



M(t) = fdT dr' G(t-T) * * GT(t-T'). (4.60) Equations (4.39) and (4.40) (the regression law and fluctuationdissipation theorem) allow us to re-express (4.60) in terms of the correlation function matrix, C(t) C ag(k,t;k',0),



M(t) - C(0) - C(t) C-1 (0) * CT(t). (4.61)



We can now immediately write down the probability and joint probability densities,



P(Z,t) = fdZ' W(Z,t;Z',0) P(Z',0)



P(Z,t'Z',t') = W(Z,t;Z',t') P(Z',t'), (4.62) where P(Z',O) is the initial probability density. From (4.61), we can determine the stationary state,





61






P (Z) = lim P(Z,t)
ss
t**


= {det[rC(O)]}-1/2 exp[- Z C 1(0) * Z (4.63)



which follows from the limits,



lim C(t) = 0 => lim M(t) = C(O) (4.64)
t** t**


and, lim G(t) = 0. Thus, as expected, because the driving process was
t.*
Gaussian and the equations linear, the distribution of fluctuations is also Gaussian. Furthermore, eq. (4.63) shows that the same asymptotic stationary state is reached regardless of the initial condition.

All of these results could have been obtained if we had worked with the Fokker-Planck equation rather than the fluctuation equation. For completeness, we write down the Fokker-Planck equation for P(Z,t) which is easily obtained. For example, by differentiating (4.62) with respect to time,



- P(Zt) J az J (Z't) + J1(Zt)] (4.65)



where the drift term is



J (Z,t) = [a k. --- Z (k) - L (k,a)Z (k)]P(Z,t) (4.66)
Oa ij i at a- aa 8





62



and the diffusion term is



J1 (Z,t) = R (k) () P(Z,t). (4.67) The matrix R a(k), the diffusion matrix, is just the correlation matrix of the stochastic forces. Equations (4.65) - (4.67) are the usual results for the relationship between a linear Langevin equation and the corresponding Fokker-Planck equation.
















CHAPTER 5
NONLINEAR FLUCTUATING HYDRODYNAMICS


In this chapter, we will consider the consequences of the

nonlinear terms occurring in the general fluctuating hydrodynamics model. We begin by considering a naive form of renormalization which illustrates in an intuitive way the concept of renormalization. Using this method, we will derive the lowest order renormalization of the shear viscosity explicitly deriving the nonanalytic dependence on shear rate. In the following sections, we will present a more formal development of renormalization which will result in a self-consistent formulation of the nonlinear problem. The chapter concludes with the connection between the naive and formal renormalization.



Naive Renormalization

Consider a simple, one dimensional, nonlinear, stochastic equation of the form



a 3
Sx + LOx + N x f(t) (5.1)



with a Gaussian, white stochastic source f(t) having correlations,



f6 (t -t2) (5.2)





63





64



and all higher order cumulants equal to zero. Furthermore, let the average of x(t), , be denoted by x0(t). Then, the equation of motion for x0(t) is found by averaging eq. (5.1),



a- Xo(t) + Lox0(t) + N = 0. (5.3)



If we write x(t) = x (t) + 6x(t), then (5.3) becomes



a X0(t) + [LO + 3NO<(6x(t)) 2>]x0(t) + No0x(t) = - No<(6x(t)) >.
(5.4)



We see that the equation for xo(t) has the same form as that for the fluctuating variable, x(t), except that the "bare" coefficient Lo has been renormalized by the nonlinearity to become the "dressed" or renormalized coefficient L,



L = L0 + 3N0<(6x(t))2>. (5.5)



If we subtract eq. (5.4) from (5.1) and solve the resulting equation for dx neglecting the nonlinear term, and use this to compute L, we find that



L = LO + 3N f /2L + o(N ) (5.6)



where we have imposed stationarity, - = O, to fix the atequal time correlation functions.
equal time correlation functions.





65



The previous calculation can be reformulated in terms of

correlation functions. The equation of motion for the two-point function, <6x(t)6x(0)> s C(t), is found from (5.1) to be



at C(t) + LoC(t) + N0<(6x(t))36x(0)> = 0 (5.7)


where we have used the condition - 0. To lowest, NO = 0, order, we find that the Laplace transform of C(t), C(z), is



C(z) = C(0)/(-z + LO) + o(N ) (5.8)



which has a pole at the bare transport coefficient L0. If we calculate the correlation function in (5.7) to lowest order, we find that


t ddd3f0 dT4 -0(t-T 1) -L0(t-2) -L (t-T3) L0 4 <(6x(t)) 6x(0)> = e e



+ o(N )


t 0 e-2L (t- ) -L (t-T2 ) = 3 jtd j d'r2 e e f + o(No)
-* 1 - 2 0 0


1 1 -L0t 2
=3 2L0 2L0 e fo + o(No)

0 (0)
C (t) + o(N0). (5.9) 2L0 0





66



So,



SC(t) + (L + 0) C(t) = o(No2) (5.10)
-t 0 2 L0 0 and


3 2
C(z) = C(O)/(-z + LO + N f /LO) + o(N ) (5.11) which has a pole at the "renormalized" coefficient defined in (5.5). This coincidence is easy to understand. Suppose that at time t=O, the distribution function, p(x), is perturbed from the steady state function, ps, in the following way


-h6x
p(t=0) = ps e . (5.12) The auxillary field, h, might be chosen such that h * ss = XO i.e., to yield some particular initial value for the macroscopic variable. For small deviations from the steady state (i.e., h small so h - ss is small), we find that



- x0 = C(t)h + o(h2)



- (C(t)/C())[h - x 0] + o(h2). (5.13) Thus, the correlation function acts as the Green's function for small





67



deviations of the macroscopic variables. They must, therefore, share the same linear relaxation times. A general .proof of this property for Markovian processes has been given by Dufty et al.45

In the same way, we can derive an expression for the renormalized shear viscosity. If we take the average of eqs. (2.45), we require that we obtain the macroscopic hydrodynamic equations. For shear flow, the only nontrivial equation is the heat equation,


8 2
e - nRa = 0 (5.14)



where nR is the renormalized viscosity. Comparing (5.14) with the average of eqs. (2.45) yields


-2
n = -a
R ii Dr 1i r tij
-2

= a


-2 8
= a <- ui(r,t)] p(r,t) u (r,t) u (r,t)> (5.15)



where we have used temporal and spatial translational invariance to set overall derivatives of the correlation functions to zero. The third line of (5.15) follows directly from the momentum equation.

Using our results from the previous chapter, eq. (5.15) can be written, to lowest order in the nonlinearity, in terms of the equal






68



time correlation functions,



nR = a p 0



-(2w) 6d k[k k y 33(k;a) +k k y44 (k;a) - k 45(k;a)]. (5.16)



This equation for the lowest order renormalization of the viscosity is an example of mode coupling. That is, the original nonlinearities in the equations of motion have coupled two different hydrodynamic modes, analogous to phonons in solid state physics, and the correction to the viscosity is due to the interaction of these modes, analogous to the phonon-phonon scattering contribution to the coefficient of thermal diffusion in solids. The coupling of the modes in (5.16) is buried in the equal time correlation functions. It can be seen by referring to eq. (4.38). Contributions also exist due to three mode coupling, four mode coupling, etc.

Upon evaluation, we find that


1/2
nR n0 + nl/2a (5.17) with

S= - kBT 2.56 + 4.(5.18) (2vB)3/2 (r0)3/2


Thus, it is seen that a nonanalytic dependence of the viscosity on shear rate arises as a result of mode coupling. This is in contrast to the analytic dependence one would obtain from the standard ChapmanEnskog solution of the Boltzmann equation (or the Enskog equation; see






69




Chapter 6). This result agrees with that of Ernst et al. and with that of Yamada and Kawasaki (although their numerical evaluations are incorrect). Thus, this result indicates the equivalence, with the present assumptions, of this model and the others appearing in the literature.



Formal Renormalization

In this section, we wish to develop a more general method of

renormalization than the naive scheme, allowing the calculation of all renormalized properties. To this end, we begin by formulating the nonlinear model in a more general notation,



- Y + vmEy] + D [y]F [y] = M [y]n (5.19)



where v [z] represents the nonlinear Euler terms in eq. (3.23) (i.e.,
m
the terms V * j = V * (pu) in (3.23) and eva and v j8 in (3.24)). The matrix Dmn[Y] is called the diffusion matrix and F ny] are the thermodynamic forces which are the variables thermodynamically conjugate to the ym's. The product DmnFn represents the dissipative terms in (3.24). The reason for returning this form of the equations is that in their derivation of fluctuating hydrodynamics, Zubarev and Morozov5 show that thermodynamic consistency imposes the fluctuationdissipation theorem,



D mn[y] = M m,] Mnn[y] Amn,; (5.20)





70



"thermodynamic consistency" meaning that the equilibrium distribution function is a solution to (5.17).

Equation (5.17) is highly nonlinear making a general analysis

difficult. However, in equilibrium it has been found that truncating eq. (5.17) at second order in the nonlinearities provides a qualitatively correct description of such nonlinear phenomena as long time tails (i.e., the algebraic decay of equilibrium correlation functions). In fact, in equilibrium, the dominant mode coupling effects arise from the Euler terms in (5.19). We shall, therefore, consider only these nonlinearities in the rest of the discussion. It can be shown that, to order (ak), this is equivalent to taking both Dmn and Mmn to lowest order (i.e., Dmn[y] - Dmn[Y 0]) thus preserving the fluctuation-dissipation theorem, eq. (5.20), in this truncated model of eq. (5.19). Furthermore, Fn[y] is expanded to first order in the fluctuations in order to preserve the usual linearized transport laws.

With these approximations, and expanding the variables ym about their steady state averages y0m, eq. (5.19) takes the form,



--z (k,t) + L (k,t) Zb(k,t) + eV (k,k k )(z (kl t) z (k ,t) t a b ab c 1 2 b -1 c -2


- )



= Ra(k,t) (5.21)



where L = -a k - + H (k;a) is the linear operator studied in
ab iji k ab





71



the previous chapter and za - Ya - YOa (details are given in Appendix B). The vertices, which arise solely from the Euler terms, have the form



Vab (k,k ,k ) = A (k,k ) 6(k-k -k ) (5.22)
abc ---2 abc--1 --1 -2


where the explicit form of Aabc(k,q) is given in Appendix B. The constant E is introduced as a notational convenience in the development of a perturbative expansion in the nonlinearity.

Equation (5.21) can be rewritten using the Green's function introduced in Chapter 4 as


(0) r
z (k;t) = z( (k;t) + eft d{G (kql ;t-T) Vabc(q,,3) a a aa 1a'bc 1 293


x (zb(q2,T) zc(q3,T) - )} (5.23)



where a sum over repeated wavevectors is implicit and where



za(0)(k;t) Gaa,(k,q;t) za,(o,0) + ft d Gaa (k,q;t-T)Ra,(q,T).

(5.24)



Using (5.23), a perturbative solution to eq. (5.21) is easily constructed.

From eq. (5.21), we can immediately construct the equations of motion of the two-point correlation functions,





72



Ca (k,k';t) = ,



C ab(k,k',t) + L (k;a) C,(k,k';t) + M (k,k';t) = 0 (5.25)
St ab aa a'b ab where we have defined



Mab(k,k';t) = Vacd N,1,q2) (5.26) and have used the fact that = 0 for t'

Mab(k,k';t) ftdT aa (,k ;t-) Cab(k ,k';r). (5.27)
a,0 aa 1 alb 1


The significance of this is that eq. (5.24) becomes a linear equation for the correlation functions and effective viscosities, etc. as would be measured in the laboratory may be identified.

Laplace transforming (5.24) in conjunction with (5.27), one obtains


z Cab(k,k';w) + Laa(k;a) k,k';w) + L ,(k,k1;w) Cab (k ,k';w) ab aa, alb ' w aa' -'-1 alb ;'1 w


= Cab(k,k';t=0) (5.28) and the Fourier transformed self-energy may be computed from (5.27),





73





ab("k' ;w) M(k,;w) C (,cb( j';w). (5.29)



As discussed in the previous chapter, Cab(k,k';t) = Cab(k,t)6(k-k'(t)) and it is easily verified that, order by order in perturbation theory, Mab(k,k';t) m Mab(k,t)6(k-k'(t)) which requires that lab(k,k',t) m Iab(k,t)6(k-k'(t)). Thus, eq. (5.28) becomes


Cab(k,t) + D(k)C (k,t) + H (k;a)C (k,t) + eM (k,t) = 0 t ab ab aa a'b ab (5.30)



where we have introduced D m -a jk.i . The equation for the selfenergy becomes



Ma (k,t) = fd aa (k;t-T)Ca, (k(T-t),t))


rt -D(t-r)
= dr (k;t-T)e C (k,T) (5.31)
0 aa' a'b


which shows the self-energy to be an operator in k-space. Laplace transforming (5.31) yields an expression for the self-energy,


wt Dt - --1 fdt e ,(k;t)e = M (k;w)Cb (k;w).



aat (k,w) (5.32)





74



and the Laplace transform of (5.30) is


Sd ; )e -(w+D)(T) (
[w+DICab(k,w) + H (k,a)C ab(k,w) + efo taa (k;())e C (k,w



- Cab(k,t=O). (5.33)



Equations (5.32) and (5.33) are the primary results of this

section. The linear mode coupling which was the unique feature of the linear problem is seen to play an important role here as well. The operator nature of the self-energy is a result of the linear mode coupling and expresses the fact that new gradient couplings, as well as the expected new hydrodynamic couplings, arise as a result of the process of renormalization. This complication is, however, removed in the context of perturbation theory in e because we have an expression, to lowest order in E, for the operator (w+D) acting on the correlation functions (c.f. eq. (5.33)). It is seen that the self-energy function shifts the poles of the correlation function renormalizing the transport coefficients as in eq. (5.11).



Application of Formal Renormalization

We now wish to apply the renormalization formalism to the

calculation of the renormalized shear viscosity and, in so doing, to make a connection with the naive renormalization method. As a simplification, we will restrict our attention to the contribution arising from the shear modes. In this case, the irreversible part of the stress tensor, tij, can, in general, be written as





75





t . j(5.34) ij ijm Jrl 5


where Vijlm is the viscosity tensor which, in the unrenormalized theory, has the Navier-Stoke's form,



S = v(6 6 + 6 6 -2 6 6) (5.35)
ijlm il jm im6jl 3 1 6ij1m) ij 1m


where v and K are the usual kinematic and bulk viscosities, respectively. The macroscopic equation for the energy density is


- -1 -1
a- e + -r- [(e + p)p j + t jl + s] = 0 (5.36)



For macroscopic shear flow, using eq. (5.34), the heating equation becomes



Se + ij aam 0. (5.37)
- e+ jlm ijalm


Thus, the renormalized viscosity calculated earlier from the heating equation corresponds to the yxxy element of the general viscosity tensor.

We now calculate the self-energy to second order in the

nonlinearity (i.e., to second order in E). Using eqs. (5.23) and (5.26), we find that





76


Ma (k,k';t) EVad(_,q,2)



+ 2E 2dVacd (k,( l 2)Gce(a, '3 t-T)Vefg(434'9 5)

x <(0) (0) (0) 3
x } + o(3 ) (5.38)


where the factor of two arises from the symmetry of the vertices. Using eq. (5.24) and causality, this reduces to Mab(k,k';t) = EVacd(k,ql92.2)Gcc,(1',3;t)Gdd,(-2'q4't) x 0+ 2E2od~Vacd , 1,2)Gce -1'9 3 t-T)Vefg 3'44'95) G f,(4 ,q4;T)Ggg' (5'9;T)Gdd'( 2';t) x } + o(3).

(5.39)


In equilibrium, the equal time fluctuations are Gaussian distributed making the o(c) term in eq. (5.39) zero and allowing easy evaluation of the o(E ) term. In the steady-state, the equal time fluctuations are not necessarily Gaussian distributed. Stationarity, in the steady state, allows us to write





77





za(k;O) - z)(k;O) + ef0 d Gaa ' (kq'-T )Vabq1' .2'3) x (zb(q2;T)Zc(3;T) - )} (5.40)



where



)(k;0) = d G (k;-)R ) (5.41)
a - - aa' -;- )Ra 1T) and



(0) (k,-t) = -td G aa(k,q;-t-T)R (q1-r) (5.42) where in (5.42), t>0. Thus, the equal time fluctations may be determined perturbatively and are Gaussian distributed to zeroth order

in :.

Equations (5.39) - (5.42) provide us with enough information to calculate the self-energy function from eq. (5.28). The result is that



aa ,(k,k";w) = e cdt e-wt IV (k,ql )cc ( 1'3;t)Gdd' (2'4;t) x ba (k',k";z)





78





4+ 4c2dt e - Vad(t ,l1,2)Vega,(3 , k4,k")



x Gde (2,3;t)God21,9;t)-



+ o(E3). (5.43)



To recover our previously calculated lowest order mode coupling contribution to the viscosity, we first replace the equal-time correlations in (5.43) with their local equilibrium values. The term proportional to c in (5.43) then vanishes and (5.43) becomes


- kBT
E (k,k";w) = fdt e A ,(k(-t)-q(-t),-q(-t)) aa' PO acd eya

-D(k)t
x Gde(k-g;t)Gcd (;t)e - }D6(k+k").

(5.44)



The momentum-momentum self-energy upon evaluation is found to have the form



2 (k,z) - kik j(k,z) (5.45)



which identifies it as the renormalization of the viscosity tensor.
(0)
If we write ijl = Vijlm + 6v'ijlm then the long wavelength, long
jlm viscositym appearing in eq. (5.37) ism
time hydrodynamic viscosity appearing in eq. (5.37) is





79






6v = lim (k,w) (5.46) yxxy k+O yXPxPy
w+O



from which the "naive" result is recovered.

The replacement of the static correlation functions by their local equilibrium values requires some comment. For a stationary process, the correlation of the stochastic sources is related through fluctuation-dissipation theorems to the static correlations. The mode coupling contributions to the static correlations may thus be interpreted as a renormalization of the amplitudes of the stochastic sources. By neglecting the equal time three-point function in (5.42), we were left with the propagator renormalization.

There is another way to understand the replacement of the static correlations in the present calculation by their local equilibrium values in order to find agreement with the "naive" calculation. The only information about the "statics" that was used in the "naive" calculation was the stochastic source correlations. But, as explained in Chapter 3, these are approximated, in the unrenormalized theory, by their local equilibrium values. It is, therefore, reasonable that the same information must be used in both calculations to obtain agreement. We speculate that if the first order renormalized forces were used in the naive calculation, one would obtain the same result as is obtained here without the local equilibrium replacement.















CHAPTER 6
SHORT TIME MODEL


In this chapter, we discuss the explicit form of the short time model of hydrodynamic fluctuations introduced in Chapter 3. We recall that this model is intended to describe hydrodynamic fluctuations on length scales as small as atomic lengths. We begin by discussing general properties of the distribution functions. This is followed by a derivation of explicit expressions for the necessary matrix elements in terms of the reduced distribution functions. The final two sections describe the approximations introduced to evaluate these matrix elements and the explicit form of the resulting equations governing the time evolution of hydrodynamic fluctuations.



General Properties of the Reduced Distribution Functions

In this section, we introduce two properties of the distribution functions which will be of use in determining the general nature of the expressions, eqs. (3.31)-(3.34), to be evaluated below and which will aid in their evaluation. These properties are the stationarity, in time, and the modified translational invariance of the distribution functions.

As discussed in Chapter 2, an external nonconservative force is applied in computer simulations of sheared fluids to counterbalance viscous heating of the system. The significance of this is that it



80






81



yields a steady state. If we denote by pss(r) the distribution function (density matrix) of the steady state system, depending on all of the phase space variables r i then we can write the steady state Liouville equation as



0 a- p (r) + �p(r) (6.1)



= � p(r) (6.2)



where � is the Liouville operator. The explicit form of � is generally given by


A N Vk 1 a
= 1 V( n-n' (6.3)
n m n�n "n 2 "n an ext
n=1n nn' n 2n


where V( -nn ,) is the pair potential. For hard spheres, the potential is singular rendering eq. (6.3) ill defined. As discussed in Chapter 3, however, eq. (6.3) can be replaced by a well defined expression which produces a statistically equivalent dynamical system


^ N ^
� = n + 1 [T(n,n') + T_(n',n)] + � (6.4)
n=1 n nsn'


where we have included a term representing the externally applied thermostat alluded to above. The explicit form of the T operator and its various adjoints and time-reversed forms are given in the literature and will not be repeated here. The point we wish to make





82



is that (6.2) allows the �+ operator to be moved in averages, i.e., for any two phase functions, A(r) and B(r), it can be shown that



= <(� A(r)) B(r)> . (6.5) s - ss



The subscript on f� refers to the fact that T+ goes to its time reversed form T..

Another important property of the distribution functions is their modified translational invariance. Clearly, because the local flow velocity varies spatially, the distribution function cannot be translationally invariant. That is, if


i *
pi,aqi , ,'i - R (6.6)



for an arbitrary displacement R , then



Pss(r) * Pss(r*). (6.7)



However, if we make the transformation



i' 4 , pi - maij Rj i - R , (6.8)



then we expect the distribution function to be invariant since the macroscopic state is characterized by the local velocity and the local velocity gradient and, while (6.6) does not preserve these properties, (6.8) does. Mathematically, it is easy to verify that (6.8) is






83



consistent with the Lees-Edwards boundary conditions while (6.6) is not. We, therefore, look for solutions of Liouville's equation which are functions of the variables



Pj s pi - maijqj



q q - q (6.9)


which are invariant under (6.8), i'* i i R'2j. (6.10) Henceforth, we will write Pss(r) P, p({p, qij).





Explicit Form of Matrix Elements

The short time model is given in Chapter 3 as



- y (k,t) + L (k,k';a)6y(k',t) - 0 (6.11) where



L (k,k';a) = <(�y a(k))6yy(k")>ss (g )y(6.12) and





84



g8(kt",k') = <6y (k")6y (k')> ss (6.13) The variables, 6y (k), are again taken to be the fluctuations of the conserved densities around the steady state:


^ ^ 1 ik*
6y1 (k) = p(k) = -1 me -iqn



6y2(k) = 6e'(k) = - k ek
2- 2m 2 Be

( = 6j'(k) - . ( eikp)n (6.14) V n



which are summarized as



6ya(k) - Ya (') e -n, (6.15) /V n


where V is the volume of the system.

We now define the reduced distribution functions,



nN! fdX' ... dX' p (r') (6.16) nnf(n)X''X~) (N-n)! n+1 n ss


where X' (n,2n) and we note that the Jacobian of the change of variables X + X' is unity. Using (6.16) and the independence of Pss on the center of mass, we find that



ga (k,k') = (2)3 V 16(k+k') g_ (k), (6.17)






85



where



ga8(k) = n fdp f() 1 a Y8


1 2 (2) ik*r
+ 2 n d dpdr2f (1 ,, )12Yy 1 ()y ()e --12.

(6.18)



The explicit form of � +y(k) is easily found to be


^ ^ 1 k En Y Eik*n
� ya(k) = i Zk -- Y, ()e a- a n
v n n


+ . - [T+(n,n') + T+(n',n)] Y(n)ei 'Q
/V nln'


- k a -j- y (k) + � y (k). (6.19)
i ij3ak a- ext aFrom this, it follows that



<(�+y (k))6y (k')> = (21r) 6(k+k')M (k)



- k.a -~- Sij k a -8 ss + <(�e y (k))y (k')> , (6.20) where





86



M (k) = ink.fdp 1 Ya(()Y1 ()f ()

2 (2)
+i- kd2 m a 1 Y )Y )f(2 ;12)


1 (2)
+ n2f dpdd 2( T+(1,2)+T+(2,1)])Y (aY 8 (2{' 12)


2 ik'r
n - -12
d+ -2fd dd 1 2([T +(1,2)+T (2,1)])Y 1 (n )Y (2)e


x f(2)


2 ^ ik*r
+ fdpdpdp dql2dq_3(E +(1,2)+T (2,1)])Y (p1 ))Y(3)e


x f(3) (1' 2Rj~I;12'13). (6.21)



Approximations

The evaluation of eqs. (6.21) and (6.18) require knowledge of the one- and two-body reduced distribution functions for the steady state. In general, these functions cannot be determined explicitly and, so, we introduce two important approximations.

The first of these is the neglect of momentum correlations, which is to say we assume the two-body distribution to have the form


f(2) 2 (2) 2



Sf() ()f )g(12), (6.22)





87



where f(1)
where f (p) is the one-body reduced distribution function, which cannot depend on 1 due to the modified translational invariance, and the function g(q12) is the steady state analogue of the pair correlation function (i.e., g(q12)dq12 is the probability of finding a particle, 2, in the volume da12 if there is a particle, 1, at position .11). In equilibrium, this approximate form becomes exact. However, away from equilibrium, we expect that particle velocities are indeed correlated (i.e., (6.22) implies that ss = 0 but in general this need not be the case). This approximation, which is analogous to Boltzmann's Stosszahlansatz,43 is standard in the development of kinetic equations. In particular, it is used to derive the Enskog equation which is known to give a qualitatively accurate description of fluids near equilibrium.43 With this approximation, the three-body term in (6.21) only contributes for 8=1. In Appendix D, it is shown that this term can be written, using an exact identity following from stationarity, in terms of two-body functions.

Having adopted the form (6.22), we must still determine the onebody distribution and the pair correlation function. The second approximation we use is to replace the steady state pair correlation function, g(a12), by the local equilibrium pair correlation function, 8LE(I 121) ' gLE.(q12). The pair correlation function is related to the density-density equal time correlation function and knowledge of one of these allows determination of the other. This approximation, which, like the previous one, is used in the derivation of various kinetic equations, may be thought of as a first guess to be "renormalized" using the density autocorrelation function calculated





88



from the resulting equations. This approximation then becomes the first iteration of a self-consistent calculation. In fact, these two approximations require that f(l)( P) satisfy an Enskog equation which is the usual dense fluid analogue of the Boltzmann equation.

With these approximations, our model is completely defined. The one-body distribution can be obtained from the first equation of the BBGKY hierarchy (i.e., by integrating eq. (6.2) over coordinates X)...XN and using eqs. (6.4) and (6.22)):



m (a f ( 1') = fdX2T (12)f )f(f () (2)gLE12)



+ sources. (6.23)



The sources in eq. (6.23) arise from the nonconservative forces in the Liouville equation (eq. 6.4). However, it can be shown that they contribute to second order in the shear rate and so are neglected in the present calculation which is only carried out to first order in the shear. While the previous approximations served to construct a model, we now make an approximation with respect to solving that model; we will only work to first order in the shear rate. Thus we expand f(,



f ( ) = fLE , )(1 + a*( ) + ...) (6.24)



where fLE(pl) is the local equilibrium distribution function for uniform shear flow,





89



fLE(p) = n(2mnkBTo )3/2 exp - (pl /2mkBTo). (6.25)


With these approximations, one is able to determine �1(P) in eq. (6.25). As this result exists in the literature, we only quote it here:



a( (P) = -ajnk I/rO(P(iPj - 1 6 )


1 4
nk = [1 + j5 n*X]nB



nB = 2 (m//80) (6.26) 16a


and X m gLE( ). The constant, nB, is the kinetic part of the viscosity as calculated from the Boltzmann equation. It should be noted that we have only sought the steady-state solution of eq. (6.22).

With these approximations, the matrix g a(k) may be determined to be



g11(k,a) = nS(k)



g22(k,a) - n(k T )



ga (k,a) = PokBT 0 a - mnk(a. +a. ) a=i+2, 8=j+2=3,4,5



gaB = 0 all others (6.27)





90





while the matrix M a(k,a) can be written as



M =-n<-- - (') YB(E')>



+ 6a2 n aij((P j 3 p'261j)/m, y 8('))



+ Ia8 - Sa8 (6.28) where <"'>E denotes an average over fE( ') truncated to first order in a. The collisional contribution, l.8, is



I g(k,a) = -n Xd'dRdp l2 Y(e) T(12)f (plf (2) ^ ik*l12 ik12 (6.29) x [Y B(g) + Y B()e - 1e (6.29) and the source term, S8a, is



SaB = <(ext (k))[k)) ) - k - ss]> ss. (6.30)
2
It is shown in Appendix D that S - a, allowing us to neglect it here.



Hydrodynamic Equations

Once the various integrals have been performed, we find the

following form for the short time hydrodynamic equations, where the





91



stars indicate dimensionless quantities with lengths scales to a and times to tE,


* 2
DtP - ixk* = 0


* * * 2 * ^^ * Dt6e + [D T(X)x + a c k k ]6e

-*
^ * ^ ^ * ^ ^ ^ *
- i[xh(x)k - a (c + y-)(k 6 + k 6 ) - a k k k a]6j
a 2 y xy 3xya a






X 2
= o(a2)


Sa a-*x ^ ^ D t6j a - i[xP (x)k - a (k 6 + k 6 )]6p t a p a 6yS(x) y ax x ay


-i[xP (x)k - a (c4 x)(k 6 ax+ k 6 )]6e
e a +y ex x sy


2 * * 2 *
+ v (x)x 6 + (v' (x)-v (x))x k k ]6j



+ a [c6(6ax 6y + 6 ay6 ) + C kx k y6



+ o8k (k6x + k 6 ) + (k ky6 x + kx6 ay)k8


^ . ^ * * * 2
+ I0 kxkyk k $16j + a 6 6 6j = o(a ) (6.31)



where x = ko and,





92


* 2 2 DT (x) = (1 - Jo(X))/x


2 2 v' (x) = (1 - J(x) + 2j2(x))/x2


2 2
v (x) = (1 - j0(x) - j2(X))/x


* It
h (x) (1 + 3yj1(x)/x)
36y2

* T S-1x)
P (x) S (x)
36y


P (x) (1 + 3yj (x)/x) (6.32)
e 3 y1(x)3yj


are wavevector dependent generalizations of the thermal conductivity, viscosity, shear viscosity, enthalpy, density derivative of pressure and energy derivative of pressure. The constants y = 2 n
-* 5w 2 -*
and v [1 + - y]; v is a dimensionless form of the kinetic
24y 5
contribution to the viscosity, nk. The functions jn(x) are nth-order spherical Bessel functions with j1(x) = (sin x)/x. The functions c = ci (x;a) are given in Table 6.1. Finally,
* 3 .* 2 * a * a p pa j = jt E/a, etc. while Dt aijki . is the at j
convective derivative in terms of dimensionless time t* = t/tE and a = atE.

At this point, we make a few comments about the general nature of these equations. The first of eqs. (6.31) is the familiar continuity equation describing the conservation of mass. The second two





93



equations describe, respectively, the conservation of energy and momentum. However, due to the fact that the effects of atomic structure are included, they are more complicated than at small wavevector. These complications disappear if we take x << 1 and expand to second order in x in which case we get a short time version of the linearized Navier-Stoke's equations. We note that in h, Pe' Pp and c2 we have picked up second order contributions to the pressure in the long wavelength (small x) limit. To illustrate the meaning of the new couplings, we have, in Table 6.2, given the expansions of the coefficients ci to second order in x. It is seen that all of the collisional contributions to the nonequilibrium couplings are of third order or higher in the gradients except for c2 (recall that these coefficients are all multiplied by the shear rate, a, which is a first order gradient). Thus, they do not appear in the ordinary NavierStokes equations but rather in the higher order, so called Burnett, equations. They are always present in the analysis to be presented below, however, as we keep all orders in ka.

Also, these equations contain wavevector dependent "transport

coefficients," eq. (6.32), which are the high frequency limit of the general wavevector and frequency dependent transport coefficients,



DT(X) = lim DT(x,w), (6.33) W-).


etc. Wavevector dependent susceptibilities such as the pressure derivatives also appear. An important feature of the transport




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NONEQUILIBRIUM FLUCTUATIONS AND TRANSPORT IN SHEARED FLUIDS BY JAMES FRANCIS LUTSKO 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 1986

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To my parents.

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ACKNOWLEDGMENTS I would like to thank Jim Dufty for his guidance, encouragement and patience throughout the time I have worked with him. His supervision and friendship have proven invaluable in the course of my work . In addition, my gratitude goes to the other faculty who have guided me through my studies as well as to my fellow travelers in graduate school. In random order, they are Dave, Brad, Pradeep, Gary, Jim, Simon, Tom, Bob and Marti. Thanks also go to Bob Caldwell for assistance with the numerical analysis. Finally, special thanks are extended to Stephanie for her support, encouragement and friendship. iii

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 2 PHENOMENOLOGY AND SIMULATION 12 Phenomenology 12 Experiments 13 Simulations 15 3 FLUCTUATION MODELS 22 Generalized Langevin Equation 22 Long Wavelength Fluctuations 29 Small Length Scale Fluctuations 32 4 LINEARIZED HYDRODYNAMIC MODEL 3H Linearized Model 31+ Green's Function 37 Correlation Functions 47 Physical Properties 52 Probability Densities 57 5 NONLINEAR FLUCTUATING HYDRODYNAMICS 63 Naive Renormalization 63 Formal Renormalization 69 Application of Formal Renormalization 74 6 SHORT TIME MODEL 80 General Properties of the Reduced Distribution Functions 80 Explicit Form of Matrix Elements 83 iv

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Approximations 86 Hydrodynamic Equations 90 7 ANALYSIS OF THE SHORT TIME MODEL 96 Equilibrium Generalized Hydrodynamics 96 Nonequilibrium Generalized Hydrodynamics 97 8 CONCLUSIONS 121 APPENDICES A LINEARIZATION OF THE LANGEVIN EQUATION 125 B GENERALIZED EIGENVALUE PROBLEM 133 C DYNAMIC STRUCTURE FACTOR 139 D ELIMINATION OF THE THREE POINT FUNCTION 142 BIBLIOGRAPHY 1H6 BIOGRAPHICAL SKETCH 150 v

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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 NONEQUILIBRIUM FLUCTUATIONS AND TRANSPORT IN SHEARED FLUIDS BY JAMES FRANCIS LUTSKO December 1986 Chairman: James W. Dufty Major Department: Physics Hydrodynamic fluctuations in a sheared fluid far from equilibrium are described by means of nonlinear stochastic differential equations. Two models resulting from two different limits of the exact Fokker-Planck equation describing hydrodynamic fluctuations in an arbitrary fluid are presented. The first model results from restricting attention to fluctuations which take place on length scales large with respect to the mean free path. The resulting Navier-Stokes-Langevin equations are specialized to shear flow and the associated linear hydrodynamic modes are determined. These are used to calculate the correlation functions of fluctuations around the steady state with the full nonlinear dependence on shear rate necessary for large shear rates. It is then shown how the coupling of the linear modes through the nonlinearities in the stochastic equations allows one to calculate the effect of the nonequilibrium state on the macroscopic transport properties. vi

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The second model results from the short time limit of the exact Fokker-Planck equation and describes fluctuations even on atomic length scales provided the mean free path is small enough. The linear modes of this model are obtained and it is found that they differ considerably from the long wavelength modes. In particular, it is found that at large shear there are two sets of propagating or sound modes. A large shear instability is also found in this model and is tentatively identified with a large shear dynamic phase transition observed in computer simulations of sheared fluids. vii

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CHAPTER 1 INTRODUCTION Statistical mechanics is a methodology for determining the physical properties of many body systems from the microscopic dynamics of its constituents. It is applicable when the properties one wishes to determine arise as a result of the interaction of a large number of the elementary constituents or when the time scales of interest are large with respect to the typical interaction time of the constituents. When this is the case, attention is shifted from the dynamics of individual constituents to the density in phase space (i.e., the space defined by the degrees of freedom of the system) of the system as a whole which is characterized by the distribution function of the system. The dynamics of the distribution function is given by the Liouville equation, which expresses the conservation of phase space density in a conservative system, together with appropriate boundary conditions, initial conditions and source terms representing the action of nonconservative external interactions. When the distribution function is constant in time, the system is said to be in a steady state. Equilibrium is the particular steady state obtained by conservative systems and, in general, is unique in that the associated distribution function is known. For this reason, the statistical mechanics of equilibrium systems is well developed and 1

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2 quite successful at predicting macroscopic properties such as transport equations and thermodynamic susceptibilities. The statistical mechanics of systems not in equilibrium is, in contrast, poorly understood. This difficulty arises from the fact that nonequilibrium states are difficult to characterize; the distribution function is not known. For systems close to equilibrium linear response theory provides a means of studying nonequilibrium systems by relating their properties to those of equilibrium systems (e.g., fluctuation-dissipation theorems relate transport properties to equilibrium fluctuations). However, far from equilibrium one has only the Liouville equation. To get a handle on systems far from equilibrium, one typically restricts attention to some subset of the degrees of freedom which obeys an approximately closed dynamics on the length and time scales of interest. Closed, in this context,, means that couplings to the degrees of freedom not in the subset being studied are negligible on the average. For example, the macroscopic equations of continuum mechanics for the local conserved densities (e.g., hydrodynamics) often provide an adequate characterization of the nonequilibrium state. The problem then becomes one of determining the average dynamics of these variables (i.e., macroscopic transport equations) and their fluctuations. At present, almost nothing is known about the transport properties of systems far from equilibrium. This semi-phenomenological approach was first proposed some fifty years ago by Onsager 1 and has essentially been verified for systems 2 3 near equilibrium. ' Recently, it has been extended to apply to

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3 systems far from equilibrium, ' D but it is difficult to evaluate the success of such an extension since it has been applied to very few problems. The reason it has not been more thoroughly tested is that its application is straightforward only if the nonequil ibrium state is characterized by a few simple macroscopic properties. One such nonequilibrium system is a simple fluid in a state of uniform shear flow. In uniform shear flow, the pressure and temperature are spatially constant while the flow velocity varies linearly with distance in a direction perpendicular to the flow velocity (Figure 1.1). In a rectangular coordinate system (denoted as x-y-z), the flow field, u^r.) , may be written u^tr) = a-r (1.1) where the shear tensor, a, is a = axy (1.2) and "a" is the shear rate. The deviation of this system from equilibrium is controlled by a single parameter, the shear rate, giving a simple characterization of the nonequilibrium state. A sheared fluid is one of the simplest nontrivial nonequilibrium systems imaginable and, indeed, may appear so simple as to be uninteresting. However, even in this simple system drastic effects due to deviations from equilibrium are present, for example in its transport properties, giving the system an intrinsic as well as

PAGE 11

4 2 C c CD E tO <*-. C 9 bO •iH

PAGE 12

pedagogical interest. In the remainder of this introduction, we will first describe three anomalies in the behavior of sheared fluids, and then we will describe the methods and physical ideas which will be used to attempt an understanding of the system. The introduction concludes with an outline of the work to be presented. Sheared fluids are of value as model nonequilibrium systems not only because of the simplicity of the macroscopic state, but also because a great deal of effort has gone into developing methods of simulating such systems on a computer . 6-1 ^ This is of prime importance because, as will be discussed in detail later, laboratory experiments involving simple atomic fluids are at present limited to the small shear rate regime (i.e., close to equilibrium). As will be discussed in the next chapter, experiments are possible involving more complex rheological fluids, such as polymeric fluids and glasses, 14 and colloidal suspensions 1 5 which exhibit many of the properties to be discussed here for simple fluids. We concentrate on simple fluids, however, to simplify the theoretical description. Thus, the only "empirical" data which exist for simple fluids at large shear rates are from simulations. In fact, it is this body of "empirical" results which has spurred most of the recent theoretical interest. 1 *' 16 " 25 The first important anomaly observed in these simulations was associated with the phenomena of shear thinning. As some fluids are sheared, their viscosity drops; they become "thinner" or less viscous. The dependence of viscosity on shear rate had been calculated in 1969 by Yamada and Kawasaki 16 as

PAGE 13

6 n(a) = n Q ^a 172 + 0(a). (1 .3) The nonanalytic nature of the dependence on the shear rate was previously unsuspected and indicates the breakdown of nonlinear response based on a formal expansion of the Liouville equation in the shear rate. In addition, the result of the simulations was that ni was measured to be two orders of magnitude larger than Yamada and 16 1 ft Kawasaki predicted. Several other authors calculated rii and obtained different numbers, although all are two orders of magnitude too small. A recent proposal by van Beijeren for the resolution of this problem will be discussed below. In 1982, a second anomaly observed in simulations was reported by 1 2 Erpenbeck. He found that, at very high shear rates, the system passed from the amorphous fluid phase into an ordered phase in which the particles line up one behind another in hexagonally packed "strings" (Figure 1.2). There has recently 27 been some debate about the actual nature of the string phase but it is generally agreed that a high shear rate instability, or dynamical phase transition, exists. We shall present below a model exhibiting an instability on atomic scales which matches up qualitatively quite well with that seen in the simulations (note that macroscopic hydrodynamics is believed to be stable for planar shear flow 28 ). A third anomaly has been reported by Evans 7 as occurring in twodimensional fluids. His simulations show the shear viscosity remaining constant as the shear rate increases until at a critical

PAGE 14

7

PAGE 15

shear rate the fluid begins to thin in the expected manner. Evans attributes this to a small shear rate instability for which no theoretical understanding exists as yet. However, the transport properties of two dimensional fluids are anomolous even in equilibrium (two is the critical dimension) leading one to believe that Evans' observations may be related to this fact. Most of the work to be presented here will consist of an attempt to construct and analyze models of fluctuations around the nonequilibrium state in dense sheared fluids. Once the fluctuations are statistically characterized, one is able to calculate time dependent correlation functions which in turn can be related to observable physical properties of the system. The phenomenological ideas behind this approach are very similar to those in the EinsteinSmolochowski model of Brownian motion. 2 ^ One considers the dynamics of the hydrodynamic variables (analogous to the position of the Brownian particle) around the steady state. The hydrodynamic variables are appropriate to describe properties on a scale which is large compared to the microscopic correlation length. Because these variables are the local conserved densities, it is assumed that they decay according to the macroscopic conservation laws (e.g., their time evolution is governed by the Navier-Stokes equations) with the exception that the equations include stochastic sources which represent the thermal noise responsible for the creation of the fluctuations. 4 ' 30 Finally, the stochastic sources are generally taken to be Gaussian white noise with correlations fixed by a fluctuationdissipation theorem.

PAGE 16

9 Having these equations, one in principle solves them for the hydrodynamic variables as functionals of the stochastic sources and thus may calculate time dependent correlation functions using the known statistical properties of the sources. As alluded to above, these correlation functions can in turn be used to calculate various physical quantities such as the effect of the steady state on transport coefficients and thermodynamic susceptibilities. While providing a closed and intuitively appealing model of fluctuations, this description is entirely phenomenological and as such represents an uncontrolled approximation. Several authors 5 '^ 1 have addressed the question of the formal derivation and validity of fluctuating hydrodynamics and we shall outline their ideas in a later chapter. The result of these studies is that the foundations and validity of this approach are much better understood. Thus, one approach to the study of nonequilibrium systems is to replace the microscopic description of the system, as characterized by the distribution function and Liouville's equation, with a hydrodynamic description. A hydrodynamic description of a system is obtained as a result of two approximations. The first is that attention is restricted to the local densities of the conserved variables (e.g., mass, momentum and energy in a simple fluid). On length scales which are large compared to the microscopic correlation length (i.e., the mean free path, i mfp ) , these variables are expected to decay slowly since their integrals over the whole system do not change at all with time. Thus, on these length scales, these variables are assumed to approximately decouple from the remaining,

PAGE 17

10 quickly fluctuating, microscopic degrees of freedom which give rise to dissipation and thermal fluctuation of the slow variables. Concurrent with this, if k is the wavevector (inverse wavelength or gradient in real space) of the conserved densities, it is required that (k£ mfp ) be small. The equations governing the dynamics of the conserved densities may then be expanded in this small parameter with truncation at second order yielding, for a fluid, the usual Navier-Stokes equations . In a sufficiently dense fluid, the mean free path is actually smaller than the typical size, a, of the atoms making up the fluid. In this case, the hydrodynamic description will include terms of all order in (ka) while still expanding in (k& mfp ). The result is a hydrodynamic description of the (dense) fluid which contains information about the structure of the fluid and is valid on atomic (i.e., ko 1) length scales although still restricted to lengths which are large relative to the mean free path (ki. mfp << 1). One might think, and indeed it was long thought, that atomic scale fluctuations require knowledge of all small scale fluctuations; after all, the reason for considering only conserved densities at long wavelengths is that they decay on macroscopic length and time scales while all other fluctuations decay on much shorter, microscopic, scales. However, while studying models of small length scale fluctuations, de Schepper and Cohen5 discovered that even in this regime the conserved densities decayed much slower than other variables and gave the dominant contribution, on these scales, to the density-density correlation function. This indicated that restricting

PAGE 18

11 attention to the hydrodynamic subspace was a valid means of studying fluctuations even on very small length scales. Thus, hydrodynamic models can be used to study even atomic length scale fluctuations provided the mean free path is small enough. In particular, van Beijeren's explanation of the discrepancy between theory and simulation results for the size of the shear rate dependent renormalization of the shear viscosity is based on this generalization of fluctuating hydrodynamics to include atomic scale fluctuations in dense fluids. The work to be presented below consists of modeling hydrodynamic fluctuations as a means of understanding the nonequilibrium state. In Chapter 2, we will discuss experimental and simulational means of studying steady state shear flow and the connection between' the two. The third chapter discusses the formal ideas of statistical mechanics used to construct models of nonequilibrium fluctuations and, in particular, to construct fluctuating hydrodynamics at long wavelengths as well as generalized fluctuating hydrodynamics. This is followed by Chapters 4 and 5 analyzing the long wavelength model, the linearized and nonlinear versions respectively, with the aim of deriving various physically observable consequences of the nonequilibrium state. The sixth and seventh chapters contain, respectively, the development and analysis of the small wavelength model and include a discussion of the hydrodynamics and large shear instability occurring on these length scales. The dissertation concludes with a discussion of the results obtained, the completeness of the present description and future directions .

PAGE 19

CHAPTER 2 PHENOMENOLOGY AND SIMULATION In this chapter, we discuss the basic phenomenology, such as length and time scales and strength of shear, which affects the observability of deviations from equilibrium behavior. We then discuss the possibility of performing laboratory experiments which measure nonequilibrium properties. Finally, we discuss the methods of simulating uniform shear flow on a computer. Phenomenology In macroscopic hydrodynamics, one typically has three types of parameters: thermodynamic derivatives such as the sound velocity, c 0 , transport coefficients such as the kinematic viscosity, v 0 , and the wavevector, k, the magnitude of which is inversely proportional to the length scale of the phenomena one is interested in. The time scale of 2 -1 hydrodynamic dissipation £s (v Q k ) (in equilibrium, momentum fluctu~ v 0 k t ations dissipate as e ) . In hydrodynamics, one usually uses perturbation theory treating the wavevector as a small parameter. The dimensionless parameter which actually occurs is (v 0 k/c Q ) thus requiring that v Q k << c Q or v Q k << c Q k. Physically, this means that the rate of hydrodynamic dissipation is small compared to the rate of sound propagation. This is consistent with the Navier-Stoke' s 12

PAGE 20

equations governing hydrodynamics since they are themselves the result of an expansion to second order in the parameter (ki, ), where mfp a mfn is the mean free P ath « and in fact c n /v n "" * • Typical values mi P 0 0 mfp 2 of these constants for water at STP 35 are v 1 .8 • 10~ 2 0 s 5 -7 0 Q 1.5 • 10 cm/s so I = 10 cm. Thus, the applicability of hydrodynamics requires that k << 8.3 • 1 0 6 cm" 1 , or ko << 10~ 1 ~8 where a 10 cm is a typical atomic radius. When a fluid is sheared, a new time scale, the inverse of the shear rate (denoted here as "a"), is introduced. When the shear rate is as large as the rate of dissipation, we expect that large deviations from equilibrium behavior will occur. For this reason, we 2 refer to a v Q k as the large shear regime. When studying the long p wavelength hydrodynamic model, we will require that a < v Q k thus including the large shear regime while allowing us to use perturbation theory (since, given this bound, a << c Q k). Experiments Various methods exist for measuring the static and transport properties of fluids. For example, neutron scattering can be used to measure the static and dynamic structure factors (related to the pair correlation function and transport properties, respectively) on atomic length scales. Light scattering samples the dynamic structure factor on hydrodynamic length and time scales. 37 These methods and others are available to measure the corresponding properties for nonequilibrium fluids. However, the present ability to test theories of sheared systems is limited by the shear rates which can be obtained

PAGE 21

in the laboratory. At present, the largest shear rates reported are less than 10 HZ.^ 0 This shear rate is in the large shear regime 1/2 2-1 when k (a/v Q ) 7.45 • 10 cm . This means that until we look at length scales as large as about 10 cm the effects of shear will be small compared to the usual equilibrium properties. Thus, light scattering, which measures properties on a length scale of the -5 incident beam's wavelength 10 cm, and neutron scattering, which is Q not practical for wavelengths above about 10 ° cm, require shear rates about two orders of magnitude larger than those attainable in the laboratory. There are systems, other than simple fluids, in which similar nonequilibrium effects occur and which have the practical advantage that they have long relaxation times. Because the large shear regime p occurs for a inverse relaxation time (v Q k in a simple fluid) it thus becomes accessible in these systems. One example of such a system is a colloidal suspension in which the relaxation 2 time t d /D for a typical interparticle spacing, d, and diffusion constant of the spheres, D. In this system, the density of spheres determines the relaxation time and the nature of the probe is not restricted as in a simple fluid. Experimental studies of sheared colloidal suspensions have been performed 1 ^ and numerous nonequilibrium phenomena are observed including a string phase similar to that seen in simulations of simple fluids. Another class of systems exhibiting easily accessible nonequilibrium phenomena is macromolecular (e.g., polymeric) fluids

PAGE 22

15 and glasses. The polymeric systems are not simple fluids because they are composed of anisotropic molecules. However, Simons et al.^ have shown that strong similarities exist between the phenomena seen in simple and complex fluids. These fluids demonstrate a wide variety of non-Newtonian behaviors including shear thinning. The accessibility of these phenomena again arise due to the fact that the constituents, the macromolecules, have long relaxation times lowering the lower bound of the large shear regime. In this work, we will restrict our attention to simple fluids. As the preceding discussion indicated, we will not have direct experimental tests of our results. The reason we do not consider colloidal suspensions, polymeric fluids or some other such system is that the results from computer simulations of simple fluids have shown that even our understanding of the simplest system is incomplete. However, because ef fects, such as shear thinning and the string phase, seen in the simulations are observed in physically related laboratory systems, we have some confidence in the simulations. Thus, we propose to attempt to understand the simplest sheared system and hope, in the process, to gain some understanding by analogy of the more complicated systems in which the nonequilibrium state has important physical consequences. Simulations Most of the data for simple fluids have been generated by nonequilibrium molecular dynamics 6-1 3 rather than in the laboratory.

PAGE 23

16 Such simulations require both the establishment of the linear shear profile as well as the maintenance of constant temperature. In equilibrium molecular dynamics, one considers a box with periodic boundary conditions containing N particles. At each time step in the simulation, one solves the equations of motion (e.g., Newton's equations) for all of the particles and moves them accordingly. The periodic boundary conditions have the effect of decreasing the size (number) dependence of the results allowing one to achieve the bulk, or thermodynamic, limit with on the order of 10^ particles. In this way, one may study static properties, such as the pair correlation function, and dynamic properties like the decay of correlations. Clearly, simulations have the advantage over experiments in that one can obtain much more data about the system since one has the microscopic state at one's disposal. They are, however, limited by the number of particles, N, and the length of time which can be simulated. In particular, the size restriction limits the smallest wavevectors which can be studied. The linear shear profile is generated by modifying the simple periodic boundary conditions of equilibrium simulations. The system remains periodic in the x and z directions but when a particle with momentum £ passes through the y = L plane at time t and with coordinates (x,L,z), it is reentered at y = 0 with coordinates (x-aLt, y, z) and momentum £ m§Lx. These boundary conditions are known as LeeEdwards' boundary conditions. They become more transparent when expressed in the local rest frame coordinates, q and jj defined as

PAGE 24

17 q = q a • £t £ ma • £. (2.1) In these coordinates, the local flow velocity is zero and it is seen that the Lee-Edwards' conditions correspond to simple periodic boundary conditions in the rest frame. The temperature, T, is a measure of energy fluctuations around the steady state and is defined by 3 1 N 2 2 Nk B T " <^ l ( £i " m i ' aj) > where kg is Boltzmann's constant. Substitution of the conditions of spatially uniform shear flow into the Navier-Stokes equations shows that a sheared fluid will undergo viscous heating. Specifically, one finds that 3e 2 ^ na , (2.3) where e is the internal energy density, leading to c(t) = e (0) + na 2 t. (2.4)

PAGE 25

18 For water sheared at 10 Hz at STP this leads to a heating rate of 0.024 K/s. For temporally uniform shear flow, particularly for the very high shears applied in computer simulations, an external, nonconservative force is necessary to hold the temperature constant. This heat extracting interaction is called a thermostat. This is equivalent to adding a drag force to the equations of motion: d HtRi " A(r) £i A(D = I L • (F a • £ ) / £ g 2 . (2.5) J J J j J where the form of A(D has been chosen to hold the temperature as defined in eq. (2.2) constant. The equation of motion form for the thermostat is important because, with the Lee-Edwards* boundary conditions for imposing the flow field, the simulations can be mapped onto a well defined mathematical problem: the solution of Liouville's equation, modified to account for the thermostat, with the imposed boundary conditions. The modified Liouville equation is obtained from the general form of Liouville's theorem,^ c !t" + iUi ' ai~ + h ' ff^ptr.t) aj: • CA(r)£.p(r ; t)] = o (2.6)

PAGE 26

19 where the repeated index, i, is to be summed and runs from 1 through N and the dot indicates an ordinary vector scalar product. So, (2.6) may be rewritten as ft P + £p = 0, (2.7) with £ = £ n + £. . + £ . (2.8) 0 int ext where the free streaming and interaction terms are, respectively, £ i„t h 4 < 2 9 > while the external term, due to the nonconservati ve thermostat, acts on the distribution function as *ext p(F;t) = " 9i~ ' ^(Dfi^r.t)]. (2.10) Because this operator includes the nonconservation of volume in phase space due to the nonconservative force (Liouville's theorem states tnat (p(r)dr) = 0 and with nonconservative forces — dr * 0) Qt dt It has a different form when acting on any other phase function.

PAGE 27

20 Specifically, if A(r) is an arbitrary phase function, we have that The bar indicates the difference between these operators and we note that £ Q = £ Q . It is easy to see that £ and £ are adjoints with respect to phase space integration. Finally, we should mention that when working with hard spheres, the potential is singular and the interparticle force, appearing in these equations ill defined. This difficulty is eliminated by replacing the interaction term with the so-called T-operators which generate the singular hard sphere dynamics. Henceforth, for hard spheres, we will write where ± refers to forward and backward time equations. The form of these operators can be found in the literature 110 and, for example, £ ext A(D Kr) £. • Jj-A(r). i (2.11) V^i'W (2.12) a • m 6
PAGE 28

and b 12 A( £i •£! ;a 2 .£ 2 ) = A^.Et " • £ 1 ;a 2 ,£ 2 + oo • £ 12 ). (2. We note here that the T-operators are not self adjoint, as is the interaction term for continuous potentials, eq. (2.9), so their adjoints are denoted with bars as well.

PAGE 29

CHAPTER 3 FLUCTUATION MODELS In this chapter, we discuss the statistical mechanical basis of the models to be analyzed. Specifically, the models we will study will be seen to be two different limits of a general Fokker-Planck equation. The detailed form of the models will also be given. Generalized Langevin Equation Our basic approach, as discussed in the introduction, to studying nonequilibrium fluctuations around shear flow is to choose a restricted set of variables whose dynamics we wish to study. At a fundamental level, any classical system of N structureless particles, or atoms, is described by the 6N coordinates, collectively called r, which are the positions, q [t and momenta p_ of the atoms (where i ranges from 1 to N). The distribution function of the system, p(T;t), which describes the density of states in phase space, satisfies the Liouville equation, |jT p(r ; t) + £ P (r ; t> = 0; (3.1) the explicit form of which was discussed in a Chapter 2. The set of restricted variables we choose to study is the set of conserved 22

PAGE 30

23 densities. These are mass density, p(r), energy density, e(r), and momentum density, j (r). A caret over a function indicates that it is a phase function. Their definitions are N p(r) = I m A(r-q ) ~ n-1 ~ ~ n N N e(£) = * (rJ A(r-a ) + ± I V( q -q ,|) A(r-q ) n=1 m n n 2 n*n* n n n N J a (n) Z £ n Mr-a ) (3.2) n=1 n where A(r) is some localized function of r (for instance, a Dirac delta function). These variables will be denoted collectively as y(r). Our reasons for choosing this particular set were alluded to in the introduction. At length scales large with respect to the mean free path, we expect that the conserved densities are the only variables with long relaxation times (i.e., long with respect to the collison frequency). Therefore, we expect that the statistical properties of the system at these scales can be adequately described in terms of these variables with the other degrees of freedom being modelled as rapidly varying sources. In dense fluids, the mean free path can be smaller than the atomic dimensions and it is found, as one might guess, that the hydrodynamic variables, or conserved densities, still have long relaxation times and so still, on the average, obey an approximately closed dynamics. Because the other degrees of freedom are more rapidly varying, they may again be modelled as stochastic sources.

PAGE 31

24 In the analysis to follow, it will prove convenient to work with the Fourier transform of the variables in eq. (3.2). This is defined to be ik t y(k) = V" 1 / dr e " ~ y(r). (3-3) We also note that the vector y(k) is explicitly, y(k) = (p(k), i(k), ji(k) ) . (3.4) For notational convenience, we will write y a ** y ct ( ) (3 * 5) where the Latin index includes both the vector index a and the wavevectorJ<_. Repeated Latin indices are to be summed. The average of y a . or y a (k), over the nonequilibrium ensemble, p(T;t), will be denoted as where Si = / dr P (r ; t)y a (3 .6) and are the macroscopic variables one would measure in the laboratory. In general, one also defines time dependent phase functions, y(k;t), as

PAGE 32

25 where y (k;t) = e" £t y (k) a — a — y a (t) (3.7) = /dr y a e" £t P (r ; o) = /dr [e' £t y ] p(r;0) CL = /dr y (t) P (r ; o) . (3.8) a So, the macroscopic variables are the average over an initial ensemble, p(T;0), of y (t). The fluctuations of y (t), denoted as 6y (t), are defined to be a 6y a (t) = y a (t) " (3.9) Because we are restricting attention to the set of variables y(t), it is useful to introduce the distribution function of these variables, p(y;t), defined to be

PAGE 33

26 p(y;t) = Jar 5(y-y') p(r';t), (3.10) where y is some possible value of the phase function y. So, for any function of the y's, say A[y], it is clear that = Jdr A[y] p(T;t) Jdy A[y] p(y;t). (3.11) Finally, for a steady state such as uniform shear flow, we have that P ag (r;t) = P gs (r;0) = P (T). So, = a ss •'a ss s y 0a . (3.12) Grabert et al. 1 show, using the projection operator technique hp introduced by Zwanzig, that the distribution p(y;t) satisfies the following exact generalized Fokker-Planck equation, 3 9 t 3t P(y;t) + — {vjy] p(y;t) J Q ds Jdy' D^Cy.y'js] P ss (y') 3y. 3y b ' p ss (y')

PAGE 34

27 where p (y) is the steady state distribution, v [y] is called the S3 £1 drift vector and D^Cy.y'js] is called the diffusion matrix. Repeated Latin indices are again to be summed in (3.13). The functionals v and D are expressed in terms of steady state averages and their detailed form is given in Grabert et al. 1 Here, we only note that — {v [y] P (y)} = 0 (3.14) 3 y a so P (y) explicitly satisfies (3.13). 3 3 Associated with (3.13) is an exact generalized Langevin equation for the variables y (t), 3t y a (t) v a Cy(t)] + /Jds {D ab [y(t-s);s] F b [y(t-s)] 3y b (t-s) D ab [y(t-s);s]} R a (t) (3.15) where the thermodynamic forces, F b [y(t-s)] are defined in terms of the steady state distribution as 3 in P (y) V'*-" ly-J(t-s) o.«) which also appears in (3.13). The functional v a is the same as that appearing in (3.13) while

PAGE 35

28 D ab [y;s] = /dy' D ab Cy,y ;s]| y _^ (3.17) The sources R (t) have the following constrained averages, a 0 = /dl" 6(y a -y a ') R a '(t) P ga (r») D ab [y:t] = / dI " 6( V y a' ) R a' (t) R b' (0) p ss (r,) (3 ' 18) where the hat on R a (t) indicates it to be a phase function and where a primed function has argument I". The first of equations (3.18) indicates that R a ,(t) is orthogonal, under the steady state average, to any function of y and indicates that the constrained average of R (t) at fixed y is zero. The second of equations (3.18) has the form of a fluctuation-dissipation theorem relating the correlation of the "fluctuating force" to the diffusion matrix. Now, eq. (3.15) together with properties (3.18) may be modelled as a stochastic differential equation if the sources R (t) are a idealized as stochastic processes, denoted as R 3 (t), which have the form, R a (t) = M mn Cy] S n (t) (3.19) where = = (2tt) 3 6(k+k') 5(t-t») A , cm ' S A aa« 6(t_t,) ( 3-20)

PAGE 36

29 where A is a constant matrix and the amplitudes M [y] are fixed by aa mn the fluctuation-dissipation theorem, eq. (3.18). Finally, the variables y (t) are replaced by stochastic variables denoted as y (t) = y (k,t) and the average over the distribution function 3. Ot replaced with that over the stochastic sources, ^a (t)> ss * V t}> R(t) (3 ' 21) since the variables y (t) may be considered to be functionals of the a stochastic sources. Long Wavelength Fluctuations If we restrict our attention to long wavelength fluctuations, then the preceding Langevin model can be explicitly evaluated. Specifically, we only consider wavevectors such that *A mfp «1. (3.22) In this case, the various functionals appearing in eq. (3.15) may be evaluated in terms of a power series in k£ . . If we truncate the mfp expansions at second order, then the model may be given explicitly, 5 in terms of the usual microscopic conservation laws, ft 9 + 1 ' 1 = 0

PAGE 37

30 ft 6 + 1 ' a = 0 It j a + jk fc a6 " ° (3 ' 23) where s a and t a g are the heat flux vector and stress tensor, respectively. These in turn each have a convective part, a dissipative part and a stochastic part, where the latter represents the stochastic source, g R s = eu + t -u. + X — T + s a a aB 6 o 3r a -a U V»B + P6 ae " Vctfn U } " *o 6 a 6 ^ + & (3-24) where we have introduced the fluctuating velocity, u = p~ 1 1 the a °a fluctuating pressure and temperature, p and T, and transport coefficients, n Q » < Q and X . These fluctuating thermodynamic functions and transport coefficients are the same functions of y as one has in equilibrium; i.e., p[y] = p [y] " P e Cp.eJ (3.25) where p is the equilibrium pressure functions and the last line

PAGE 38

31 indicates that p g depends only on the (local) density and internal energy density, e, where e is given by e e p j . (3.26) The dissipative parts of the heat flux vector and stress tensor are seen to be given by a fluctuating version of Fourier's law and Newton's viscosity law, respectively, where Finally, as previously mentioned, the stochastic sources are modelled as Gaussian white noise with covariances determined by the fluctuation-dissipation theorem, eq. (3.18), S* = (k R A T 2 ) 1/2 s (r,t) a Bo a — fc ae = (k B TTl o )1/2 W^' t} + (k B TK o )1 /2 (3 ' 28) = = = 0 a lag 2ag <3 a (r 1 ,t 1 )s e (r 2 ,t 2 )> = 2 6 a6 Sir^) Sit^) < WVV S 2 l i V (£ 2» t 2 )> = 2 6 a6 6 yv 6( W 6( VV (3 ' 29)

PAGE 39

with all other cumulants equal to zero. Equations (3.23) (3.29) completely specify the fluctuating hydrodynamics model which may be summarized as a Langevin equation for the conserved densities, y (t), with a deterministic part given by the Navier-Stokes equations, local equilibrium transport coefficients and susceptibilities supplemented by the stochastic sources, eqs. (3.28) and (3.29), which model the excluded degrees of freedom. Finally, we note that this model is given in the language of eq. (3.15) in Appendix A where it should be noted that — D a Jy;t] o (3.30) 8y b as is shown by Zubarev and Morozov. Small Length Scale Fluctuations In a dense fluid, the mean free path may actually be smaller than the typical atomic length scale, to be denoted here as a (for a gas of hard spheres, o would be the hard sphere diameter). In this case, one may study atomic length scale fluctuations with the present methods. However, the evaluation of eq. (3.15) requires care for while we still expand in (k£ mfp ) we must keep all orders in (ka). Quantities dependent on (ka) carry information about the small scale structure of the fluid. We can evaluate (3.15) while retaining the desired information regarding the small scale structure by taking its short time limit to obtain

PAGE 40

33 ft y a (t) vJyU)] = R a (t). (3.3D The drift vector is defined to be v a Cy(t)] = (<6(y-y) £ y a > ss /<5(y-y)> ss }| y= ; (t) . (3.32) In general, this is a complicated nonlinear functional of y(t). However, as is explained in Chapter 5, we will only be interested in this functional expanded to second order in the deviations from the stationary state. Thus, we expand as v Cy(t)] = v [y (k)] + 6y.(t) • [ -J v [y(t)]] „ . etc. 3 ° • b «y b (t) 3 y 0 ( ^ (3.33) The first order term is sr — v Cy(t)] = (<<5y~l) ho <5y„£y > . (3.3*0 ,5y ( t ) a ss be J c a ss This model is explicitly evaluated in Chapter 6 and analyzed in Chapter 7. While it may seem strange to study the short time limit of the generalized Langevin equation, it should be noted that similar short time equations have been found to provide qualitatively accurate descriptions of dense fluids even out of the short time regime (an excellent example is the Enskog equation*^) •

PAGE 41

CHAPTER 4 LINEARIZED HYDRODYNAMIC MODEL In this chapter, we linearize the Navier-Stokes-Langevin equations described in Chapter 3 and use them to calculate various statistical properties of the system. We will do this in such a manner that the results apply for shear rates up to and including the large shear regime as previously defined. Of particular interest will be the hydrodynamic propagators and the two time correlation functions. We will discuss the applications of our results to light scattering experiments and will make contact with the small shear results of other authors. The chapter concludes with a description of various formal statistical properties of the system highlighting the difference of this model from local equilibrium. Linearized Model The complete hydrodynamic model was given in Chapter 3, eqs. (3.23) (3.29), and will not be repeated here. The variables we choose to work with are the mass density, total energy density and momentum. So, y a * (p.e.j) a (4.1) 3^

PAGE 42

35 and we denote the steady state averages of these quantities with subscript naught, y 0a~ (p 0'VVa {H ' 2) and the deviations from the steady state, the fluctuations, are denoted by Z a , Z a ( -* t) " y a ( -' fc) " y 0a ( ,o) (4 ' 3) where we also use internal energy density and velocity in place of total energy density and momentum density. Finally, we scale and Fourier transform Z a (r,t), to obtain the variable Z(k,t) in terms of which we wish to work, Z a (k,t) = /d 3 r e 1 ^ Z a (r,t) (4.H) Z a (k,t) «*• Cc^pCk.t), c 2 6e(k,t), e (a) (k).u(k,t)] (4.5) -2 8p 0 2 -1 8p n where p Q = P LE Cp 0 »e 0 ] is the steady state, macroscopic pressure; h 0 = e 0 + p 0 is the enthal Py density and {e (a) (k ) } are a set of pairwise orthonormal vectors with e (1) (k) = k/k = k and e is the internal energy density and u = p" 1 j is the velocity. The scaling,

PAGE 43

36 (4.6), will simplify the form of the equations below and we emphasize that c-j and C2 are equilibrium thermodynamic susceptibilities. In the course of Fourier transforming and linearizing the equations, we find that we must transform the convective terms which include the operator Pq 1 ^ • V which on an arbitrary function f(rO looks like P~ 0 \ ' Vf(r) = a. .r. JL f (r). (4.7) The transform of which is (4.8) Thus, it turns out that the form of the linearized equations is ft V*.t) " a. .k. Z a (k,t) + H a6 (k;a) Z g (k,t) = R^k.t) (4.9) 3 where R a represents the stochastic sources which, at linear order, do not depend on the fluctuations, Z 0 . The details of the linearization including the explicit forms of and R a are given in Appendix A. Equation (4.9) is the primary result of this section. In the following sections, we will analyze the hydrodynamic (i.e., normal) modes described by this equation. For zero shear, H is the usual

PAGE 44

matrix of linearized hydrodynamics the eigenvalues of which describe sound propagation, heat diffusion and momentum diffusion. An important feature of eq. (4.9) is the coupling of variables of different wavevectors implicit in the gradient term. This unusual feature has important consequences on the evolution of fluctuations and is known as linear mode coupling. Green's Function To solve equation (4.9), we introduce the Green's function G a g(k.,k.' ;t) defined to satisfy ft W^' ;t) a ij k i W~. W^' :t) + H aa ( ^ a)G a6 ( ^ ,;t) * 0 with G (k,k';0) = (2tt) 3 6(k k') 5 0 . (4.10) In equilibrium, i.e., a = 0, the solution of (4.10) requires nothing more than the diagonal ization of the hydrodynamic matrix H . Before ag solving (4.10) for nonzero shear, we recall the equilibrium solution. 39 Let the eigenvectors of H ag (k_,a=0) be denoted as l£ (l) (k) } i=1 >5 and the corresponding eigenvalues be {A^QO}^ 5 so that H a0 (k.O) S^Oc) = X (i) (k) ^ i} (k). (4.11)

PAGE 45

38 Furthermore, as H a g is in general nonhermitian, we introduce the biorthogonal set of vectors, [y, (k ) } , defined to satisfy i = i , b n (l) (k) • £ (J) (k) = 6 . (4.12) In terms of these functions, the Green's function can be immediately determined to be 5 G ae Qi.i<';t) = ! ^ l) (k)ng i) (k')(2 1 r) 3 5(k-k')exp [A . (k) (t-t Q )] (4.13) and t 0 is a constant which can be fixed by an additional boundary condition. The solution to eq. (4.9) is then given by dk' V*' fc) = J 7TT3 G a6 ( -'-' ;t) V*'»V (2ir) dk' (2tt) dk ' + It dT / — T G (k,k' ;t-T) R.(k',T) (4.14) where the wavevector integrations extend to infinity. Because the hydrodynamic matrix H ag is the result of a truncation at second order of an expansion in (k4 mf ), the eigenvalues can be determined by perturbation theory in this same small parameter. In equilibrium, one obtains two sound (propagating) modes, a heat mode and two degenerate shear modes,

PAGE 46

39 * 2 (k) 2 (4.15) where explicit forms for the speed of sound, c Q , the sound damping constant, r Q , and the thermal diffusivity, D QT , will be given later. For the present, we note only that the modes all decay like exp-( constant) k t and that these results can be used to calculate the intensity of scattered light from an equilibrium fluid (as described below). Upon doing so, one obtains the classic Landau-Placzek 44 formula which is perhaps the most successful application of fluctuating hydrodynamics. In fact, comparison of this result and experiment provides one of the best known means of measuring the equilibrium transport coefficients. Returning to the nonequilibrium problem, eq. (4.10), we introduce a shear rate dependent generalization of eqs. (4.11) (4.12) as follows:

PAGE 47

40 ' a ij k i W~. ^^ a) + H aa ( * 5a) 4* )( * }a) ^ (A) («Li*)ci A) (|ci«) n (l) (k;a) • i (J) (k;a) = 6 . (4.16) The presence of the gradient operator in (4.16) greatly complicates the problem. However, as discussed in Chapter 2, we restrict the size of the shear rate by requiring that the rate of convection be no larger than the rate of dissipation yielding the condition, a < v Q k 2 . (4.17) This choice is what allowed us to neglect heating in these equations, since the rate of heating goes like a 2 k 4 (see eq. 2.3), and makes possible a perturbative solution of eq. (4.16). The details of this calculation are given in Appendix B. The eigenvalues so obtained are A = -ic k + (r k 2 + ak k /k 2 ) l 0 2 0 x y A = ic k + (r k 2 + ak k /k 2 ) ^ U d 0 x y X 3 ' D 0T k ' 2 2 An v n k ak k /k 4 0 x y A 5 v Q k 2 . (i4 18 )

PAGE 48

41 We see that the degeneracy of the shear modes has been lifted while the sound damping constant has been shifted and the heat mode remains unaffected at this order. If we now write the Green's function as 5 O ag (k,k';t) = I n^ i) (k;a)5g i) (k';a)G (i) (l<,l<';t), (4.19) then we find that the function satisfies the following equation: C lt ' a ii k i 9k~ + * U) G U) (k,k';t) = 0 J G v '(k,k';0) = (2ir) J S(k-k'). (4.20) This scalar equation is easily solved and if we introduce a time dependent wavevector as k.(t) = k. k. aj .t = k. 6. y k x at, (4.21) then the solution of (4.20) is G (i) (k,k';t) = (2tt) 3 5(k-k'(t))E (1) (k,t) where E (l) (k,t) = exp \\ dtA.(k(T )). (4.22) z 0 1 The explicit form of the propagators in (4.22) is

PAGE 49

42 E (1) (k,t) = [k/k(-t)] 1/2 exp{-ic 0 ka(t) | r Q k 2 B(t)} (2) (1 )* E^ ; (k,t) = E U; (k,t) E (3) (k,t) = exp{-D QT k 2 8(t)} E v ; (k,t) = [k(-t)/k] exp{-v Q k 6(-t)} E (5) (k,t) = exp{-v 0 k 2 g(-t)} (4.23) where a(t) = (2ak k)'^ {(k (-t)k(-t) k k) x y y k(-t) + k (-t) sgn(ak ) + k ln[ k ? / sgn(ak ) X ] 1 y x B(t) = (ak x k 2 )" 1 |k 2 (k y (-t) k y ) +1 (k 3 (-t) k 3 )} , 2 , 2 ,2 k k x + \> (4.24) These functions behave for long times as o(t) (ak /2k)t 2 x 8(t) * ] (ak x /k) 2 t 3 . (4.25)

PAGE 50

Because lim f5(t) > 0, the modes are all stable as expected (see Drazin and Reid and Case for further discussion of the stability of macroscopic shear flow). In order to illustrate the physical significance of some of the shear rate dependent effects described by these equations, consider the evolution of a purely hydrodynamic sound excitation along the direction of flow. Specifically, we take Z (k,0) = £ (k) 6(k-k„) a — a — — -0 with k^ = k Q x. The evolution of hydrodynamic excitations is given by eq. (4.14) when conditionally averaged over the stochastic force with fixed initial condition Z^Ck.O). Denoting this average by a double bracket we have that dk' «Z (k,t)» = / G „(k,k<;t) Z Q (k',0) (2w) J op 3 SOS-k^t)) ^ 1) (k) E (1) (k,t). (4.26) Inverting the Fourier transforms yields, (1) 2 ? 1/U -r n k n 0(t,-a)/2 «Z a (r,t)» = £y ; (k Q (t))(1 + a 2 t 2 ) 17 ^ e 0 0 x cos{k Q [x(t) c Q a(t,-a)]} (4.27) where a(t,a) and 8(t,a) were defined in (4.25) and x(t) = x + ayt. The time dependent coordinate x(t) is due to the expected Doppler shift in the sound wave which travels in a moving medium. This kinematic effect may be suppressed by restricting attention to the

PAGE 51

44 y = 0 plane and without loss of generality, we may chose x = 0 as well. Thus, the remaining shear effects are due to gradients of the velocity field and not the magnitude of the velocity. Figure (4.1) shows <>/<
PAGE 52

a c cO CO 3 -a c 3 C CO 4h o bO C •H • Q. E I CO I O I ^ •o s° II c CC\I o H J* o bO \ CO CO a. o t, £O hO

PAGE 53

46

PAGE 54

Correlation Functions The correlation functions of the deviations of the fluctuations from the steady state are useful for a number of reasons. From them one may calculate the dynamic structure factor, which describes the intensity of light scattered from the system and transport properties such as the shear rate dependent renormalized viscosities and they characterize the deviation of the system from local equilibrium. In this section, we calculate the two point correlation functions. The two point correlation functions are defined to be where the brackets denote the nonequilibrium average which in the present case corresponds to an average over the stochastic forces. Time reversal invariance and stationarity require that where P Q = ±1 is the parity of Z a under time reversal. Therefore, may consider t _> 0 without loss of generality. If we define C aB (k,fk' f t';a) = (4.3D c „ ntk.tjk'.t' ;a) = C o (k,t-t';k,0;a) C af} (k.t;k',t';a) = P^C^Ck' ,t;k,f -a) (4.32) F (i) (k,t) = n (i) (k;a) R (k,t), — a a — (4.33)

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48 then, using eq. (4.14), we find that C a6 (k,t;k\0;a) = C (k,k» ;t) I /" dT/;dxM^ i ^l<)^ J) (k')E (i) ( ! <,x)E ( J ) ( J <., T .) i t J l (4.34) where we have taken t Q + -« in eq. (4.14) and used the boundary condition that Mk,00 ) is finite. It is clear that (2Tr) 3 6(k+k')6(t-f)F (iJ) (k) (4.35) where the form of F (lJ) (k) is easily determined. Thus, C a(} (k.t;k') = ( 2l r) 3 5(k + k'(t)) I } (k ) ) (-k . ) E ( 1 } (k, t)C 0 (-k';a). (4.36) The functions C ag (k;a) are related to the equal time condition functions because, in particular, c ae Q<.t=0;k' ;a ) = (27r) 3 5(k+k')C ae (k;a). (4.37) The explicit form of these functions is

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49 V!<;a) = i; dx I {^ i) (k)^ J) (-k)E (i) (l<,x)E (J) (-k,x) i.j x F (iJ) (k(-x))}. (4.38) The nonzero elements of C 0 (k;a) are given in Tables 4.1 and 4.2. Sore immediate information may be gained from these expressions. In particular, eq. (4.36) shows that the two body correlations decay in the same way, i.e., with the same propagator E *^(k t t), as the macroscopic modes. This represents a generalization of Onsager's regression hypothesis to nonequilibrium states. Furthermore, eq. (4.38) is a fluctuation-dissipation theorem for the nonequilibrium system. These statements are more obvious when cast in the form of differential equations: C lt " a ij k i g^WH'H'it) + H aa (k;a) C oe (k,k',t) = 0 (4.39) C_6 aa a ij k i air + H aa (k ;a)] C a6 (k ;a) + [ V _k ;a)] C aa (k -' a) = V* J (4.40) where R ag (k_) is the amplitude of the fluctuating forces, = 6(k+k») 6(t-f ) R 0 (k). (4.41) 0t p Equation (4.39) shows more clearly the linear regression law while (4.40) has the usual form of a fluctuation-dissipation relation.

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50 Table 4.1 Equal time correlations. C pp ( * ;a) =k B0 0 X T [1 + A^k.a)] V* :a) = k B T oVo X T [1 -n^ T 0 + V* a)] C £e (k;a) = k B T 0 x T [x;V v + [h 0 " T 0 ^ + *W±' M C 33 (k;a) = k B T Q PQ [1 + YA^ (k;a)] C 41| (k;a) = k B T QP Q 1 [1 A 2 (k;a)] C J45 (k;a) = I< b T 0 Pq 1 A 3 (k;a) C 55 (k;a) = k B T oP Q 1 [1 A 4 (k;a)]

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51 Table 4.2 Nonequilibrium contributions to equal time correlations, -irkk v k v ( t) -r 0 k 2 e(-t) A (k;a) = aY 1 J dt — ^ e ° , , fk x k y (_t) 2 V 2 S(-t) A 2 (k;a) = 2aJ 0 dt e 0 k A 3 (k ;a) " a /o dt hrfer F(k -' t} + fk y k v (_t) k , -2v n k 2 6(-t) A^kja) = 2aJ 0 dt F(k,t)[ \ y ( . t) F(k,t) jj£]e ° F(k,t) = A(k(-t)) A(k) k k k A(k) = 1 tan" 1 (-I) k k k x k(t) k k • a t k = k + k x z

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52 Physical Properties We now discuss two physical properties which may be determined from the correlation functions. They are the long range nature of the correlations and the light scattering spectrum. The density-density equal time correlation function in real space is easily seen to have the form dk C n J r ' a) m \ a e " C (k;a) = Jl~ 3 k B T 0 p^x T [6(r/£) + %/r F(r/4)] (4.42) where the isothermal compressibility, x T = P~ 1 (|^) T , and where T 9p T 1 /2 I = (r Q /a) is a typical length scale associated with the shear rate. The limit F(r/J,)=0 yields the usual equilibrium result where the 6-function is an idealization of the atomic scale correlations which actually exist in the fluid. The part porportional to r/fl, has been obtained before via an expansion to first order in the shear 20 rate. Because of the form of F, this actually requires that r/l « 1 and yields F(0) = -a r.r./8iTYar 2 (4.43) * J ^ J where Y = c p /c y is the ratio of specific heats. This does not, as has 20 been claimed, however, allow us to conclude that the correlations are long-ranged because it is the product of a short distance expansion. To determine the true nature of the correlations, at large

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53 distances, we consider a somewhat simpler problem than the large-r expansion of F(r,l). Specifically, we consider Jdy/dz C pp (r) = r 1 k B T 0 p^x T C6(x/Jl) + YH(x/i)]. (4.44) The part of the correlation function due to the deviation from equilibrium, H(x/4), is given by HCx/4) = (2A)" 1 r Q dtf(l tV 3/2 t" 1/2 (1 1 t 2 )" 1/2 x exp[-(x/Jl) 2 (4t(1 +jt 2 ))" 1 ] (4.45) and for large (x/4) behaves as H(x/4) * 0.828(x/A)' 5/3 . (4.46) At short distances, this function is Gaussian and the complete function is shown in Fig. 4.2. It is thus clear that the correlations are in fact long ranged, decaying as a power law. The second physical property we wish to study with the correlation functions is the intensity of light scattered from the fluid. It can be shown 39 that the intensity of light at the point r with frequency a due to the initial scattering of a coherent beam of wavevector and frequency u 0 is given by

PAGE 61

bo

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55 I(r,w) JI Q a) 1, (a/2TrCQr)sin * S(kjQ) (4.47) where I Q is the spectral intensity of the incoming beam, a is the polarizability of the particles composing the medium, • s Eq cos<}>, Eq is the amplitude of the coherent beam, j£ kg ~ (u Q /C Q )r and = oj w Q and where the structure factor, S(k_,Q), is defined as dk dk S(k,0) J-dt J Te(k-k )e(k + k ? )C n (k 1 t;k„,0;a)e iQt . (4.48) Thus, we see that the density-density correlation function determines the intensity of scattered light. The functions, 6(k), are Fourier transforms of form factors which limit the volume of integration to that irradiated. In Appendix C, it is shown that S(k,Q) can be written in terms of nonequilibrium Brillouin and Rayleigh peaks: S(k;Q) = S R (k;fl) + S B (k;fl) + S B (k;-Q) (4.49) where t&iQ) " 2Re{[-io + Xl (k;a) \ ^"Vft 1 >n< 1 \ (k;a) } S R (k;«) = 2Re{[-iQ + A 3 (k;a) i 2 a ij k i 3 kj J S c i \ c al (k ; a)j. (4.50)

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56 In equilibrium, the gradients are not present and these expressions reduce to simple Lorentzians. Expansion of eqs. ( 4 . 19) to first order in the shear rate yields previously obtained results; however, the general evaluation of (4.19) is difficult. A quantity which can be calculated directly is the Landau-Placzek ratio, R, defined as R(k;a) * Jdw S R (k , u ) /Jdw[S B (k , M ) + S (k,-u)] (4.51) where a = P Q o^/h Q c 2 . In particular, the integrated Rayleigh intensity is unchanged from equilibrium, Jdw S R (k,u>;a) = ( Y 1)/Y (4.52) where Y = ratio of specific heats. An explicit form of (4.51), obtained using the expressions for the equal time correlation functions given in Appendix C, is R(k,a) = (Y 1 )/[1 + YA (k;a)] (4.53)

PAGE 64

57 where A = [C (k;a) C (k;0)]/C (k,0). In Fig. 4.3, we plot i pp pp — pp AR = [fHk;a) -jUk ;0) ] /R(k ;0 ) as a function of 9 for 2 k k Q (cosex + siney) and a/r Q k 0 = 1. The small shear rate expansion of these results is in agreement with earlier work by Kirkpatrick et 20 al. The result, eq. (4.53), is unexpected in that it states that the integrated intensity of the Rayleigh peak is unaffected by the shear. Probability Densities In this section, we recast our previous results in the language of the probability distributions of the fluctuations. In particular, the probability distributions are more useful than the description in terms of fluctuations for the calculation of higher order correlation functions or for averages of more general functions. The probability and joint probability density are defined by P(Z,t) = <6(Z(t) Z)> P(Z,t;Z\t') = <6(Z(t) Z) 6(Z'(f) Z')> (4.54) where the averages are taken over the stochastic forces, {R a }, and over a specified ensemble of initial values, {z(0)}. The 6-functions are short for <5(Z(t) Z) = n 6(Z (k,t) Z (k)). . a — a — a,k (4.55)

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We begin the evaluation of eq. (4.51) by calculating a related quantity— the conditional probability density defined by W(Z,t;Z\t') = P(Z,t;Z\f)/P(Z',f). (4.56) Because the system is being modelled by a Markovian process, the conditional distribution is independent of the initial values, l z a (0)} because of the neglect of heating, the system is stationary so W(Z,t;Z\t') = W(Z,t-t' ;Z » ,0) . So, without loss of generality, we consider W(Z,t';Z',0) = <6(Z(t|Z») Z> R (4.57) where, as indicated, the average is only over the stochastic forces and Z(0|Z') = Z'. From eq. (4.14), z a (t|Z') = G(t)Z« + /J d-r G(t-x) R( T ) (4.58) where we have suppressed internal summations. An integral representation of the 6-function in (4.57) and use of (4.58) yields W(Z,t;Z',0) = JdA e U,CZ-G(t)Z,] = JdA expjiACZ G(t)Z'] A 2 M}

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60 = ldet[TTM(t)]}" 1/2 exp{|[Z-G(t)Z'] • M~ 1 (t) • [Z-G(t)Z-]} (4.59) where the second line follows from the first upon use of the Gaussian statistics of the source term. The matrix M(t) is given by M(t) = /Jdi/Jdr' G(t-t) • • G T (t-T'). (4.60) Equations (4.39) and (4.40) (the regression law and fluctuationdissipation theorem) allow us to re-express (4.60) in terms of the correlation function matrix, C(t) = C D (k,t;k' ,0) , otp — — M(t) = C(0) C(t) • C _1 (0) • C T (t). (4.61) We can now immediately write down the probability and joint probability densities, P(Z,t) = JdZ« W(Z,t;Z',0) P(Z\0) P(Z,t'Z',t') = W(Z,t;Z',f) P(Z'.t'), (4.62) where P(Z',0) is the initial probability density. From (4.61), we can determine the stationary state,

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61 P (Z) = lim P(Z,t) ss {det[TrC(0)]}" 1/2 exp[| Z • c" 1 (0) • Z] (4.63) which follows from the limits, lim C(t) = 0 => lim M(t) = C(0) (4.61) t+°° t +o> and, lim G(t) = 0. Thus, as expected, because the driving process was t+<*> Gaussian and the equations linear, the distribution of fluctuations is also Gaussian. Furthermore, eq. (4.63) shows that the same asymptotic stationary state is reached regardless of the initial condition. All of these results could have been obtained if we had worked with the Fokker-Planck equation rather than the fluctuation equation. For completeness, we write down the Fokker-Planck equation for P(Z,t) which is easily obtained. For example, by differentiating (4.62) with respect to time, It P(Z ' U = "I STlky CJ 0c t (Z ' t) + J 1a (Z ' t)] (M 65) OL f K 0t where the drift term is J oo (z t) = Ca ij k i atT L ae (!l^)Z B (k)]P(Z,t) (4.66)

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and the diffusion term is J 1a (Z ' t} "5 P(Z ' t) (4 ' 67 6 The matrix R (k), the diffusion matrix, is just the correlation up matrix of the stochastic forces. Equations (4.65) (4.67) are the usual results for the relationship between a linear Langevin equation and the corresponding Fokker-Planck equation.

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CHAPTER 5 NONLINEAR FLUCTUATING HYDRODYNAMICS In this chapter, we will consider the consequences of the nonlinear terms occurring in the general fluctuating hydrodynamics model. We begin by considering a naive form of renormalization which illustrates in an intuitive way the concept of renormalization. Using this method, we will derive the lowest order renormalization of the shear viscosity explicitly deriving the nonanalytic dependence on shear rate. In the following sections, we will present a more formal development of renormalization which will result in a self-consistent formulation of the nonlinear problem. The chapter concludes with the connection between the naive and formal renormalization. Naive Renormalization Consider a simple, one dimensional, nonlinear, stochastic equation of the form It X + L 0 X + V 3 = f(t) (5.1) with a Gaussian, white stochastic source f(t) having correlations, = f 0 6(t r t 2 ) (5. 2) 63

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64 and all higher order cumulants equal to zero. Furthermore, let the average of x(t), , be denoted by x 0 (t). Then, the equation of motion for x Q (t) is found by averaging eq. (5.1), ft X 0 (t) + L 0 X 0 (t) + N 0 = 0. (5.3) If we write x(t) = x (t) + 6x(t), then (5.3) becomes ft X 0 (t) + CL 0 + 3N 0 <(6x(t)) 2 >]x Q (t) + N Q x 3 (t) = N Q <(6x(t)) 3 >. (5.4) We see that the equation for x Q (t) has the same form as that for the fluctuating variable, x(t), except that the "bare" coefficient L Q has been renormalized by the nonlinearity to become the "dressed" or renormalized coefficient L, L = L Q + 3N Q <(6x(t)) 2 >. (5.5) If we subtract eq. (5.4) from (5.1) and solve the resulting equation for 6x neglecting the nonlinear term, and use this to compute L, we find that L = L 0 + 3N 0 f 0 /2L 0 + o(N 0 } (5.6) where we have imposed stationarity , |^ = 0, to fix the equal time correlation functions.

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65 The previous calculation can be reformulated in terms of correlation functions. The equation of motion for the two-point function, <6x(t)6x(0)> = c(t), is found from (5.1) to be |^ C(t) + L Q C(t) + N 0 <(6x(t)) 3 6x(0)> = 0 (5.7) where we have used the condition = 0. To lowest, N Q = 0, order, we find that the Laplace transform of C(t), C(z), is C(z) = C(0)/(-z + Lq) + o(N Q ) (5.8) which has a pole at the bare transport coefficient L Q . If we calculate the correlation function in (5.7) to lowest order, we find that <(6x(t))3 5x( 0)> J^dx^/ 0 .^ e'Vt-V^-^-Vt-VV* + 0(N ) 3 ft j fO j ^V^V ' L 0 (t T 2 ) 2 3 )-J^).J^ 2 e e 2 f Q 2 + o(N Q ) 1 1 " L 0 t 2 3 2L^2L^ e f 0 +0( V 3f 2L^ c( ° )(t) + o( V" (5.9)

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66 So, N f |^ C(t) + (L Q + \ -—) C(t) = o(N Q 2 ) (5.10) and C(z) = C(0)/(-z + L Q + | N Q f 0 /L 0 ) + o(N Q 2 ) (5.11) which has a pole at the "renormalized" coefficient defined in (5.5). This coincidence is easy to understand. Suppose that at time t-0, the distribution function, p(x), is perturbed from the steady state function, p , in the following way s p(t-0) = p g e' h6x . (5.12) The auxiliary field, h, might be chosen such that h * ss = x Q , i.e., to yield some particular initial value for the macroscopic variable. For small deviations from the steady state (i.e., h small so is small), we find that n s s x Q = C(t)h + o(h 2 ) = (C(t)/C(0))[ h x Q ] + o(h 2 ). (5.13) , the correlation function acts as the Green's function for small

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67 deviations of the macroscopic variables. They must, therefore, share the same linear relaxation times. A general proof of this property for Markovian processes has been given by Dufty et al. In the same way, we can derive an expression for the renormalized shear viscosity. If we take the average of eqs. (2.45), we require that we obtain the macroscopic hydrodynamic equations. For shear flow, the only nontrivial equation is the heat equation, where n R is the renormalized viscosity. Comparing (5.14) with the average of eqs. (2.45) yields where we have used temporal and spatial translational invariance to set overall derivatives of the correlation functions to zero. The third line of (5.15) follows directly from the momentum equation. Using our results from the previous chapter, eq. (5.15) can be written, to lowest order in the nonlinearity , in terms of the equal (5.14) -2 3 n R = ~ a J = a' 2 < Ui (r,t) ^t. j (r,t)> J = a' 2 j J -2 3 = a <[ 3r~ u i ( H' t)] P(£.t) u^r.t) u (r,t)> (5.15)

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68 time correlation functions, n R = a 1 p Q = -(2ir) Jd kCk x k y c 33 (k;a) + k^c^k ;a) k^Ckia)]. (5.16) This equation for the lowest order renormalization of the viscosity is an example of mode coupling. That is, the original nonlinearities in the equations of motion have coupled two different hydrodynamic modes, analogous to phonons in solid state physics, and the correction to the viscosity is due to the interaction of these modes, analogous to the phonon-phonon scattering contribution to the coefficient of thermal diffusion in solids. The coupling of the modes in (5.16) is buried in the equal time correlation functions. It can be seen by referring to eq. ( -4 . 38) . Contributions also exist due to three mode coupling, four mode coupling, etc. Upon evaluation, we find that 1 /2 \ = n Q + r^a (5.17) with 0 v 0 Thus, it is seen that a nonanalytic dependence of the viscosity on shear rate arises as a result of mode coupling. This is in contrast to the analytic dependence one would obtain from the standard ChapmanEnskog solution of the Boltzmann equation (or the Enskog equation; see

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69 Chapter 6). This result agrees with that of Ernst et al. 16 and with that of Yamada and Kawasaki 16 (although their numerical evaluations are incorrect). Thus, this result indicates the equivalence, with the present assumptions, of this model and the others appearing in the literature. Formal Renormalization In this section, we wish to develop a more general method of renormalization than the naive scheme, allowing the calculation of all renormalized properties. To this end, we begin by formulating the nonlinear model in a more general notation, It y m + v m [ y ] + D mn Cy]F n Cy] = M mn Cy]5 n (5 ' 19) where v m [z] represents the nonlinear Euler terms in eq. (3.23) (i.e., the terms V • j = V (pu) in (3.23) and ev and v j in (3.210). The Ot Qt 3 matrix D^Cy] is called the diffusion matrix and F [y] are the thermodynamic forces which are the variables thermodynamically conjugate to the y m 's. The product D mn F n represents the dissipative terms in (3.24). The reason for returning this form of the equations is that in their derivation of fluctuating hydrodynamics, Zubarev and Morozov show that thermodynamic consistency imposes the fluctuationdissipation theorem, (5.20)

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70 "thermodynamic consistency" meaning that the equilibrium distribution function is a solution to (5.17). Equation (5.17) is highly nonlinear making a general analysis difficult. However, in equilibrium it has been found that truncating eq. (5.17) at second order in the nonl inearities provides a qualitatively correct description of such nonlinear phenomena as long time tails (i.e., the algebraic decay of equilibrium correlation functions). In fact, in equilibrium, the dominant mode coupling effects arise from the Euler terms in (5.19). We shall, therefore, consider only these nonlinearities in the rest of the discussion. It can be shown that, to order (ak), this is equivalent to taking both D mn and M mn t0 lowe st order (i.e., Djy] D mn [y Q ]) thus preserving the fluctuation-dissipation theorem, eq. (5.20), in this truncated model of eq. (5.19). Furthermore, F n [y] is expanded to first order in the fluctuations in order to preserve the usual linearized transport laws. With these approximations, and expanding the variables y m about their steady state averages y Qm , eq. (5.19) takes the form, 3t z a (k,t) L ab (k,t) z b (k,t) + eV ab0 ) R (k,t) (5.21) where L ab + H ah^' a ) is the linear operator studied in

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71 the previous chapter and ~ y & (details are given in Appendix B). The vertices, which arise solely from the Euler terms, have the form = *abo ( *»v 5( *-w (5 22) where the explicit form of A^k.gJ is given in Appendix B. The constant e is introduced as a notational convenience in the development of a perturbative expansion in the nonlinearity . Equation (5.21) can be rewritten using the Green's function introduced in Chapter 4 as « a (k,t) Z^W) «jSdT{0 Mf ) J (5.23) where a sum over repeated wavevectors is implicit and where Z a° )( * ;t) " G aa' ( ^ ;t) z a ' ( 4>°> + /Jdx O^.C^t-t)^, (£.,). (5.24) Using (5.23), a perturbative solution to eq. (5.21) is easily constructed. From eq. (5.21), we can immediately construct the equations of motion of the two-point correlation functions,

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72 C . (k,k';t) = , ao a — b — ft C ab (fc,K'.t) + L aa ,(k;a) C a , b (k,k';t) + M^Oc.k' ;t) = 0 (5.25) where we have defined M ab (k,k';t) = V^Ck.q^q^ (5.26) and have used the fact that = 0 for t*b (k,k';t) = /JdT I aa ,(k,k i; t~r) C^U^ ,k' ;x) . (5.27) The significance of this is that eq. (5.24) becomes a linear equation for the correlation functions and effective viscosities, etc. as would be measured in the laboratory may be identified. Laplace transforming (5.24) in conjunction with (5.27), one obtains Z 5 ab ( ^' ;w) + L aa' ( ^ ;a) £ a'b ( ^' ;w) +
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73 E a b ( ,k -1 JM) = \c ( ^'^ ;w) C~ ob^S.k'iw). (5.29) As discussed in the previous chapter, C (k,k»;t) = C (k , t )<5(k-k ' ( t) ) ab — ab — — — and it is easily verified that, order by order in perturbation theory, M ab ( -'-' ;t) S M (k,t)6(k-k'(t)) which requires that lab^.k'.t) s I ab (k,t)6(k-k'(t)). Thus, eq. (5.28) becomes ft C ab ( ^' t} + D ^>C ab (k,t) + H aa ,(k;a)C a , b (k,t) + eM^Oc.t) = 0 (5.30) where we have introduced D = -a ^ -|. The equation for the selfenergy becomes M ab ( ^ t} = /o dT I M .^t-T)C aIb (k(T-t) f T)) which shows the self-energy to be an operator in k-space. Laplace transforming (5.31 ) yields an expression for the self-energy, J 0 dt e ^ aa ( ii ;t)e M ab ( * ;w)5 ba' ( * ;w) 3 ^aa' ( -' w) (5 -32)

PAGE 81

74 and the Laplace transform of (5.30) is [wD]C ab (k,w) + H aaf (k,a)C a , b (k.w) + e / 0 "dtI aa ,(k;(O)e(w+D)(T) 5 a , b (k,w) = C ab (k,t=0). (5.33) Equations (5.32) and (5.33) are the primary results of this section. The linear mode coupling which was the unique feature of the linear problem is seen to play an important role here as well. The operator nature of the self-energy is a result of the linear mode coupling and expresses the fact that new gradient couplings, as well as the expected new hydrodynamic couplings, arise as a result of the process of renormalization. This complication is, however, removed in the context of perturbation theory in e because we have an expression, to lowest order in e, for the operator (w+D) acting on the correlation functions (c.f. eq. (5.33)). It is seen that the self-energy function shifts the poles of the correlation function renormalizing the transport coefficients as in eq. (5.11). Application of Formal Renormalization We now wish to apply the renormalization formalism to the calculation of the renormalized shear viscosity and, in so doing, to make a connection with the naive renormalization method. As a simplification, we will restrict our attention to the contribution arising from the shear modes. In this case, the irreversible part of the stress tensor, t , can, in general, be written as

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75 " v ijlm 9^ J m ' (5 ' 3J4) where v m is the viscosity tensor which, in the unrenormalized theory, has the Navier-Stoke's form, V ijlm " ^ 6 ii 6 jm + 5 im 6 jl § + (5.35) where v and k are the usual kinematic and bulk viscosities, respectively. The macroscopic equation for the energy density is ft e + 3^ C(e + p)Po 1 j. + t. 1 p* 1 j 1 s.] = 0 (5.36) For macroscopic shear flow, using eq. (5.34), the heating equation becomes It 6 + V ijlm a ij a lm = °(5.37) Thus, the renormalized viscosity calculated earlier from the heating equation corresponds to the yxxy element of the general viscosity tensor. We now calculate the self-energy to second order in the nonlinearity (i.e., to second order in e). Using eqs. (5.23) and (5.26), we find that

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76 M ab (k,k';t) = eV acd (k,q 1 ,q 2 ) t)z b (k' > 0)> + 2e2 /o tdT{V acd ( ^a 1 .a 2 )G ce (a 1 ,a 3 ;t-T)V efg (q 3 ,q 4 , % ) x <4 0)( ^' T)z g 0)( %' T)z d 0)( %' T)z b ( ^'' o)> J + ° (e3) (5.38) where the factor of two arises from the symmetry of the vertices. Using eq. (5.24) and causality, this reduces to M ab (k,k';t) «V Md (M 1 .fl e )0 oe ,(fl 1 .fl 3 lt)O ddi ( flB . a4 «t) x 0)z d ,(q ljf 0)z b (k',0)> G ff .(£l 4 .a j ;;T)G gg ,(q 5 ,q^; T )G dd ,(q 2 ,q 2; t) x } + o(e 3 ). (5.39) In equilibrium, the equal time fluctuations are Gaussian distributed making the o(e) term in eq. (5.39) zero and allowing easy evaluation 2 of the 0 (e ) term. In the steady-state, the equal time fluctuations are not necessarily Gaussian distributed. Stationarity , in the steady state, allows us to write

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77 z a (k;0) = (k;0) «/!.dt{0 M( (k, ai »-T)V afbo ( ai . fl2 . a3 ) x (z b (q 2 ;x)z c (q 3 ;T) 0. Thus, the equal time fluctations may be determined perturbatively and are Gaussian distributed to zero tn order in e. Equations (5.39) (5.42) provide us with enough information to calculate the self-energy function from eq. (5.28). The result is that £ aa .Q<.i<";w) e / 0 -dt eWt {V acd (k, 2l , a2 )G cc ,( ai , a3 ;t)G dd ,( a2 , ai|; t) * <2 c .(%.0)z d ,( 24 ,0)z b (k',0)>}c^ a ;" 1 (l<',k«;z)

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78 P r oo — wfc r + Wo dte { V acd ( ^ r a 2 )V ega ,(q 3 ,q 1( ,!<") x G d e ( a 2 'a3 ;t)G cd' ( ^'^;t)}< Zd ,(q^0)z (q^,0)> + o(e 3 ). (5.H 3 ) To recover our previously calculated lowest order mode coupling contribution to the viscosity, we first replace the equal-time correlations in (5. 43) with their local equilibrium values. The term proportional to e in (5.43) then vanishes and (5.43) becomes k T E aat (k,k«;w) --f^Jo" e " Wt l A acd (l<,q)A eyat (k(-t)-q(-t),-q(-t)) -D(k)t x ^e^'S't^Qd.^.'tJe }6(k+k"). (5.44) The momentum-momentum self-energy upon evaluation is found to have the form y-* z) k i k j w-* z) (5 -_*« which identifies it as the renormalization of the viscosity tensor. If we write v. = v<°> m Sv'..^, then the long wavelength, long time hydrodynamic viscosity appearing in eq. (5.37) is

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79 «v ww = lim I (k,w) (5.16) k+0 w+0 yxxy k+Q "yxp x p y from which the "naive" result is recovered. The replacement of the static correlation functions by their local equilibrium values requires some comment. For a stationary process, the correlation of the stochastic sources is related through fluctuation-dissipation theorems to the static correlations. The mode coupling contributions to the static correlations may thus be interpreted as a renormalization of the amplitudes of the stochastic sources. By neglecting the equal time three-point function in (5.42), we were left with the propagator renormalization. There is another way to understand the replacement of the static correlations in the present calculation by their local equilibrium values in order to find agreement with the "naive" calculation. The only information about the "statics" that was used in the "naive" calculation was the stochastic source correlations. But, as explained in Chapter 3, these are approximated, in the unrenormalized theory, by their local equilibrium values. It is, therefore, reasonable that the same information must be used in both calculations to obtain agreement. We speculate that if the first order renormalized forces were used in the naive calculation, one would obtain the same result as is obtained here without the local equilibrium replacement.

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CHAPTER 6 SHORT TIME MODEL In this chapter, we discuss the explicit form of the short time model of hydrodynamic fluctuations introduced in Chapter 3. We recall that this model is intended to describe hydrodynamic fluctuations on length scales as small as atomic lengths. We begin by discussing general properties of the distribution functions. This is followed by a derivation of explicit expressions for the necessary matrix elements in terms of the reduced distribution functions. The final two sections describe the approximations introduced to evaluate these matrix elements and the explicit form of the resulting equations governing the time evolution of hydrodynamic fluctuations. General Pro perties of the Reduced Distribution Functions In this section, we introduce two properties of the distribution functions which will be of use in determining the general nature of the expressions, eqs. (3. 31 ) -(3. 34) , to be evaluated below and which will aid in their evaluation. These properties are the stationarity , in time, and the modified translational invariance of the distribution functions. As discussed in Chapter 2, an external nonconservati ve force is applied in computer simulations of sheared fluids to counterbalance viscous heating of the system. The significance of this is that it 80

PAGE 88

81 yields a steady state. If we denote by p (r) the distribution ss function (density matrix) of the steady state system, depending on all of the phase space variables r = lSLi»£ilJ.i« tnen we can write the steady state Liouville equation as 0 " ft p ss (r) + £ "-P (r) = «.p(r) (6.2) where £_ is the Liouville operator. The explicit form of £ is generally given by n =1 m£n 9 a n 2^ n , 9 q n 3 £n + £ ext (6 ' 3) where V(|q J1 -q Jl ,|) i S the pair potential. For hard spheres, the potential is singular rendering eq. (6.3) ill defined. As discussed in Chapter 3, however, eq. (6.3) can be replaced by a well defined expression which produces a statistically equivalent dynamical system N 1 £ ~ = h £ n * 3q~ + 2 ^ ^_(n,n') + T_(n',n)] f (6.4) n=l -^n n*n* ext where we have included a term representing the externally applied thermostat alluded to above. The explicit form of the T operator and its various adjoints and time-reversed forms are given in the literature and will not be repeated here. The point we wish to make

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82 is that (6.2) allows the £ + operator to be moved in averages, i.e., for any two phase functions, A(r) and B(r), it can be shown that = <(£ a(D) B(r)> . (6.5) ss ss The subscript on £_ refers to the fact that T + goes to its time reversed form T_. Another important property of the distribution functions is their modified translational invariance. Clearly, because the local flow velocity varies spatially, the distribution function cannot be translationally invariant. That is, if Ei^i " R (6.6) for an arbitrary displacement R , then However, if we make the transformation * * Ei-S-i S. L ~ ma ijRj. l t R , (6.8) then we expect the distribution function to be invariant since the macroscopic state is characterized by the local velocity and the local velocity gradient and, while (6.6) does not preserve these properties, (6.8) does. Mathematically, it is easy to verify that (6.8) is

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83 consistent with the Lees-Edwards boundary conditions while (6.6) is not. We, therefore, look for solutions of Liouville's equation which are functions of the variables a 1<3 ~= £i aj (6.9) which are invariant under (6.8), £1'% * £i*'a*j(6.10) Henceforth, we will write p (r) a p (f D '. a }) ss K ss l ^i' % j J Explicit Form of Matrix Elements The short time model is given in Chapter 3 as atV^t) + L a6 (k,k';a)6£ 6 (k',t) =0 (6.11) where V^' ;a) <(£ + y a (k)>*y Y (!<")>33 (g ~\ 6 ,k«,k' (6 12) and

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84 g YR (k",k') = <5y v (k")6y Q (k')> . (6.13) IP T p — SS The variables, 5y (k ) , are again taken to be the fluctuations of the conserved densities around the steady state: fiy, (k) = P(k) = — I me 1 -'^ /V n 6y 2 (k) = 6e'(k) — I | k T )e^^n 2 /V n 2 B o «y 2+6 (!<) = 6j:(k) = 4 I (p:) r cs.14) /V n n 3 which are summarized as 6y a (k) = 4 I J (j.) e^-Sn, (6 . 15) V n where V is the volume of the system. We now define the reduced distribution functions, nnf(n)(X 1 X n } " (MhnTT H +1 dX n p ss (r,) (6 ' l6) where « (a n »£ n ) and we note that the Jacobian of the change of variables X r x; is unity. Using (6.16) and the independence of p gs on the center of mass, we find that B oB " (2 l r) 3 V _1 6(k + k') g ae (k), (6.17)

PAGE 92

where g ag (k) n /dp' f (1) ( £ .) ; a ( £ .) y £ .) The explicit form of £ y (k) is easily found to be + a — £ + C (k) i 4 I H • ^ i (R')e^^n a /v n m a n + ?4 I [T (n,n*) + T (n',n)] Y (p')e^" k i 3 ij 3T J a ( ^ + = (2TT) 3 «5(k+k')M Q (k) a p ss — — ag — " k i a ij 9kT + ^xtV^V^^ss' where

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86 M a6 (k) = lnk./d E }^i o ( E [)i B ( Ef )f (1 >( E{ ) + \ n 2 /d £ 'd^d£ 12 (CT + (1,2) + T + (2J)])i a ( E |)y E .)f (2) ( E . >£ . ;ai2 ) °2 / d Ei d E2 d ai2 (CT + (1 ' 2)+T + (2 ' 1)]) ^(Ei)V £ 2 )el ~ 1112 x f (2) (£ 1 ',E';a 12 ) + §7 J d Ej d^d^dq, 2 dq 1 3 ( CT + ( 1 , 2) + T + ( 2 , 1 ) ] ) Y a (£• ) ) y 6 ) e 1 " £l 3 (3) x f (a, .£2 '£3 ; a 12 -a 13 )« (6.21) Approximations The evaluation of eqs. (6.21) and (6.18) require knowledge of the oneand two-body reduced distribution functions for the steady state. In general, these functions cannot be determined explicitly and, so, we introduce two important approximations. The first of these is the neglect of momentum correlations, which is to say we assume the two-body distribution to have the form f (2) (X',X') "= f (2) ( £l ', E2 ; £l2 ) = ^"(P^f^Cp^gCajg). (6.22)

PAGE 94

87 where f 1 ^(j>' ) is the one-body reduced distribution function, which cannot depend on due to the modified translational invariance, and the function g(q. 12 ) is the steady state analogue of the pair correlation function (i.e., 2^ d Sl 2 is the Probability of finding a particle, 2, in the volume dq^ 2 if there is a particle, 1, at position £ 1 ). In equilibrium, this approximate form becomes exact. However, away from equilibrium, we expect that particle velocities are indeed correlated (i.e., (6.22) implies that <£{£^> sa = 0 but in general this need not be the case). This approximation, which is analogous to Boltzmann's Stosszahlansatz , ^ is standard in the development of kinetic equations. In particular, it is used to derive the Enskog equation which is known to give a qualitatively accurate description of fluids near equilibrium. 1 * 3 With this approximation, the three-body term in (6.21) only contributes for 6=1. in Appendix D, it is shown that this term can be written, using an exact identity following from stationarity, in terms of two-body functions. Having adopted the form (6.22), we must still determine the onebody distribution and the pair correlation function. The second approximation we use is to replace the steady state pair correlation function, g(q 12 ), by the local equilibrium pair correlation function, 8 LE (|l 12 |) = 8le. (q 12^ The pair correlation function is related to the density-density equal time correlation function and knowledge of one of these allows determination of the other. This approximation, which, like the previous one, is used in the derivation of various kinetic equations, may be thought of as a first guess to be "renormalized" using the density autocorrelation function calculated

PAGE 95

88 from the resulting equations. This approximation then becomes the first iteration of a self-consistent calculation. In fact, these two approximations require that f (1) (p_.f) satisfy an Enskog equation which is the usual dense fluid analogue of the Boltzmann equation. With these approximations, our model is completely defined. The one-body distribution can be obtained from the first equation of the BBGKY hierarchy (i.e., by integrating eq. (6.2) over coordinates X£...X^ and using eqs. (6.4) and (6.22)): ^4" f(1)(1,) = / d V-^ 2 ) f(1) ^)f (1) (2)g LE ( ai2 ) + sources. (6.23) The sources in eq. (6.23) arise from the nonconservati ve forces in the Liouville equation (eq. 6.4). However, it can be shown that they contribute to second order in the shear rate and so are neglected in the present calculation which is only carried out to first order in the shear. While the previous approximations served to construct a model, we now make an approximation with respect to solving that model; we will only work to first order in the shear rate. Thus we expand f ' 1 \ f "teP = f LE ( £i )(1 + •!<£{) + -..) (6.24) where f LE (£») is the local equilibrium distribution function for uniform shear flow,

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89 -3 /2 2 f LE (Rj) n(2miTk B T o ) J exp (pj c /2mk B T Q ) . (6.25) With these approximations, one is able to determine ^(jaj) in eq. (6.25). As this result exists in the literature, 3 we only quote it here: a/ 1 V = -^jVoVPliPij " I Pl ,2
PAGE 97

90 while the matrix M 0 (k,a) can be written as dp — ik-£». M = -n< y (p') y (p')> + X oB " S ag (6.28) where <...> E denotes an average over f E ( £ *) truncated to first order in a. The collisional contribution, I ag , is I a6 (k,a) = -n 2 X /d £l 'd£'d£ 12 y a ( £ .) T_(12)f (1) ( £ pf (l) ( £j .) x ^ B ( £ ') + Y 6 ( £ ')e 12 5 31 e ~ H2 ] (6.29) and the source term, S , is a8 S a 6 = ^extV^V"^ ~ V'^ss^as' (6.30) It is shown in Appendix D that a 2 , allowing us to neglect it here. Hydrodynamic Equations Once the various integrals have been performed, we find the following form for the short time hydrodynamic equations, where the

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91 stars indicate dimensionless quantities with lengths scales to a and times to tg, D p ixk«j_ = 0 * * # 2 * ~ " * D.6e + [D T (x)x + a c,k k ]6e t T 1 x y iCxh(x) ka a*(c 2 * %)(k y 6 ax k x 6 ay ) a% 3 k x k y k a ]6j^ o(a 2 ) — * D*6j* i[xP (x)k a* ^st-t (k 6 + k 6 )]6p* c 01 Pa oyS(x) y ax x ay F -* iCxP (x)k a*(c, + X )(k 6 + k 6 )]6e* e a 1 tt y ax x ay + | [v*(x)x 2 6 a6 + (v'*(x)-v*(x))x 2 k a k e ]6j* + a [0,(6 6„ +5 6„ ) + c k k <5 6 ax By ay flx' 7 x y afl + c ft k (k {. + k 5„ ) + n (k 6 + k 6 )k 8 a y Bx x By 9 y ax xay B + C 10 k x k y k a k 6 ]6J B + a \x 6 By 6J B = ^ < 6 -3D where x = ka and,

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92 yx) 1 d j 0 (x))/ x 2 v'*(x) = I (1 j.(x) + 21 (x))/x 2 v*(x) = I (1 J 0 ( X ) j 2 (x))/x 2 h (x) = r (1 + 3yj 1 (x)/x) 36y^ 1 P*(x) = -2-^ S"\x) P 36y^ P e (x) = | (1 + 3yj 1 (x)/x) (6.32) are wave vector dependent generalizations of the thermal conductivity, viscosity, shear viscosity, enthalpy, density derivative of pressure and energy derivative of pressure. The constants y = Tma 3 x and v = 2Hy ^ 1 + 5 v is a diroensionless form of the kinetic contribution to the viscosity, The functions j n (x) are n th -order spherical Bessel functions with j^x) = (sin x)/x. The functions c. = c.(x;a) are given in Table 6.1. Finally, * 3 .* 2 * a * s P = Pa , j = jt E /a, etc. while D fc a^k. la the convective derivative in terms of dimensionless timeV = t/t and * E a = at £ . At this point, we make a few comments about the general nature of these equations. The first of eqs. (6.31) is the familiar continuity equation describing the conservation of mass. The second two

PAGE 100

93 equations describe, respectively, the conservation of energy and momentum. However, due to the fact that the effects of atomic structure are included, they are more complicated than at small wavevector. These complications disappear if we take x << 1 and expand to second order in x in which case we get a short time version of the linearized Navier-Stoke's equations. We note that in h, P p ' e' p and c 2 we have picked up second order contributions to the pressure in the long wavelength (small x) limit. To illustrate the meaning of the new couplings, we have, in Table 6.2, given the expansions of the coefficients Cj to second order in x. It is seen that all of the collisional contributions to the nonequilibrium couplings are of third order or higher in the gradients except for c 2 (recall that these coefficients are all multiplied by the shear rate, a, which is a first order gradient). Thus, they do not appear in the ordinary NavierStokes equations but rather in the higher order, so called Burnett, equations. They are always present in the analysis to be presented below, however, as we keep all orders in ka. Also, these equations contain wavevector dependent "transport coefficients," eq. (6.32), which are the high frequency limit of the general wavevector and frequency dependent transport coefficients, * D T (x) = lim D (x,w), (6.33) etc. Wavevector dependent susceptibilities such as the pressure derivatives also appear. An important feature of the transport

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Table 6.1 Coefficients appearing in hydrodynamic equations. c 1 * C 2 " 2 1 -« J 3 ) * C 3 = -2C1 | v*]J 3 * 12 2 -* 57 y ci 4v ] (Jl + J 3 ) * C 5 = 12 2 -* -~ y ci nv ]j 3 * c 6 " 2 2 -* f yd + v ](1 2 J 2 ; X °7 = -6y[1 + | v ] j / x * c 8 " * °7 * °9 * C 7 C 10 " -6y[1 + | v*][j 2 7 • i -x J 3 ] Table 6.2 Small wavevector limits of the hydrodynamic coefficients. w v 20 -* 2 3tt * 2 r 1 -* °2 * " T5 [1 + 2 v ]x * + 0 * 4 2 r -* 2 175^ y C1 + 4v ]x c c » 0 * 1 2 -* 2 °6 * ~ 35 yCl + 7 v ]x * 2 2 — * 2 C 7 * " 35 y[1 + 7 v ]x # * c 8 * c 7 C 9 * c 7 c 10 * 0

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95 coefficients is that they decay faster than x~ 2 for large x which contributes to the mode softening of generalized hydrodynamics which was alluded to in the introduction and which shall be explicitly presented in the next chapter.

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CHAPTER 7 ANALYSIS OF THE SHORT TIME MODEL In the previous chapter, we obtained the equations of a short time hydrodynamic model valid at large wavevector. In this chapter, we study the hydrodynamic modes of this model and compare them to the more familiar long wavelength hydrodynamic modes. The primary characteristic of equilibrium generalized hydrodynamics is a softening of the modes at large wavevector and we shall find that this is further enhanced at large shear rates to the point of creating an instability. We shall also find the appearance of new propagating modes at large shear rate. Equilibrium Generalized Hydrodynamics Before studying the effects of shear on the small wavelength hydrodynamics, we review equilibrium generalized hydrodynamics and compare it to ordinary hydrodynamics. If the shear rate is set equal to zero, the hydrodynamic matrix calculated in the previous chapter is found to be a function of ko, where k is the magnitude of the wavevector and o the hard sphere diameter, with no dependence on the direction of k. it is also easy to see that the shear mode, momentum fluctuations in a direction perpendicular to the wavevector, decouple and is doubly degenerate as in the small-k limit (c.f., eq. (4.15)). 96

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97 However, while the small k modes grow as k , the generalized modes soften at large k and, in fact, the generalized heat mode is two orders of magnitude smaller at ko 7 than an extrapolation of its small k quadratic behavior would predict. For this reason, the relaxation time of hydrodynamic fluctuations on these length scales is much longer than one would expect and this explains how these microscopic modes can affect macroscopic properties. Figure 7.1 is a plot of the real parts of the equilibrium sound, heat and shear modes respectively for no 3 = 0.88. Figure 7.2 shows the equilibrium sound velocity as a function of (ka). These results agree well with the more detailed calculation of deSchepper and Cohen and support the extension of the present model to nonequilibrium situations. The most striking feature of the generalized modes aside from their softening is that the sound mode becomes purely dissipative for 5.75 < ko < 6.5. In other words, for this range of wavevectors, sound does not propagate. Finally, we should note that these modes were calculated using the Carnahan-Starling expression for x = g(a) and the Percus-Yevick expression for the static structure factor, S(k) , with an approximate Verlet-Weiss correction. 116 Nonequilibrium Generalized Hydrodynamics If we now vary the shear rate, we find several striking changes in the hydrodynamic modes. Before discussing these, however, we must note that the results to be presented here were obtained for the case k = ky. This choice of wavevector eliminates the gradient term in eq. (6.20) and reduces the problem of determining the modes to a simple

PAGE 105

H

PAGE 107

91 -D O E o c § CO 5. O *J o CD > CD 5 3 0) 60 CO Cm O ±S CO Q. >, L CO C •H M CO E C\J En

PAGE 108

101 g os o n a h i m oc o wo m sio m o to o ft o HI

PAGE 109

102 eigenvalue problem. The case of wavevectors not along the y-axis can be studied using the methods presented in Chapter 4. However, the hydrodynamics at large wavevector is complicated even for the choice of wavevector used here and is representative of the effects to be found for other wavevectors so that the simplifying choice seems best. What is more, the results of Chapter 4 indicate that the slowest decaying modes will be those with k along the y-axis and so any instability in the system should occur for a lower shear rate than for any other direction of wavevector. Thus, we will henceforth take k = ky unless otherwise stated. Figures 7.3 7.8 show the real parts of the sound, heat and shear modes for a* = at E = 0.1 through 0.64 for n* = na 3 = 0.88. A peculiar feature of these modes is that above a* = 0.05, there are two sets of propagating, or sound, modes for certain ranges of ko. One set evolves from the equilibrium sound modes and is a shear rate dependent version of these modes. The other pair of propagating modes is the result of a shear induced mixing of the heat mode and one of the shear modes. As the figures show, the propagating modes appear and disappear and mix with one another as ka is varied. For the density no 3 0.88, Fig. 7.8 shows that an instability exists for ka * 6.0 and a* = 0.64. m Fig. 7.9, we have plotted the critical shear rate (i.e., the minimum shear rate for which 'an instability exists) as a function of density. We believe this instability to be associated with the transition to the so-called string phase observed in computer simulations. Also shown in Fig. 7.9 are the two values of the critical shear rate as reported by Erpenbeck. 12

PAGE 110

o II * a) £O D II X so o > 2 CO > Cd CO 0) -o c E C >> •o o c o >> js t, o o > > id 3 CD bO !CD
PAGE 112

CM O II * cc t, o »H D II X c3 o * > CO 3 > Ed cr CO CD c E O -a >> x: c3 j-> o > CD 3 0 SO L CO O 00 CO a) o £_ II CO a * c CO CD cr to Eh

PAGE 114

on o ii * to U o o o CD > CD > as 2 > Ed cc CO CD a o £ I c >> a o u -o £ O o CO > CD > g CD bO TO 4-i 00 O OO CO O 4-3 i. II td a * c TO CD cc C TO •H

PAGE 116

to L C O it £O 4-1 O > > CO 3 Si > n a; -o o E B SL. TJ >> -C -P o > CD > CO
PAGE 117

K'l Oi l 001 M O 00 0 01 0 09 0 OS 0 Ok 0 Of 0 »' 010 ft 9

PAGE 118

in o * 03 L O 4-1 D c o J-> o > > CO CO > CO > a o b a >> js b +3 a > CD & 3 CD s 0 CO CO CO o 4-> JII cd a * c CO ai H

PAGE 119

112

PAGE 120

on * b ii x o *J o CD > CD > 03 2 CO > CO 0 -o o E O -i § c >> t. T3 >i JC O +J o CD > CD > CO CD bo L TO O oo 00 02 O £II CO Q. * C 03 CD CO bo

PAGE 121

114 Oi l 01 1 00 1 06 0 0( 0 01 0 09 0 OS 0 Ok 0 Ot 0 02 0 010 3a

PAGE 122

Fig. 7.9 Critica^ shear rate, a c *, as a function of reduced density, n* = no . Also shown are Erpenbeck's values for the transition (o).

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117 Numerically, Erpenbeck's values are about a c * = 0.4 and 0.8 for n* 0.704 and 0.88, respectively. (There is some uncertainty in these numbers as the transition is not sharp.) Our values for these densities are a c * = 0.63 and 0.85, respectively. Furthermore, S. Hess, in private conversation, has reported that the transition to the string phase is preceded by a transition from the amorphous fluid phase to a layered phase in which the hard spheres are partitioned into planes lying parallel to the x-z plane. This is consistent with the critical wavevector pointing in the y-direction since this indicates homogeneity in the x-z plane with variation in the ydirection. Finally, the eigenvector associated with the unstable mode is, in our calculation, almost a pure density mode while the partitioning into planes could be viewed as a phase transition in which the macroscopic density goes from being a constant, p(r) p , to being constant in the x-z plane and varying periodically in the y direction, p(r) p^ sin iry/d (where d is the interplane spacing). The instability we observe is attributable primarily to a single element in the hydrodynamic matrix: the coupling of momentum fluctuations in the x-direction into the equation for momentum fluctuations in the y-direction (i.e., a coupling of transverse to longitudinal momentum fluctuations since k = ky). This is easily ascertained in that when this coupling is set to zero, by hand, the minimum in the equilibrium heat mode remains unchanged as the shear rate is increased. From eq. (6.31 ), the coupling in question is seen

PAGE 125

118 to be a*(c 6 *+c 9 *) which, from Table 6.2, is found to be proportional to a*(ka) in the long wavelength limit. It thus is of third order in the gradients in this limit (n.b., a |VuJ) and does not occur in the Navier-Stoke's equations but, rather, in their generalization to include third order gradients which are known as the Burnett equations. This explains why the instability is not seen at small wavevectors (c.f., Chapter H in which the long wavelength equations are shown to be stable). There is, however, a second instability at small (although not asymptotically small) wavevectors as seen in Figs. 7.3 7.9. We believe that this second instability is not seen in the simulations due to the fact that our calculations were performed for an infinite system while the simulations are necessarily done for finite systems. Applicability of our results requires that Ak/k << 1 where Ak 2n/L, where L is the length of the box (for a finite system, the wavevector must be integral multiples of the "unit wavevector" Ak). For the simulations, L 3 N/n where N is the number of particles and n the density. Thus, applicability of our results requires that (ka)»0.25 for n*=0.88 and N=4000 (the smallest Ak occurring in Erpenbeck's simulations). For wavevectors on the order of Ak, we expect that the boundary conditions, which we have not explicitly included in our calculation, become important. In fact, it is easy to understand physically both the nature of the instability and the reason it is not seen. It is found that this instability arises from the inertial (k-independent) momentum-momentum coupling in eq. (6.31). However, one can show that a zero-wavevector instability

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119 arises from this coupling which is associated with center of mass motion. This zero-wavevector instability is unphysical, being prohibited by boundary conditions, and we conjecture the same is true of the small wavevector instability. A similar calculation has been performed by Kirkpatrick^ 7 and Nieuwoldt (hereafter, KN) . Two primary differences exist between our calculation and that of KN. The first is that KN neglect temperature fluctuations. We find in our model that the neglect of temperature fluctuations drastically alters the hydrodynamic modes and, in particular, gives rise to an unstable mode at small shear rate and both large and small wavevector. Thus, this approximation is unjustifiable in our model. The second difference between our calculation and that of KN is that KN begin with the Enskog equation rather than the Liouville equation. They therefore approximate the nonequilibrium pair correlation function with its equilibrium form from the beginning. This means they have not built in information about the stationary state and as a consequence cannot use stationarity to eliminate the three-point function as we do. They are thus forced to use an approximation to the three-point function (specifically, they use the Kirkwood superposition approximation) which we do not need. The result of these approximations is that KN sees an instability but at second order in the shear rate. There is an inconsistency, however, in that KN do not evaluate the effects of the thermostat which appears explicitly, at second order in the shear rate, in the

PAGE 127

120 equations. Thus, the present calculation uses fewer assumptions than that of KN. Finally, it would be very useful to obtain some idea of the completeness of the description of fluctuations presented here. The simplest check would be to study how well the hydrodynamic modes described here conform to those generated in the simulations. For instance, qualitatively, one might check whether there are indeed two sets of propagating modes at intermediate shear rates.

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CHAPTER 8 CONCLUSIONS The object of this thesis has been a description of fluctuations and transport in a sheared fluid. As was previously discussed, this system is particularly amenable to theoretical studies, both analytical and simulational , but is difficult to study experimentally. Experimental studies of more complicated systems are possible and many of the phenomena seen in the simulations of sheared simple fluids are also seen in laboratory experiments on these more complex systems. This agreement lends credence to the simulations which we thus use in place of experiment as a source of empirical data. We began our theoretical study by restricting our attention to long wavelength fluctuations. These were described using fluctuating hydrodynamics generalized to the nonequil ibrium state. The linear problem was solved under less restrictive conditions than had previously been used which allowed us to include states far from equilibrium in our analysis. Using our knowledge of the hydrodynamic modes, we investigated various statistical and physical properties of the fluid including the behavior of the time dependent correlation functions, the static correlation functions, the light scattering spectrum, and the distribution function. The most important consequences of the shear were due to the appearance of a gradient in 121

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122 wavevector space in the linear equations (the linear mode coupling term). This term brings about a much faster decay of fluctuations and correlation functions than in equilibrium (an e decay as opposed to e t ). We also find long ranged correlations in the fluid. Specifically, the equal time one dimensional correlation of densities decays as x" 5/3 indicating that the radial distribution function decays as r " 11/ 3 f or i arge r . We then preceded to analyze the effect of the nonlinearities. These manifest themselves through a renormalization of the microscopic, bare, transport coefficients and susceptibilities. The renormalized shear viscosity was calculated using a "naive" method and found to be in agreement with previous calculations. This important concurrence indicates the equivalence of the presence model with those studied previously. A formal discussion of renormalization in this model was also presented. The formal scheme allows us to calculate renormalized properties other than the viscosity as well as providing a systematic means of studying higher order contributions. In the final two sections, we studied the derivation and analysis of a model of small wavelength phenomena. It was found that the hydrodynamic modes were drastically modified by the shear and, in particular, that a large shear instability was found. The instability was proposed as the mechanism behind the transition to the string phase observed in simulations. There is still much work to be done on this problem. In the fluctuating hydrodynamics problem, the nonlinear problem is still relatively unexplored. At a practical level, the complexity of the

PAGE 130

123 correlation functions as determined from the linearized model makes the calculation of the renormalizations, even to lowest order, a difficult numerical problem. It would be of particular interest to calculate the renormalized shear viscosity from the formal theory without using local equilibrium correlation functions (c.f., the end of Chapter 5). More generally, the shear induces non-Gaussian statistics for the static fluctuations making this problem qualitatively different from most of those previously studied. The consequences of this difference remain unexplored. In the short time model, several questions remain. One of the most important concerns the shear rate expansion. It is found that the shear rate is scaled to two different times: the Enskog time, t E , and the Boltzmann time, t Q = xt £ , where for a dense fluid x 5. The highest shear rate considered is atg 0.6 (near the instability) or at E 0.1. Thus, the expansion in at E is quite reasonable while that in at B is questionable. However, the at B expansion arises solely from the expansion of the local equilibrium distribution function. If this expansion were not made, the elements of the hydrodynamic matrix would have to be calculated numerically. Such a calculation would provide a more quantitative description of the instability. It would also be useful to understand the importance of momentum correlations. When we assumed a product form for the distribution function (c.f. eq. 6.22), we implicitly neglected momentum correlations. In equilibrium, this leads to a 50$ error in the viscosity as calculated from the Enskog equation. We expect that an inclusion of these effects would produce quantitative changes in our

PAGE 131

124 results. However, because the instability is due to the appearance of a new coupling, not present at long wavelengths, one might expect the qualitative result of the existence of an instability to be robust while the quantitative results as to the critical shear rate must be expected to be sensitive to changes in the model. Finally, the methods used to study the long wavelength model could also be applied to the short time model. One might then calculate correlation functions, renormalized transport coefficients, etc. These calculations can be expected to produce results which may be compared to data compiled from simulations.

PAGE 132

APPENDIX A LINEARIZATION OF THE LANGEVIN EQUATION The fluctuating hydrodynamics model was detailed in eqs. (3.23) (3.29). We begin the linearization of this model by casting it in the form given in eq. (5.12), Tt\ + + D mn [ y ] F n [ ^ ] = M mn Cy] h' U ' 1) This is accomplished by identifying, V [y] = V j p a a V e [y] = V a (ej a /p) V y] " V p \& + W P) ' (A. 2) FpCy] = (uCy] \ j 2 /p)/T[y] F e [y] = 1/TCy] F^Cy] j a /( P T[y]) (A . 3) 125

PAGE 133

126 and all elements of D mn zero except D ee " VM{T[y]ACy]« ai| + Wylj^/pjv D. D , J e ej v v In these equations, T[y] = Tg[y] and y[y] = Mg Cy] where TgCy] is the equilibrium temperature functional, etc. The y-dependent transport coefficients are similarly defined in terms of their equilibrium analogs. Finally, in (A.4) we have abbreviated n aBuv = nA aBlJV + < « a6 « yv (A.5) With A a6yv 33 in eq * ( 3-^). The matrix M mn is zero except M e,1a6 " ' V J« T ^ U2 V P M e,2a 6 = " V«W T ^ U2 VP M e,3a = "

PAGE 134

127 and 5 1ctB = S 1a8 C 2aB = S 2aB C 0 S . (a 7) 3a a v.*. f ; From eq. (3.29), we find A =0 except mn 1aB,1uv aBuv A 2a8,2yv = Vuv A 3a,3B = V < A -8) With these definitions, it is easy to verify that the fluctuationdissipation relation, eq. (5.13), is verified. Although this model could be studied as it stands, we wish to consider a restricted model in which only quadratic nonlinear! ties arising from the convective terms, i.e., those in (A. 2), are kept. Thus, the restricted model is obtained by making the substitutions;

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1 D [y] D [y_] mn J mn J 0 6F F m Cy] + F[y ] + —21 | 6y m m 0 jy 'y M mn [y] * M mn Cy O ] < A -9> where Sy = y y Q and the convective terms to second order (except the pressure which, as in equilibrium calculations, is only expanded to first order). With these substitutions, it is found that the fluctuation-dissipation theorem is preserved up to and including o(k ). The resulting equations are ft z a " a ij k i 3iT7 z a + H ab (k ;a) z b + >VV 10 ) where H ab ( * ;a) "^OO + * 2 *%M *™M

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129 ab 0 0 ip n c i 0 0 0 0 ih„c 0 2 o o lp 0°1 ih 0 C 2 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 3 0 4 0 0 o o 0 0 0 0 0 0 0 v o 0 0 0 0 0 v o 0 0 0 0 0 0 0 0 0 0 0 0 r 11 1 1 1 2 13 0 0 F 22 f 23 0 0 f 31 r 32 r 33 L^ ; (k) = r.. r p ( A. 1 1 ) where A Q is the thermal conductivity, v Q = n B /p Q is the kinematic viscosity, V( J | v Q + p~ 1 Kq and m The vertices are V abc (k -'2 1 .2 2 ) " 5( ^ara 2 )(A abc (k. ai ) * A acb (k fflg ))

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130 "2 6 ab e a" ( ^ )k a 3=1 ' 2 b ' c=3 " 5 W*-a> " " 5 •i* ) «!i)«i b) (a>vi 0) (J£-a) -=3-5 b,c= 3 5 0 b,c < 3. (A. 13) and where we have introduced a change of variables y -*• w z. The change y + w is defined by replacing the momentum density, j , with the local velocity fluid, u , and the total energy density, e, with the internal energy density, e, where A a a J = p u a a * 1 ^ ^2 A e = p u + £. (A.14) We note that <6> = e 0 = 2 P 0 1 J 0 + e 0 ( A 1 5) so,

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131 e£ = e Q + . (A. 16) The second change of variables, w + z, is defined by z 1 " c l (p " P 0 } Z 2 = C 2 (e ~ e 0 } : 3 + i = e^ ) (k)(u a u Qa ) (A.17) where {e 1 (k ) } is a set of three mutually orthogonal unit vectors with e (k) k k/k and 2 2 ? 3 Pn V'V,' ' -^la 0 2 -2 3p 0 , °0 r "t '" P ° ^ '
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132 coefficient, c p is the constant pressure specific heat, c Q is the sound velocity and h Q is the enthalpy density. For later reference, we also define °2 9T 0 , C 2 rT -1 h 0 ° 3 " °1 3p 0 £ 0 = C 1 Cp 0 a T " p 0°v ] 9T 0 where c y is the constant volume specific heat and Y = c /c P u Finally, the fluctuating forces appearing in (A. 10) have correlations, = 0 a — (2,) 3 (o* 2 kgT^yk 2 «
PAGE 140

APPENDIX B GENERALIZED EIGENVALUE PROBLEM The generalized eigenvalue problem is We solve this equation perturbatively recalling that 2 ck >> vk a. Thus, we expect the eigenvectors and eigenvalues as »«" * »{» , 2 »«•> . .... The lowest order equation is H r (oi) ,W r (a) lab^Ob A 1 ^Oa The eigenvalues and eigenvectors of H 1 are found to be 133

PAGE 141

with -ic = A (2) ^ o tf> ^ 5 > ,(1) 1 , = 7T~ p °° 1, h o c 2« V °» 0) Vd c Q ,(2) 1 . * " ~J~ (p O c 1» h 0°2» -° 0 ' °» 0) Vd c Q (3) 1 * = ^ (h 0 C 2' " P 0 C T °' °» 0) (*0 * = (0, 0, 0, 1, 0) (5) * = (0, 0, 0, 0, 1). (b.6) These vectors are, in fact, orthogonal. Because of the degeneracy, we find that r (D ,(1) r (2) (2)

PAGE 142

1 where A is to be determined at next order. The second order equations are H r (ct) i (ot M a ) + ,(00.(0) ^ -2 , 3 (a) H 1ab 5 1b " X 2 5 0a + A 1 Sa + k a ij k i »T V + (H 2ab +k " 2aH 3ab ) 4b ) ' ( B. 8 ) Multiplying through by <|/ 6) and summing on a, we find immediately that X 2 = X 2 "2 (F 0 + ak r n } ^ B -9) where V 0 C 2 °0 — (p oCl c 3 h 0 c 2Cj( ) + V( J = ^ D T + V( » (B.10) is the sound damping constant and D ? is the thermal diffusivity. For a»3 3-5, one obtains (B.11) Now,

PAGE 143

136 So one solution of eq. (B.12) for a=3 is A (3) = D A 1 D T A 3a = V (B. 13) Equation (B.11) then becomes lx\ a \ 2 v Q k 2 ar 22 ak x JL]A = 0 y EX 1 k v Q k + ak x — ]A q5 ar 32 A ai< 0 (B.14) which is solved by A HU = 1 \ 5 A(k) A c), = 0 A = 1 k k A(k) = Z~1T tan~ 1 (k /ki) k x k l y 1 k l" k x +k y' (B.15) This is not the general solution to (B.14) and is clearly not defined at k x = 0. However, it is possible to show that the Green's function

PAGE 144

137 is independent of the particular solution to (B.I 4) chosen and is well defined even at k x = 0. Thus, to order k , A = -ick + r k 2 + 1 a k k /k 2 I 2 0 2 x y » 2 » » 3 D T K 2 2 2 X,. v„k a k k /k l 4 V x y A 5 = v 0 k2 (B. 16) and, to lowest order, M .(a) 5 * a = 1-3 (4) 5 = (0, 0, 0, 1, A(k)) (5) 5 (0, 0, 0, 0, 1) (B>17) and the set of vectors bi-orthonormal to {^ {a) } , { n a } where a ^a = a8' 13 low est order,

PAGE 145

(a) ,(a) = 9 (4) = (0, 0, 0, 1, 0) (5) (0, 0, 0, -A, 1). a = 1-3 (B.18)

PAGE 146

APPENDIX C DYNAMIC STRUCTURE FACTOR The dynamic structure factor, S(k,u) , was defined in eq. (4.47) to be dk dk, ~1 —i (2tt) 6 dk.dk. s( k'^ = /!. dt / — ] — | Z(k-k,)Z(k+k 2 )C pp (k 1 ,t;k 2 ,0)e lwt 2Re 11 dt / S(k-k )5(k+k ,)C (k 1f t;k ? ,0)e ia)t , (2ir) * (C.1) Using eqs. (4.36) and (4.39), this may be written in the form . dk. S(k.«) = 2Re / — Ls^-k )c a) (a) (k> (2-ir)5 iii -i = a ^ S 3)( !i'^ (C.2) (a) with o the solution to Huu(a,( v a '-vu^ (a) ^^ = 1j < 2 5(JcH£ 1 )i^ a> (k 1 )o a1 (^ l «) (C.3) 139

PAGE 147

140 (a) Multiplying (C.3) by ^ (k)o(k-k 1 ) , integrating over kj and integrating by parts the gradient term yields dk J o 2 (k-k ){[-ia) A (a) (k ;a) \ a. k. ^-]a (a) (k, ,co) (2ir) 3 1 2 ij i 3k -1 where we have defined 0 (a) (k ;u>) = oCk-j^ ) a (a) (k ;u ) and we have used the fact that the shear modes do not contribute so the eigenvectors appearing here are _kindependent. Equation (C.4) can be written as 3 / ^3 o + A (a) (k + k i; a) ^ a. J (k + k 1 ) i ^-]a (a) (k^ ,„) n^c^^k+k^a)} = 0. (c.5) In terms of o (a) (k,u>), S (a) (k,a)) is S (a) (k,c) = J-S-^j, j (a)(k ( 2tt) 1 --1 Now, the wavelength of the light is usually much less than the length of the illuminated volume so, in (C.5) and (C.6), k » k 1 (C.7)

PAGE 148

141 yielding the approximate equations, c (a), . -(a),, . S (k,a)) = o (k;u)) (C.8) The right side of (C.8) is the same for a 1 and 2 so S (1) (k,u>) = S (2) (k,-a,) = S B (k,u>),

PAGE 149

APPENDIX D ELIMINATION OF THE THREE POINT FUNCTION Upon making the factorization approximation in eq. (6.22), it is clear that only the 6=1 term contributes. Returning to eq. (6.21) and using the adjoint of the Liouville operator we find that <(£ + y a (k)) 6y g (k))> ss = ss (D.1) and for 6=1 , we have « ^ „ £_ fiy^k) = (£J 0 6 yi (k) UJ lnt 6y^k) + (£.) ext «y 1 (k) (d.2) where (£_) 0 6 yi (k) = ik j 5y 2+J (k) (£_)6y 1 (k) = 0 ^Jext^!^ " °' (D.3) So, 142

PAGE 150

1 43 3 <(£ + y o (k))«y 1 (k)> = -i z k g (k) (d.U) j=1 J J and, from eq. (6.12) , L a1 (k,k';a) = L a1 (k;a) 6(k-k') ( D .5) where 3 L a1 (k;a) = -i Z kj g a 2+j (k)/ gl1 (k). (D.6) So, finally, using the explicit form of g (k), eq. (6.27), L n (k;a) = L (k;a) = 0 L a1 (ii;a) = " nib" {p 0 k B T 0 k a -2 " mn 0 U a t3 *y + (D ' 7) We also note here that the source in eq. (6.30) is easy to evaluate to lowest order in the shear if we note that £ ext " 0 cx-1 = 2A(D[y 2 (k) + (| k B T) yi (k)] a=2 = A(r) a =3 (D.8)

PAGE 151

1 44 and because A(r) eq f^XAtr^ (k)«y e (!<)> eq o(a 2 ) S a6 * eq + ^ ' °=3~5 (D.9) where, as indicated, the averages are now done in equilibrium and y o 0O is evaluated at a=0 for all a. Now, to lowest order in the shear, so the angular averages require that S 28 =° S *B " " a eq (6 a3 6 ^ + 4 a i» a « 3 > (D ' 11 > to first order in the shear. This is easily evaluated to give S a6 = (4 m N K B T) a (6^ ) af6=3 _ 5 (D>12)

PAGE 152

145 and, the contribution to the hydrodynamio matrix, is aB ' (1 m 2 » k B T) a (S o3 S ol) * Vl 4 <,3 )(g ~\e * 0 ,3) But, to lowest order in the shear, (g~ 1 ) = 5 for a = 3 or H. Thus, H fi o(a ). (D. 1 4)

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BIOGRAPHICAL SKETCH James Francis Lutsko was born on March 9, 196-4, in Tampa, Florida. There he received his primary and secondary education. In 1978, he entered the University of South Florida. In 1980, he transferred to the University of Florida from which he received bachelor's degrees in physics and mathematics in the spring of 1982. In the fall of that year, he began graduate studies in physics at the University of Florida. 150

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. J.Wl Dufty, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. P . Kumar Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. N.S. Sullivan Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the deer of Doctor of Philosophy. P,.
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Engineering Sciences This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1986 Dean, Graduate School