Spatially nonlocal fluctuation theories: Hydrodynamic fluctuations for simple fluids

Physica l15A (1982) 301-338 North-Holland Publishing Co.

SPATIALLY NONLOCAL FLUCTUATION THEORIES: HYDRODYNAMIC FLUCTUATIONS FOR SIMPLE FLUIDS Joel KEIZER and Magdaleno MEDINA-NOYOLA* Chemistry Department, University of California, Davis, California 95616, USA Received 25 March 1982

Our development of the effect of spatially nonlocal thermodynamic correlations on fluctuations is continued, focusing here on the hydrodynamic description for simple fluids. We use the canonical fluctuation--dissipation formalism and show that short ranged thermodynamic correlations are systematically included in the hydrodynamic description. In general the theory requires a knowledge of the second functional derivatives of the entropy-or equivalently-the two, three, and four point equilibrium correlation functions. For hard spheres this degenerates to a knowledge of the equilibrium radial distribution function. Within the context of the hydrodynamic theory, we present an exact calculation of the intermediate scattering function for hard spheres and compare with recent results from computer molecular dynamics. At liquid densities quantitative agreement is found as a function of time for distance scales up to the order of the hard sphere diameter. As expected, the time dependence of the fluctuating hydrodynamics results is only qualitatively correct for wavelengths much shorter than interactomic spacings or at interatomic spacings in the domain of gas densities. The implications of spatially nonlocal effects far from equilibrium is discussed.

I. Introduction T h e t h e o r y o f near equilibrium fluctuations that was d e v e l o p e d b y Onsager 1) and others 2"3) is b a s e d on the p h e n o m e n o l o g i c a l t r a n s p o r t equations. O n s a g e r ' s regression h y p o t h e s i s states that a s p o n t a n e o u s fluctuation satisfies the p h e n o m e n o l o g i c a l e q u a t i o n s on the average, but otherwise is a G a u s s i a n r a n d o m p r o c e s s . C o n s i s t e n c y with the equilibrium t h e o r y is a c h i e v e d b y appealing to the B o l t z m a n n - P l a n c k f o r m u l a for the equilibrium distribution of fluctuations. T h e O n s a g e r t h e o r y is limited in its application to fluctuations w h i c h o c c u r in the n e i g h b o r h o o d of equilibrium, and its predictions h a v e b e e n verified b y the direct m e a s u r e m e n t s o f noiseS), b y light scatteringS), and a v a r i e t y o f o t h e r techniques6). In the last d e c a d e a t t e m p t s have b e e n m a d e to generalize the near equilibrium t h e o r y so that fluctuations far f r o m equilibrium can also be described. T h e B o l t z m a n n - P l a n c k f o r m u l a is clearly of no help f o r this, and a c o n c e p tually different sort of t h e o r y is required. One a p p r o a c h is b a s e d on a * Present address: Depto. de Fisica, CIEA-IPN, Apdo. Postal 14-740, Mexico 14, DF, Mexico. 0378-4371/82/0000-0000/$02.75 O 1982 N o r t h - H o l l a n d

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conception of transition rates between molecular thermodynamic states (the "master equation" approachT). Our own work is more closely related to Onsager's approach and is based on a postulated connection between dissipation and molecular fluctuationsS-12). It is by now well understood that this so-called fluctuation-dissipation theory is the thermodynamic limit of both the master equation theory 12'~3) and related theories based on nonlinear stochastic differential equations~2'~4). To properly describe both dissipation and fluctuations it is necessary to characterize the underlying molecular events. In the macroscopic domain these events manifest themselves as elementary processes. Each elementary process has a forward and reverse rate depending on the intensive thermodynamic variables conjugate to the extensive variables that are changed by the molecular process. The form of these rate expressions is similar for all elementary processes and so this form has been called "canonical". The appropriate interpretation of these rate expressions leads to a consistent mechanistic theory of fluctuations which involves the extensive and intensive thermodynamic variables and the transport coefficients. Near equilibrium the transport coefficients disappear from the static fluctuation formula and appear only in the phenomenological expressions for the relaxation rates. Indeed near equilibrium the theory reduces to the Onsager theory,8-~°). The Onsager theory can be used to examine the correlation of fluctuations in space. Although this seems to be well known, the theory has seldom been applied to properly account for the equilibrium fluctuations. Similarly the fluctuation-dissipation t h e o r y - o r what we shall sometimes call "mechanistic nonequilibrium t h e r m o d y n a m i c s " - c a n be used to describe the spatial extent of fluctuations. Although early applications of these ideas were to ideal solutionsg'tl'ts'~6), we have recently shown that spatial effects in strongly interacting solutions associated with diffusion can be describedlT). It is our purpose in this paper to extend that analysis to the hydrodynamic level for simple fluids. A particular advantage of this approach is that it gives a well-defined procedure for calculating the static correlation functions at nonequilibrium steady states. This is based on the generalized fluctuation-dissipation theoremH). A novel aspect of our hydrodynamic theory is that the static equilibrium spatial correlations are properly included. Consequently the theory can be used to calculate short ranged spatial correlations at steady states which arise from the interplay of equilibrium and nonequilibrium spatial effects. Such features are missing in the classical theory of Landau and Lifshitz~8), as we discuss in the following section. To help justify the appearance of spatially nonlocal effects in our theory, we have carried out calculations of hydrodynamic correlations for a hard

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sphere fluid. In this paper our results for hard spheres are limited to spacetime correlations near equilibrium, where we can compare with the computer molecular dynamics results of Alder, Yip, and Alley~9'2°). The comparison is quite good in the region of liquid densities even for a length scale comparable to the hard sphere diameter. The quality of this agreement suggests that the present techniques should be useful in obtaining spatial correlations at far from equilibrium stationary states"'27). The plan of this work is as follows. In section 2 we discuss the background and write out explicitly the nonequilibrium hydrodynamic fluctuation theory. Section 3 is devoted to a treatment of spatially nonlocal thermodynamic relationships, enlarging on ideas expressed earlier by Schofield2~), Zubarev22), and others23). We examine spatially nonlocal hydrodynamic effects at equilibrium in the fourth section. There we derive the matrix Ornstein-Zernicke equation for the equilibrium convariance 22) from the equilibrium fluctuation-dissipation theorem and show that near equilibrium our results are equivalent to the spatially nonlocal linear theory. In sections 5 and 6 we obtain the local approximation to our theory, and verify that the local theory is identical to the Fox-Uhlenbeck theory3). Section 7 is devoted to formal expressions in terms of particle distribution functions for the equilibrium correlation functions which appear in our theory. For hard spheres we obtain explicit expressions for these correlations in terms of the radial distribution function. Using these results we carry out calculations of the spatial Fourier transform of the time correlation functions for hard spheres in section 8. Our results for the hard sphere intermediate scattering function are given in section 9 along with a comparison to the molecular dynamics calculations of Alder, Yip, and Alley~9). In the final section we outline how the theory can be applied away from equilibrium.

2. Hydrodynamic fluctuations The hydrodynamic level for describing fluctuations originated with Green 24) and Landau and Lifshitz is) although, undoubtedly, it was known to Onsager. The idea is that in a simple fluid the local mass density, momentum density, and internal energy density form a set of extensive densities which solve a closed set of hydrodynamic equations. Consequently the hydrodynamic equations can be linearized around the equilibrium state, and the Onsager theory applied to describe the space-time correlation of the fluctuations. This program seems first to have been carried through explicitly by Fox and Uhlenbeck3). Indeed, earlier Landau and Lifshitz Is) had made a conceptually

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incorrect application of the Onsager theory, although Fox and Uhlenbeck verified that the Landau-Lifshitz correlation expressions are correct. We call this theory a hydrodynamic "level" description because it can be correct only for a time and length scale on which the basic elementary processes are hydrodynamic, i.e. only for times and distances where the processes of shear and bulk viscosity and heat conduction actually give a good description of molecular change. When more molecular detail is needed, a deeper level of description is required. For example, to treat molecular level correlations in a dilute gas collisional effects necessitate use of the fluctuating Boltzmann equationS'25). For the linear theory this hierarchical point of view has been explored by Fox and Uhlenbeck3'2s), and it forms an important aspect of our ideas regarding the mechanistic nonequilibrium thermodynamic theory of fluctuationsS'9'26). In previous work one of us has derived the hydrodynamic level description from mechanistic nonequilibrium thermodynamics27). In that theory the conditional average solves the usual hydrodynamic equations. The conditional fluctuations, on the other hand, satisfy equations gotten by linearizing the hydrodynamic equations with the addition of a random stress tensor and heat flux vector. Our derivation yields correlations for the random stress tensor and heat flux vector which are identical to the Landau-Lifshitz formulae~8), except that the variables are to be evaluated at the conditional average value. We write down these results explicitly. Calling the mass density p, the momentum density Pi (i = x, y, z), the velocity u~, the internal energy density e, and letting overbars represent the conditional average, the equations governing the hydrodynamic level fluctuation theory are f)/5/D t = -/5(Of~j/igxj),

(la)

[ ) ~ / D t = -g(OF~i/Oxj) + ~(OujlOx~) - 2 + 2rleiieij -o , - Og(;gT-~/Oxj)/Oxj,

(lb)

f)p~[Dt = F~ - (ape/ax~) + 2 a ~ J a x j

+ a((a~j/oxj)/ox~ - p~(a~/axj);

l ) S p [ D t = - g(pOuj[ Oxj ) - ( 0/5[ (gxj) 8uj, f)~e/Dt

(lc) (2a)

= 5 [ - h(c~uj/~gxj) + I~(Ouj/axj) 2 + 2"O~ij~j - OK(c~T-~/axj)lc~xj] - (aP./Oxj),Suj + aglj/Oxj,

I),Spi/Dt = Og(rij/ Ox i - O~i,Suj/ Oxj - ( a~i/ Oxi)Spi + a6rJ Oxj.

(2b) (2c)

In these and other equations we use the summation convention on repeated indices and the notation 8p, Be, ~p~ represents deviations around the conditional average, e.g. ~p = p -/5; ~ is the coefficient of shear viscosity, ~ the coefficient of bulk viscosity, F~ is the external body force per unit volume, x = T : K , (where K is the thermal conductivity), T is the temperature,

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305

h = e +pC (the enthalpy density), pe is the thermodynamic pressure (a function of p and e). Other notation is o'ij, the stress tensor, defined by orii = --pe6ij + 2rl~.ij + I~(aUk/OXk)gij,

(3)

~,j = ~[(au,/ax~) + (auj/ax,)] - ~' ( a u d a x 0 6 , , ,

(4)

I)/Dt = a/at + &a/ax~, 6~j is the random stress tensor and qr is the random heat flux vector. Since eqs. (2) are gotten by linearizing eqs. (1), the linearization operator 8(.), defined by 8G = (OG/O~)Sp + (OG/OO~)Sp~ + (OG/O~)Se appears in eqs. (2). As we discuss in subsequent sections, the full interpretation of this equation is as the functional differential. We recall that the bar over a function of p, e, and p~, e.g. i6e, signifies that the function is evaluated at the conditional average values of its argument, i.e. Oe = p"(tS, ~). Finally the random stress tensor and heat flux vector are purely random Gaussian processes with the following correlation formulaZ7): (Clk(r, t)6"ij(r', t')) = 0,

(5)

(o'ij(r, t)6"kl(r', t')) = 2ka~f['FIAil, kj + ~6ifil~]6(r - r')6(t - t'),

(6)

(cli(r, t)Oj(r', t')) = 2kaff6ijS(r - r')6(t - t'),

(7)

where A is the fourth rank tensor defined by Ail, kj = 6il6kj + 6ik6li --(2/3)61i61k.

(8)

In what follows we shall be primarily concerned with the fluctuations in eqs. (2). Those can be written as linear, generally nonantonomous, equations in the fluctuations 6p, Be, 8p~- although the right-hand sides of eqs. (2) have not been written out explicitly in that fashion. Actually it is also convenient to use the variables thermodynamically conjugate to p, e, and p~ as variables on the right-hand sides of eqs. (2). The conjugate variables are -ix~T, liT, and -u~lT, where ix is the mass based chemical potential. Symbolically changing to these variables we can write 08plOt = L l t S ( - ix~T) + L t z 6 ( l / T ) + L i f t ( - uJT),

(2a')

age/at = L 2 j S ( - i x l T ) + L226(1/T) + L z j 6 ( - u J T ) + fie,

(2b')

a6p~lcgt = L,,6(-ix~T) + Li26(l/T) + L~,6(-uJT) + f~.

(2c')

Clearly /e = aCt,lax, and ~ = asgox,. The quantities L ~ are linear differential operators with a time dependence inherited from the conditional averages in

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J. KEIZER AND M. MEDINA-NOYOLA

eqs. (1). In obtaining the explicit form for the operator L ~ one needs to recall the Gibbs-Duhem relationship ape/ax~ = - Tha(l/T)/ax~ - T p a ( - pJT)/Ox~ which allows the pressure to be eliminated in the momentum equations in favor of lIT and -I~/T. Even more symbolically we will write ~n for the column vector (~p, ~e, ~Px, ~Py, ~Pz) and ~F for ( 8 ( - ~ / T ) , ~(1/T), 8 ( - u x / T ) , ~(-uy/T), ~(-uz/T)). Then eqs. (2') become

O~n/at = LSF + ]

(9)

with L the 5 x 5 matrix of time dependent differential operators implied by eqs. (2'). We will also need eq. (9) when the right-hand side is expressed in terms of 8n. This involves a linear change of variables, which we write symbolically as ~F : E[Sn],

since, as we shall see below, E is generally a linear functional of ~n(r, t) which is to be evaluated at the conditional average values h(r, t). Thus eq. (9) is transformed to the linear functional stochastic equation

aSh~at = LE[Sn] + ].

(9')

Eqs. (9) and (9') with the explicit form for L, E, and ] obtained from eqs. (2) are the starting point for our treatment of hydrodynamic fluctuations. It is the spatially nonlocal character of the operator E in eq. (9') that permits a full treatment of spatial correlations of fluctuations in #, e, and p~.

3. Spatially nonlocal thermodynamic relationships For a macroscopic system all thermodynamic relationships are contained in the entropy as a function of the extensive variables. One useful generalization of these functions to nonequilibrium states is obtained by using the functional form of the equilibrium state functions (entropy, chemical potentials, etc.) but evaluating them at nonequilibrium values of the extensive variables. These are called the "local equilibrium functions"22). For example, in eq. (lc) pe is the local equilibrium pressure, and it is the local equilibrium temperature and chemical potential that appear in eqs. (2') and (9). The local equilibrium entropy, S, of a simple fluid can generally be written as an integral over the local equilibrium entropy density, s. In the hydro-

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307

dynamic level description s depends on p, e, and p~. Thus

sto, e, p,] = f s(p(r, t), e(r, t), pi(r, t)) dr,

(10)

where the integral is over the volume and the square brackets remind us that S is a functional of p, e, and pi. Thus s depends only on the spatially local values of p, e, and pi. Indeed, locally in space one has all the usual thermodynamic relationships, such as

ds = -I~ d p / T + de/T - uj dpJT,

(11)

and so familiar relationships like (as/ae)p.p, = 1/T, or more generally,

(Oslan~),, = Fj,

(11')

where Fj is the intensive variable thermodynamically conjugate to n i. Actually eq. (11') can be expressed more succinctly using the functional derivative:E), which gives the change in S when the forms of the functions nj(r) are changed by an infintesimal amount Anj(r). More explicitly

J\~nj(r)/

J

Fj(r)Anj(r) dr,

(12)

where the final equality uses eqs. (11) and (11'). This is the spatially dependent generalization of the well-known formula a s -- F j a N j ,

(AN~ - f A n j dr) which is recovered from eq. (12) when the intensive variables F~ are independent of position. Eq. (12) implies that

,~S[n]

8ni(r) = Fi[n; r],

(13)

where the functional dependence of F~ on n has been made explicit. Thus the first functional derivatives of S give the local equilibrium values of the intensive variables at the position r. The second functional derivatives are more complicated and depend on two spatial locations,

,32S[n] 8ni(r),3nj(r') = Eii.(r, r'),

(14)

where we have dropped the square bracket notation for ease in writing. Now just as S is a functional of p, e, and p~, so is Fj. Consequently, eqs. (13) and (14) can be combined to give

~F~(r) ~nj(r') = Eli(r, r').

(15)

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J. KEIZER AND M. MEDINA-NOYOLA

Recalling the meaning of the functional derivative, eq. (15) implies that

AFi(r) = f

Eij(r,

r')Anj(r')

dr'.

(16)

In other words, AF = E[An], so that any deviation in AF is related in a spatially nonlocal way to a deviation in An. When eq. (16) is specialized to average deviations around the equilibrium state, then AF t - - F t - F~=-Xj, a thermodynamic force. Thus one has

xj(r) = f

E,(r, r')at(r') dr',

(17)

where aj = ~j - n~, the average deviation from equilibrium. The functional representation contains all the usual thermodynamic ideas. For example, one can write a functional Taylor expansion 28) of the entropy around a reference state as S--S°+

.

(,8,

If the reference state is an equilibrium state and the system is isolated, the entropy maximum principle implies that

AS = f F~(r)Ant(r ) dr

= 0

(19)

for all variations Ant(r) consistent with fAnt(r) dr = 0 (the condition of isolation). Under these conditions eq. (19) will hold if and only if F~(r) is a constant, which is the familiar condition of equilibrium29). Consequently for an isolated equilibrium reference state /-p

S = S e + l/2JJaT(r)E¢(r,

r')a(r') d r d r ' + . . . .

S ~+ (1/2)AmS + . . . .

(20)

Applying the maximum principle again shows that E~(r, r') is a negative definite matrix. We can also obtain an expression for the entropy production near equilibrium from eq. (20). Taking the total time derivative and using eq. (17) gives

dS/dt = f xr(r, t)(aa(r, t)/at) dr =

f faT(r, t)F.T(r, r')(aa(r', t)/at) dr

dr'.

(21)

Thus the entropy production density is XT(Oa/at). All these results are familiar in thermodynamics :9) and are really no more

SPATIALLY NONLOCALFLUCTUATIONTHEORIES

309

than the continuum extension of the more familiar discrete variable formulation. However, when spatial effects are being considered, explicit use of the functional formulation is necessary. The importance of this sort of thermodynamic formulation for generalized hydrodynamics has been recognized previously21'22), although in a somewhat different guise. Finally we point out that for a uniform system the expressions for the functional derivatives have a local character in wave number space. To see this we define the Fourier transform by +~

~(k) = f eikrg(r)dr. --o¢

Recalling the convolution theorem and that for a monatomic fluid Eij(r, r')= E i j ( r - r'), the Fourier transform of eq. (16) is Ai~(k) =/~j(k)At~j(k).

(16')

Consequently it is meaningful to define /~,i(k) = (COPi/COtij)(k), the wave vector dependent thermodynamic derivative. For convenience we will usually drop the redundant k-vector notation and write eq. (16') as AFi = (0Pi/0tij)Atij.

(16'~

This is a local expression in the sense that no integrals appear in eq. (16'9. Similarly the Fourier transform of eq. (12) will be Ag = FjA~ij - (0g/0rij)Atij,

(12')

since Fj is independent of space in a uniform system. Eqs. (16'~) and (12') are identical in form to the spatially local expressions normally encountered in thermodynamics. Consequently for a uniform system relationships among spatially dependent thermodynamic quantities in wave number space exactly parallel the usual thermodynamic relationships. Indeed for a uniform system all the classical expressions which relate thermodynamic derivatives, e.g.

( CO(1/T)/ Oe)p = - l I C i T 2,

..

also hold as their k-dependent analogues, e.g.

(c9 IlT/CO~)~ =/~22(k) = l/C~(k)T 2. Clearly to make eqs. (16) or (16'9 useful, one needs to obtain expressions for the second functional derivatives of S. Since functional measurements of the entropy are not normally made, we appeal in the following section to the hydrodynamic fluctuation theory.

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J. KEIZER AND M. MEDINA-NOYOLA

4. Hydrodynamic fluctuation theory near equilibrium In this section we look at the special case of the hydrodynamic fluctuation theory around equilibrium. In doing so we obtain the relationship between the matrix E(r, r') and the covariance matrix of the equilibrium fluctuations. In addition we show that the equilibrium fluctuation-dissipation theorem is equivalent to the Ornstein-Zernike integral equation for the covariance3°). Our starting point involves the average equations ( l a ) - ( l c ) and the equations for the fluctuations as written in the form of eq. (9). Taking for simplicity a fluid in an isolated, stationary container of volume V in the absence of external body forces, the average equilibrium state is found by solving eqs. ( l a ) - ( l c ) when the partial time derivatives on the left-hand sides are set equal to zero. This gives rise to the constant values p -- M] V, e = E / V , and pi = 0, where M and E are the total internal energy and mass, respectively. Because of the dynamic stability of this statel°'~l), on the average, a sufficiently aged ensemble will have these mean values. Indeed the rate of relaxation of the conditional average towards equilibrium can be gotten from eqs. (la)-(lc), linearized around equilibrium. That gives Oa(r, t)lOt = L~x(r, t),

(22)

where L ~ is the matrix L evaluated at ~5= M / V , ~ = E~ V, and pi = 0. Similarly the conditional fluctuations have a simple d e s c r i p t i o n - a s long as the system is away from an equilibrium critical point or phase boundary1°'"). The fluctuations in the aged ensemble then satisfy eq. (9), i.e.

(23)

08n/St = L ~ g F + f.

From eqs. (5)-(7) and the definitions below eqs. (2'), the correlation formula for ] can be calculated to be (f(r, t ) f r ( r ', t)) --- ~/e(r, r')8(t - t') = kB[L~8(r - r') + (Le8(r - r'))T]8(t - t'),

(24)

where

L° = (

0

p eTeO/Ox

-- ~eO~lOXiOXj hOTeO/Ox

heTe(3lOx) T ) L~m /.

(25)

In eq. (25) the entry peT~3/Ox at the bottom of the first column presents 3 row entries and peT¢(O/Ox)T at the top of the third column represents 3 column entries, etc., and the momentum block matrix is (L~)il = - Te['qeAil, gj + ~eSijSkl]C32[3Xic3Xk.

(26)

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311

oT)

Using eqs. (24) and (25) and recalling that 0 6 ( r - r')/Oxi = - O6(r - r')/Ox~, one easily finds that

3,e(r, r') = 2ks

--Ke02

8(r - r').

0T

(27)

OxiOx ~

L~m

o

Eqs. (22)-(27) are the hydrodynamic level description of fluctuations in a simple, monoatomic fluid near equilibrium. They can be compacted by writing a = fi + 6n, X = Y¢ + 8 F and using eq. (16)

o oy)

aa/ at = H e [ a ] + .f,

-KeO 2 0 T OxjOx~

(](r, t)]T(r ', t)) = 2ks

0

6(r - r'),

(28)

(29)

L~

where H~[ .1 = LeEe['].

(30)

The equilibrium fluctuation-dissipation theorem is a relationship among ~[.], H~[.], and the covariance matrix of 6a at equilibrium9-H). Its form follows from eqs. (28) and (29) and is

f[

H~(r,

r")ore(r ", r ' ) +

tre(r, r ' 9 ( H e ( r '',

r')) T] d r " = -~,e(r, r'),

(31)

where tr~j(r, r') - ( ~ a i ( r ) S a j ( r ' ) ) e,

(32)

with the average taken over the aged (equilibrium) ensemble and 3,~(r, r') given by eqs. (27) and (29). Consistent with the fluctuation-dissipation theory 8-~) we interpret eq. (31) as a linear integral equation for o-L To solve it, eq. (31) is rewritten using eqs. (24) and (30) in the form 2 sym ( ~ L e ~ ( r

- r ' ~ E e ( r ", r")tr~(r '', r') dr" dr'" -- - 2 k s sym(Le6(r - r')),

(33) where " s y m " means the symmetric part of the operators, including the spatial variables. In the form of eq. (33), the solution for tr e is obviously ire(r, r') = - k a E e - l ( r ,

r').

(34)

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J. KEIZER AND M. MEDINA-NOYOLA

Here E e-t the functional inverse of E e in the sense that

f cre(r '', r')[-k~lE¢(r, r")] d r " = I~(r - r'),

(35)

with I the 5 x 5 identity matrix. Eq. (35) can be simplified, somewhat, since it is well known from classical statistical mechanics 22) that cr~ has the form

(

0w )

ere(r, r') =

~ ( r , r')

0T

0

peksTelmS(r - r')

0

.

(36)

In this equation the 2 × 2 covariance matrix of p and e appears in the upper left-hand corner and Im is the 3 x 3 identity. Thus

Ee(r, r') =

E~(r, r')

0T

0

--Ira peT~3(r - r')/

0

,

(37)

with E~(r, r') satisfying

2(r, r'gE~(r", r') d r " = I2~(r - r')

-u~'f

(38)

(12 is the 2 x 2 identify matrix.) Because the momentum density decouples from the mass and energy densities, we need only concern ourselves with the 2 × 2 matrices cr~ and E~. According to eq. (37) we will have the matrix E after we obtain E2 by inversion of the energy-density correlation function. In section 7 we use this approach to derive formal expressions for E2 for hard spheres. To help understand the spatially nonlocal contributions to hydrodynamic fluctuations, we explore the equilibrium fluctuation-dissipation theorem further. In particular we show that the Ornstein-Zernicke integral equation for the direct correlation function can be derived from the fluctuation-dissipation theorem, eq. (31). In the present context we define the matrix direct correlation function22), c(r, r'), by analogy to the usual direct correlation function3°). Our definition is

Ee(r, r') = E~8(r - r') + kBc(r, r'), (39) where E ~ ( r - r') is the 5 × 5 matrix corresponding to the functional E for a hypothetical ideal monatomic fluid. Specifically

/-5k~12mo ° E ~ I=I / ~ T

1/o°T e

0T

-2m/3pCksTe2

0T

)

0

- Ira/peT ¢

,

(40)

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313

where m is the mass of the molecules. We have gotten eq. (40) by inverting the ideal fluid covariance matrix

/ rope tr~ = ~ 3kBT~9"[2

\

o

3kBT~p~[2 15#~(kaT')2[4m 0

OT 1 0T . kapeTelm/

(41)

We then introduce the matrix generalization of the indirect correlation function, h~2~(r,r') = g(2)(r, r ' ) - 1: crY(r, r') = ~r~g(r -- r') + cr~h(r, r')o~.

(42)

With these definitions it is now easy to obtain the Ornstein-Zernicke equation. We first recall that the equilibrium fluctuation-dissipation theorem implies eq. (35), so that

-k~fE°(r, r")tr~(r ",

r ' ) ( o ' ~ ) -1

d r " = (tr~)-16(r- r').

(43)

We then substitute the definitions of the matrices c and h (eqs. (39) and (42)) into eq. (43) to obtain

h(r, r') = c(r, r') + f

c(r, r")cr~h(r", r') dr",

(44)

which is the matrix Ornstein-Zernicke equation22). Actually because the momenta decouple from the mass and energy densities, the direct correlation function for the momenta and the associated h are zero. Thus eq. (44) is a 2 × 2 equation (cf. eqs. (36) and (37)). Usually the Ornstein-Zernicke correlation function is defined in terms of the covariance matrix of the density and temperature3°). Although our definition in eq. (39) is related to the usual definition for the number density, it involves the mass and energy densities rather than the number density and temperature. Consequently our Ornstein-Zernicke equation for the direct mass density correlation function is coupled to the direct energy density-mass density and direct energy density-energy density correlation functions. However, as we demonstrate in appendix A, by changing variables to number density and temperature eq. (43) is transformed to the usual scalar OrnsteinZernicke equation for the direct correlation function c~2)(r, r'), i.e. h(2)(r, r') = c(2)(r, r') +

f c'2)(r,r")(p e/m)ha)(r '', r') dr".

(44')

The equivalence of the spatially nonlocal fluctuation-dissipation theorem to the Ornstein-Zernicke integral equation makes it apparent that the spatial nonlocality which enters into the fluctuation equations (28) and (29) through

314

J. KEIZER AND M. MEDINA-NOYOLA

c(r, r') is short ranged in space. Indeed, at liquid densities the scalar direct correlation function falls to zero with one or two molecular diameters~. Actually the matrix direct correlation function is required for the full treatment of hydrodynamic fluctuations, w h i c h - w e will show in section 8 requires a knowledge of the two, three, and four point correlation functions. We close this section by writing out eq. (39) for the matrix direct correlation function in somewhat more detail. Employing eq. (14)which defines E, eq. (39) becomes

82S

82SI

8nk(r)Snl(r') - 8nk(r)Snl(r')

+ kaCkl(r, r'),

(45)

where S~ is the associated ideal fluid entropy functional. Eq. (45) is related to the hierarchy of equations which has been used in the equilibrium theory of classical nonuniform fluids23). In particular defining the part of S caused by interactions as 0 gives

S[n] = St[n] + O[n].

(46)

Taking the first functional derivatives then gives, for example, 8S[n] = - / x [ n , r] = kB ln(A3p(r)) ~ 8o[.] 8p(r) •

8o(r)

T[n; r]

where X is the thermal de Broglie wave length (a function T[n; r]). Carrying out the next functional derivative with respect to the density yields the following expression of the direct mass density correlation function:

820[n] = _ k ~ l S ( - I x / T [ n ; 8p(r')

Cu[n ; r, r'] = k E18p(r)dp(r')

r])

(47)

This should be compared to the expression for the scalar direct correlation function23)

826 = k~lS(-ix/T[o, l/T; r]) c(2}[n; r, r'] = -(kBT)-lSo(r)Sp(r, ) 80(r, ) ,

(48)

where q~ is the interaction part of the Helmholtz free energy functional. Clearly the difference between the two direct correlation functions is the nature of the thermodynamic potential which generates them and the corresponding variable which is held fixed. We emphasize that it is the interactive part of the entropy functional that gives rise to the nonlocal coupling of the fluctuations in mass and energy density. Because temperature fluctuations are independent of density fluctuations in classical equilibrium statistical mechanics, one might suppose that a simpler structure in the hydrodynamic level theory could be achieved by changing

SPATIALLY NONLOCAL FLUCTUATION THEORIES

315

variables from energy density to temperature. While this can be done, the structure of the resulting dynamical equations is inconvenient. Specifically recalling eqs. (28) and (30), the matrices 3,~ and L e are local operators when the variables are p and e. H o w e v e r , when p and T are used, both these operators are nonlocal, and even though E is diagonal, it too is nonlocal. 5. The local approximation In the usual development of the equilibrium hydrodynamic fluctuation theory, the relaxation operator in eqs. (28) and (30) is a local operator. This is achieved by averaging the spatial correlations which are inherent in the operator Ee(r, r'). To see this we write out eq. (28) as

Oa/at = LeE~[Sa] + ],

(49)

where L ~ is a local differential operator and, from its definition, (Ee[Sa])i =

f E,,(r, r')Saj(r',

t) dr'.

(50)

The local approximation consists in replacing this expression by

(E[,Sa])'i°:"l=- (f Eij(r, r') dr')Sa,(r, t),

(51)

so that

(Eij(r, r'))'°ca'=- (f E,~(r, r') dr'),~(r- r').

(52)

= Eq(r)~(r - r'). Referring to eq. (50), it is seen that the local approximation is exact only for fluctuations 8a which are spatially uniform. Nonetheless we anticipate that in some circumstances the approximation will be a good one because of the short ranged character of the direct correlation functions cq(r, r'). In particular when only distances long compared to direct intermolecular correlations are important (such as in quasi-electric light scattering measurements) we expect the local approximation to be goodS). The coefficient E~j(r) which appears in the local approximation can be given a simple thermodynamic interpretation. Referring to eq. (16) again we see that for a spatially uniform change in the variables An AFi(r) =

Eij(r)Anj.

(53)

Thus for a spatially uniform system one has (OF~/anj) = E,j.

(54)

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J. KEIZER AND M. MEDINA-NOYOLA

This feature of the local approximation can be expressed simply in the k-dependent thermodynamic formalism. Referring to eqs. (16') and (16'3 and the definition of Eij in eq. (52), it is evident for a uniform system that

(OF~lOnj)= lim OFil~nj(k). k-,0

(55)

In other words, the short wave vector limit leads to the usual homogeneous thermodynamic derivatives. An example of this sort of result is +0¢

(O(-tzlT)lOp)e = f Etl(r, r') dr'.

(56)

Eqs. (54)-(56) have the same character as the Ornstein-Zernicke expression for the compressibility +~

kml(Op/Op)w = (1 + (p/m) f

h(2)(r) dr).

(57)

Indeed the compressibility equation (57) is easily derived using the matrix E(r, r') and the wave vector dependent thermodynamics outlined at the end of the previous section. We show how to do this in appendix B.

6. Local versus nonlocai theory at equilibrium

For an equilibrium ensemble, the hydrodynamic fluctuation theory outlined in previous sections is identical to the spatially nonlocal version of the linear Onsager theory. The nonlocal version of that theory has seldom been applied-even to monotomic fluids-presumably because of the feeling that the hydrodynamic level description is incapable of correctly describing the longer time, nonzero wave vector fluctuationsS'33). In this section we write out the equilibrium spatially nonlocal theory in detail and compare it to the more familiar local theory which has been so successful for interpreting light scattering experimentsS). We begin by Fourier transforming eq. (28) to obtain ~a(k, O/at = tT(k)t~°(k)~,~(k, t ) + f(k, O.

(58)

The Fourier transform /~e is gotten from eq. (25) and /~e(k) has already been discussed in some detail. It is convenient to separate the momentum density into its transverse and longitudinal parts according to 33) = Pt + Pl,

(59)

SPATIALLY NONLOCAL FLUCTUATION THEORIES

317

where /il × k = 0,

Pt" k = 0.

(60)

When this is done, the transverse momentum decouples to give OOt[Ot = - ( ~ k : [ p

~)~t + i[kt~ - (k , ~ . k)k/k2],

(61)

with the Gaussian random term )~1expressed in terms of the Fourier transform of the fluctuating stress tensor t~. Eq. (61) is easily solved using standard techniques 34) and the explicit representation of # in eq. (6). This leads to the wave vector time correlation function ^

(80,(k, t)616t~(k', t')) = kBTe o ~( 6ii - klki/k:) e-(n°~:/°°)lH'l(27r)36(k + k').

(62)

Thus near equilibrium the spatially nonlocal theory does not modify the transverse momentum fluctuations33). Since the transverse momentum has two independent components, only three equations of the original five remain. They are

0A~(k, t )/ Ot

= -ikO~( k, t ),

OA~(k, t )/ Ot = Kek2E~l( k )Af)(k, t) + Kek~E~2( k )A~( k, t) + ik(h"/p~)O)(k, t) + f~(k, t),

(63)

OO~(k, t)/Ot = ( i k T e h ~ E ~ ( k ) + T~peE~(k))Af~(k, t) + (ikT~h ~E~z(k) + T"p~E~2(R))A~(k, t)

+ ff~kZi6)(k,t) + fl(k, t), with (fl(k, t)fl(k', t')) = 2 k B T ~ p ~ 8 ( t

- t')(27r)38(k + k'),

(f~(k, t)fe(k', t') -- 0,

(64)

(f~(k, t)f~(k', t')) = 2kakZS(t - t')(27r)38(k + k').

Here ~'~= (4~e/3 +,~')/pe, the superscript " e " emphasizes that the quantities are evaluated at equilibrium, and Ate(k, t) = ~(k, t ) - (2~)3pe6(k), etc. Using a prime to denote the column vector of these three variables and the notation of eq. (16'~), we write eqs. (63) and (64) symbolically as O~'(k, t)lSt = I~'(k )E~'(k)fi'(k, t) + f ( k , t),

(65)

with

(o ° l~e'(k ) =

o

ikT~p ~ ~:ek 2 ikTeh~ I , \ i k T ~ p ~ i k T ' h ~ ~epeTek2,/

(66)

318

J. KEIZER AND M. MEDINA-NOYOLA

a(-_$/T)/ae(k) a(I/T)/a@k) 0

a(-j/T)/+Xk)

I?(k)

=

a(I/T)/+(k) 0

0 (67)

and

(jk

t)J(k’, t’)) = 2kB : C,o

0 0

K$ 0

S(t - t’)(2r)3S(k + k’).

(68)

&CTek2

Eqs. (65)-(68), along with eq. (61), give the spatially nonlocal hydrodynamic level description of fluctuations in an equilibrium ensemble. The local approximation is obtained by setting k equal to zero in eq. (67). To compare our results with those of the local theory, it is simplest to change variables from A&k, t) to Af(k, t) = f(k, t) - (2r)3T”S(k) (the temperature deviation around T”). To make this change in eq. (63) we employ the following identity in the second of eqs. (63): -&(k)/&(k)

= -(a(i/T)la~)l(a(i/T)la&) = (he- T’(ap^/a~),)/p’,

= (a8/a& (69)

which is well known3) when k = 0, and thus true for kf 0. We also need to use the wave vector dependent Gibbs-Duhem relationship in the third of eqs. (63) (see also (67)) A$‘= -T”h’A(i/T)-

T”p”A(-G/T).

(70)

Making use of eqs. (69), (70), and the variable change aA%lat = -Te2aA(i/T)/at

= -Te2(8;,,Afi/,,

+ &ah&/at)

(71)

in eqs. (63), then yields aAfi/at = ikfi,, &aAf/at

= (ikT’/p’)(ap^/af),@,- (Kek2/Te2)AF + fe,

ajS,/at = -ik(afi/a&Afi

- ik(afi/a?),Af

(72)

+ [‘k’p, +fe.

(All the thermodynamic derivatives are to be evaluated at equilibrium.) The equations in (72) are easily compared with the Fox-Uhlenbeck LandauLifshitz fluctuating hydrodynamic theory3V34) by setting k = 0 in the thermodynamic derivatives. In this way our spatially local equations are seen to be identical with those equations. We also point out that there is no special advantage in using Afi and A?’ as variables since (as eq. (69) shows) a knowledge equivalent to the matrix E is required in (72). We have not before seen the spatially nonlocal Onsager theory written out for fluctuating hydrodynamics, although several theories with nonlocal ther-

SPATIALLY NONLOCAL FLUCTUATION THEORIES

319

modynamic derivatives have been considered previously35). Explicit representations of the Onsager theory are given by any of the equivalent groups of equations (61)-(64); (61), (62), (65)-(68); or (61), (62), (68), and (72). This is what the mechanistic fluctuation theory 27) in eqs. (1), (2), and (5)-(7) reduces to at equilibrium.

7. Particle distribution function expression for E(r, r')

The thermodynamic expressions which we have given for E(r, r') in the previous sections are not particularly useful for obtaining the functional form for ~E. However, as given in eq. (34), E~ is proportional to the inverse of the equilibrium covariance matrix of the energy and mass densities. Consequently, it is possible to obtain E~ directly by inverting try. In this section we pursue this approach by writing out expressions for tr~ in terms of moments of the two, three, and four particle equilibrium distribution functions. The resulting expressions demonstrate the way in which many particle correlations enter into the dynamics of fluctuations. The inversion of tr~ in terms of familiar functions, (i.e. other than in terms of the direct matrix correlation f u n c t i o n s - s e e eq. (39)) is possible only under special conditions on the intermolecular potential. For hard spheres we carry out the inversion explicitly. In order to obtain distribution function expressions for p~, we use the classical definitions for the mass and energy densities 2~) N

nl(r) =--p(r)

= m ~

8(r - ri),

(73)

i=l

n2(r) ~- e(r) = - ~ v ~ 8 ( r

- rl) +

~'~(r,-

rj)~(rj - r),

(74)

where 4~ is the intermolecular pair potential, vi is the velocity of molecule i and the summations are over all N molecules. Using the classical canonical ensemble and the definition of ~r~, we need to calculate le (tr2)ij(r , r') = (ni(r)nj(r')) le -- (ni(r))le(ni(r')) je,

(75)

where (g)'e = f f PN(P t~, rN)g dp ~ dr N.

(76)

Here P~ is the local equilibrium 22) canonical N-particle probability density and the integrals extend over all 3N coordinates and 3N momenta. Substituting eqs. (73) and (74) into (75) and using the identity of the particles, the

320

J. K E I Z E R A N D M. M E D I N A - N O Y O L A

integrals are found to be straightforward, if rather tedious for the energyenergy term. To simplify the integrals we use the standard definitions 36) of the n-particle correlation functions p " g ( " ) ( r l - r2, r z - r3 . . . . .

r.

,-- r.)=--f f PN(pN, rN) dv ~' d r ~-"

(77)

and h(Z)(r - r') = g(2)(r - r ' ) - 1.

In terms of these functions we find ( S p ( r ) S p ( r ' ) ) l~ = m p S ( r - r') + p2h(2)(r - r'),

(78)

(ao(r)ae(r')) I~= [3OkBT ~m f g(2)(r')cb(r'3 dr''] ,5(r + ~mTp

2h(E)(r -

r')

r') +-~m ~(r- r')g'a)(r - r') (79)

+ p312m2fg(3)(r - r', r')~b(r') dr",

( ~ e ( r ) a e ( r , ) ) , ~ = [ P (15)iv ,r,2 3 2 ~ f a . . . . p2 Lm ~ ~,~B-J + p 2 g ( 2 ) ( r ' ~ ( r " ) u r -r 4~m x

f

g(E)(r")cbE(r')dr"+~--~m j u t

jut

x a(r - r')+

h(E)(r - ,-")~-"~ - - [3kR ~ , ~ - r O , 4~(r4- r')) 3kBZp3[ f

× 4 ) ( r - r')g(Z)(r- r')+ ~ [ j -

g tr,r")4~(r')4a(r'") .....

dr"g(2)(r')d~(r '

+ p

d r " g ( 3 ) ( r - r', r")g)(r')

d r ,,g (3)( r -

r ,' r")

3

x , ( ~ " ) ~ ( ~ " + ~ - ~')+ 2 - ° ~ ( ~ - ~')I d~"

x g'~'(~-~', ~")4,(~'~+~-~f d~"f d~" X g(4)(r -- r', r" - r', r'" -- r')4)(r"-- r)4~(r .... r') 40-~m( f dr'g(2)(r')4~(r')) 2.

]

SPATIALLY NONLOCAL

FLUCTUATION

THEORIES

321

In using these expressions in the fluctuations-dissipation theory one must recall that the densities and temperatures are to be interpreted as their c o n d i t i o n a l a v e r a g e values.

We have written eqs. (78)-(80) in such a way that cancellations when r and r' are well separated in space are more obvious. For example, let I r - r ' [ be large in the double integral appearing in the final term of eq. (80). Because of the potential terms, the integrand is non-zero only when r ~ r" and r' ~ r ' . But since I r - r ' l is large, this implies independent correlations between the points r, r" and r', r". Thus g(4) c a n be replaced in the integrand by g ( Z ) ( r " - r ) g ( 2 ) ( r . . . . r'). Clearly, then, when I r - r'l is large, the double integral is cancelled by the product integral in the final term. In a similar fashion all the terms in eqs. (78)-(80) can be shown to vanish when I r - r'l is sufficiently large. Eqs. (78)-(80) are exact and require a knowledge of the local equilibrium two, three and four particle correlations. These correlations are difficult to measure or calculate analytically, although it may be that computer molecular dynamics can be helpful in obtaining these expressions. For hard spheres, however, the fact that g(")~bm vanishes everywhere can be used to greatly simplify the distribution function expressions for Or~. Indeed, one finds that eqs. (78)-(80) become /

OrE(r , r ' ) = Or21t~(r --

1o2

r') + \ll2kaTpZ'2m

3kBTp2/2m \ ~(2)/ ( 3 k a T p [ 2 m ) Z ] n Hstr - r').

(81)

Similarly using eqs. (40) and (41), our definition of the correlation matrix h ( r , r') in eq. (42), and eq. (81) we find for hard spheres that (hHs(r,r'))ii = ~th(2)t Hs~ r -- r')/m2)SliSli.

(82)

We can now complete the program of obtaining E2(r, r') for hard spheres. According to eq. (39) we need the matrix of energy and density direct correlation functions. This can be found explicitly by Fourier transforming the matrix Ornstein-Zernicke equation (44) and solving for Cns in terms of hHs, as given by eq. (82). The manipulations are straightforward and yield

(CHs(r, r'))ij = (Crjs(r (2) -

r')/m2)61i6ji,

(83)

where c(ffs is the usual scalar direct number density correlation function. Thus for hard spheres the direct correlation function c (2) contains all the information necessary to describe the spatially nonlocal behavior of hydrodynamic fluctuations. Eq. (83) also confirms our earlier remark that the matrix direct correlation function is very short ranged3°).

322

J. KEIZER AND M. MEDINA-NOYOLA

8. Hard sphere fluid near equilibrium

We have used the hard sphere fluid to test the nonlocal spatial features which the mechanistic nonequilibrium thermodynamic theory predicts for hydrodynamic level fluctuations. Thus far our calculations are restricted to a neighborhood of equilibrium, and there our results are equivalent to the spatially nonlocal Onsager theory. Indeed, we have already verified using eq. (72) that at small wave vectors our theory reduces to the usual local hydrodynamic fluctuation theories3). Near equilibrium there are two basic differences between the local and nonlocal theories. First the nonlocal theory gives the correct equilibrium covariance matrix, ere; or said in another way, it gives the correct equal time correlation functions. The local approximation does not. Second, the spatially nonlocal theory requires the use of space (or E-vector) dependent thermodynamic derivatives to describe the temporal relaxation of fluctuations. The local theory uses the corresponding spatially averaged thermodynamic quantities. The first difference in the theories is easily checked. By combining eqs. (14) and (34) we see that the mechanistic theory implies ( S n i ( r ) S n i ( r ' ) ) ° = cr ij(r, r') = - k ~

8ni(r)~nj(r')

'

(84)

which, according to statistical mechanics22), is an exact expression in the thermodynamic limit. In the local theory, on the other hand, E e is given by eqs. (51) and (52). Consequently in that approximation the equal time correlations as described by eq. (34) will be (S n ~(r )S n i(r ')) ~= - kB(Oni/OFj)e8(r - r').

(85)

Written in wave vector space, eqs. (84) and (85) become + k'),

(84')

crTi(k, k') = -kB(Oh,/OFj)~(O)(27r)3rS(k + k').

(85')

cr~ii(k, k') = -kB(Oft,/aff'j)e(k)(2zr)38(k

Thus there will be a significant difference between the local and nonlocal theories for the static correlations whenever (ariJaPj)~(k) depends strongly on k. We can estimate this dependence for hard spheres using eq. (82). This shows that the local and nonlocal theories will differ whenever h<2)tk HSk )~ depends significantly on k. For example, let us call R the hard sphere diameter. Then O ( ~ / a ( - ~ / T ) gives the k vector dependence of the static density fluctuations. According to molecular dynamics calculations~9'3°), at liquid densities this varies by a factor of about three between k = 0 and k R = 4. At higher k values the variation is much greater.

SPATIALLY NONLOCAL FLUCTUATION THEORIES

The nonlocal part of the k-vector dependence of the the fluctuations is given by the matrix E e in eq. (68). As (82), /~ for hard spheres depends only on d~2~(k). In fact and (83) we can write out eqs. (65)-(68) for hard spheres is

323

relaxation matrix of we have seen in eq. using eqs. (39), (40) in detail. The result

O~t'(k, t)/Ot =/4'(k)ti'(k, t) + f(k, t),

(86)

with

I?t'(k ) =

0 pe TeKek:/ ~[pe/p~_ l/SHs(k)]

0 - 2mKekE/3peka -2ikp~/3P~

-ik

-ikhe/Oe ~ _~ek 2 ; .

(87)

For hard sphere h e = 5kBTp~/2m, p~ = pekaT/m, and S,s(k) = 1 + (pe/m)l~t~s(k) = ((pe/m)e~s(k) - 1)-1,

(88)

the static structure factor. The local equilibrium pressure, p~, is a function of p~ and TL The simplest stochastic quantity to calculate for hard spheres is the two-time correlation function

C(k, k', t, t') -- (~(k, t )~T(k ', t')).

(89)

For the transverse momentum, this correlation function has already been exhibited in eq. (61). For the remaining variables we appeal to the fact that the mechanistic theory gives rise to a stationary, Gaussian, Markov process near equilibriumg-U). Thus for Ate, A~, and/51 it follows that 2)

C'(k, k', t, t') = exp[H'(k)(t' - t)](-kaE¢-t(k))(2~r)38(k + k'), t' >! t =C'X(k',k,t',t), t'<~t. It suffices, then, to focus on the quantity 6-'(k, t) --- exp[/4'(k)t](- kBE~-I(k)).

(90)

For hard spheres eqs. (82) and (88) imply that [ mpe(SHs(k)- 1) -kaE~-I(k) = tr7 + ~3gaT~pe(S.s(k) - 1)/2 \ 0

3kaTepe(S.s(k)- I)/2 (3kaT~/2)2pe(S.s(k)- 1)/m 0

0 u

\

) •

(91)

kaTep'/

For purposes of calculation we have written out eq. (90) in terms of the eigenvalues A~(k) of IgI'(k) as 3

6"(k, t) = ~ e-X'tk~tcJi(k)E~-l(k).

324

J. KEIZER AND M. MEDINA-NOYOLA

The exact form of the matrices (~ are easily obtained using the characteristic polynomial 37) o f / 4 ' . They involve only the matrices/4'"(k) for n = 0, 1, 2 and the eigenvalues hi(k). Thus the time dependence of &'(k, t) is given by a linear combination of three k-dependent exponentials. Alternatively we have also considered the time Fourier transform +~

~'(k, oJ) - f 6-'(k, t) e i~t dt,

(92)

an expression which is often used to describe the results of neutron and light scattering experimentsS). For small k one of the roots of H'(k) is strictly positive (hi, the thermal diffusion root) and two are complex (X2 = h~ = AR+ ih0, where ;~i is related to the sound speed and )tR to the sound wave damping rate38). Under these conditions &'(k, ~o) is the sum of three Lorentzians A(k) + B(k)((o~_ ~'(k, ~o) = ~oz+ ~.~(k)

1 1 + )tl(k))2 + )t ~(k) +(to - )q(k))2 + h i ( k ) ) '

(93) the first term giving the Rayleigh peak, Brillouin side bands. At k values for appear we will retain this nomenclature The equation governing the roots o f / 4 '

while the final two are the symmetric which two complex conjugate roots even in the spatially nonlocal theory. can be gotten from eq. (80). It is

h 3 + [(2T~K3ml3k~) + ~]k2h 2 + [ ( 2 T e K ~ m ~ / 3 k a ) k 2 - a(k)]kEh +/3(k)k 4= 0,

(94)

where or(k) = (kBTe]m)[~(Pem]P~) z + 1/Sus(k)],

(95)

[3(k ) = - 2KCTe2/3S~s(k ).

(96)

To complete the calculation we need p e = p(p~, T~), i.e. the equation of state for hard spheres, the thermal conductivity K ~ = K ( p ~, T~), the bulk and shear viscosities (-0(p e, T ¢) and ~(p¢, T~)), and Sias(k), the static scattering function. None of these quantities are known analytically for hard spheres in the dense fluid regime. However several excellent approximations-e.g, the Percus-Yevick approximation for both the equation of state and S,s(k)39), and the Enskog expressions for the transport coeflicients4°)-are available. Similarly computer molecular dynamics estimates of all the necessary quantities are available4~). In calculating the eigenvalues from eq. (94) we have relied on the molecular dynamics calculations for both the equation of state ~9) data and transport coefficients41).

SPATIALLY NONLOCAL FLUCTUATION THEORIES

325

In o u r c a l c u l a t i o n s w e have used the dimensionless variables ~*(k, t) --- (zrR3/6m)~(k, t), ti*(k, t) = (zrR3/9kaT)~(k, t), ti*(k, t) = (zrR3/6(mkaT)I/2)~l(k, t), t h e d i m e n s i o n l e s s t i m e t* = t(kBT/mR2) 1/~, a n d t h e d i s t a n c e r* = r/R. W i t h t h e s e s c a l i n g s , n e i t h e r t h e m a s s n o r t h e t e m p e r a t u r e e n t e r e x p l i c i t l y into o u r r e s u l t s .

9. Results for hard spheres R e p r e s e n t a t i v e r e s u l t s o f o u r c a l c u l a t i o n s a r e g i v e n in figs. 1-9, a l o n g w i t h t h e r e s u l t s o f c o m p u t e r d y n a m i c s c a l c u l a t i o n s p e r f o r m e d b y A l d e r , Yip, a n d Alley19). T o f a c i l i t a t e c o m p a r i s o n w i t h t h e m o l e c u l a r d y n a m i c s w e h a v e c a l c u l a t e d t h e i n t e r m e d i a t e s c a t t e r i n g f u n c t i o n 2°)

F(k, t) =- ~1(k, t)lm 2 a n d its F o u r i e r t r a n s f o r m

S(k, to) =- d~l(k, o~)lm 2.

1.0

~'~

v/ vo --~.6 °.

kR

= .38

0.5

°.e

0.1

0

I

I

I

l

2

3

t Fig. 1. The intermediate scattering function for hard spheres of diameter R, density V]Vo = 1.6, wave vector kR = 0.38, and S(k)= 0.026. The time t is measured in units of (mRE/kaT) ~I2.The line is based on eq. (90), and the dots and error bars are the computer molecular dynamics results of Alder, Yip, and Alleyt9'2°). The values of F(k, t) have been normalized to F(k,O)= I by dividing by S(k), the hard sphere structure function.

326

J. KEIZER AND M. MEDINA-NOYOLA

T h e s e are equivalent expressions for the n u m b e r density correlation functions and the ones which are accessible by neutron or light scattering. T h e y have also b e e n calculated by Alder, et alJg). In figs. 1-3 we show both our calculated and the c o m p u t e r dynamics results for F(k, t)/S(k) for t >i 0 and three values of kR all at a density corresponding to V/V0 = 1.6 (V0 = NR3[~/2). This density is near the transition density of the hard sphere fluid 42) and c o r r e s p o n d s to a dense liquid. Fig. 1 includes confidence limits on the molecular dynamics calculation, and our analytic calculations and the molecular dynamics results are essentially superimposed for kR = 0.38. Figs. 2 and 3 show a similar c o m p a r i s o n for kR = 0.759 and kR = 2.277. A disparity b e t w e e n the spatially nonlocal h y d r o d y n a m i c calculation and the c o m p u t e r dynamics appears at these higher kR values. This, of course, is to be expected since these higher values of k correspond to a distance scale which is very close to the hard sphere diameter R. Indeed details such as the hard sphere b o u n d a r y condition are missing f r o m the h y d r o d y n a m i c level calculation. N o n e t h e l e s s the agreement is r e m a r k a b l y good for kR ~< 1. Figs. 4-6 show similar calculations at the intermediate density V/Vo = 3. This density corresponds to a low density liquid and, again, as long as kR <~ 1 the a g r e e m e n t b e t w e e n the spatially nonlocal h y d r o d y n a m i c s calculation and I.O

V/Vo = 16 kR = .759

0.5

! 0

' z

i

L

2

Fig. 2. The hard sphere intermediate scattering function for V/V0= 1.6, kR = 0.759, S(k) = 0.027. See fig. 1.

SPATIALLY NONLOCAL FLUCTUATION THEORIES

327

1.0 ~=1.6 kR = 2 . 2 7 7

o.5

o

•

o

m

0.5

1.0

1.5

t Fig. 3. The intermediate scattering function for V/Vo = 1.6, kR = 2.277, and S(k) = 0.0365. See fig. 1.

,.oL kR = . 5 0 8

~1 ~

0

0

I

I

I

5

I0

15

t Fig. 4. The intermediate scattering function for V/Vo = 3, kR = 0.308, and S(k) = 0.145. See fig. 1.

328

J. KEIZER AND M. MEDINA-NOYOLA

1.0 v/ vo = 3 kR = .6/6

0.5

eo

0

0

I

2

o

o

5

4

5

Fig. 5. The intermediate scattering function for V]Vo= 3, kR = 0.616, and S(k) = 0.149. See fig. 1.

the c o m p u t e r dynamics is good. We emphasize that the agreement here is for the entire time scale accessible to the molecular dynamics calculation as is illustrated clearly in fig. 4. The oscillations in that figure came f r o m the two c o m p l e x roots, )~2(k) and )~3(k), of /~'(k) and the period of the oscillation is essentially the time for a sound w a v e to travel the hard sphere diameter. The amplitude of the oscillations decreases rapidly as kR increases and they are barely visible in fig. 6. At a value of kR = 1.232 a g r e e m e n t with the molecular dynamics remains good for t* ~ 0.5, and the two results are qualitatively the same for t* > 0 . 5 . Nonetheless, it is again clear that the nonlocal hydrodynamic theory breaks d o w n for kR ~ 1. A further decrease in density is illustrated in fig. 7. There we give results for S(k, to) for a typical dense gas (V/Vo= 10). The units of S(k, to) are arbitrary, and because the molecular dynamics calculations are for finite to, we h a v e b e e n unable to normalize the two calculations so that the area under the two curves is the same. This would c o r r e s p o n d to setting SHs(k) = F(k, 0) the same for both calculations. Instead we have set the two values of S(k, to) equal at to = 0. At V/V0 = 10 the Brillouin and Rayleigh p e a k s overlap in the

SPATIALLY NONLOCAL FLUCTUATION THEORIES

329

.0

W~:3 kR = 1 . 2 3 2

0.5

t Fig. 6. The intermediate scattering function for V/Vo= 3, kR = 1.232, and S(k) = 0.165. See fig. 1.

molecular dynamics calculation and have merged into a single broad peak with a long shoulder in our calculation. While the results are qualitatively similar, the hydrodynamic results are no longer accurate. This is not a surprise because of the absence of events in the hydrodynamic level description which occur on a time scale of the mean free time43). Compared to the higher densities, the mean free time at V/V0 = 10 is increased significantly. To accurately calculate this effect would require a molecular collisional level description of fluctuations (e.g,, fluctuating Enskog equation). From figs. 7, 8, and 9 one can glean the effect on S(k, to) of increasing the density for kR near 0.5. Clearly the Brillouin peak moves to higher f r e q u e n c y as the speed of sound increases and is reduced in amplitude as the density is increased from WV0 = 10 to V/V0--1.6. In fig. 6 the Brillouin peak is completely lost due to the increased damping of the sound waves. The improvement of the nonlocal over the local hydrodynamic fluctuation theory for the kR values in figs. 1-9 comes primarily in giving the correct value of F(k, 0 ) = SHs(k). Indeed, when we plot F(k, t)/S~s(k) for the local theory, the curves are similar to the nonlocal results given in figs. 1-6 if SHs(k) is the correct nonlocal structure factor. Thus for this range of k values, the

330

J. KEIZER AND M. MEDINA-NOYOLA 5

".•

wvo

= ~o

4

5¸ 3 v

oo 2

00

0.5

1.5

t.O Fig. 7. The frequency spectrum of the hard sphere dynamic scattering function for V/Vo = 10, kR = 0.412 and S(k) = 0.569. The frequency ~o is measured in units of (kBTlmR2)~j2and the units of S(k, co) are arbitrary. To compare the calculated results based on eq. (93) (full line) with the computer molecular dynamics calculations of Alder, Yip, and Alley~9'2°),(dots), the two values of S(k, 0) have been set equal. time d e p e n d e n c e of the local and n o n l o c a l theories is hard to differentiate. B e c a u s e of the e x a c t n e s s of the n o n l o c a l t h e o r y at all k values for t ~ 0 , it will give distinctly better values f o r F(k, t) f o r higher kR values and s h o r t times. O n the basis of our calculations w e c a n estimate the limits of validity of the spatially n o n l o c a l h y d r o d y n a m i c t h e o r y . First we h a v e seen that the w a v e v e c t o r s m u s t satisfy kR ~ 1.

(97)

F u r t h e r m o r e as s h o w n in fig. 7, the m e a n free time, ~'m, m u s t be sufficiently small. To estimate h o w small, w e utilize the b e h a v i o r of F ( k , t) in the a b s e n c e of collision. T h a t is well k n o w n 33) to be

F(k, t) = exp[-½(kvot)l12]F(k, 0),

(98)

w h e r e Vo = (kBT[m) 1/2. C o n s e q u e n t l y the effect of the m e a n free time will be

SPATIALLY

NONLOCAL

FLUCTUATION

THEORIES

331

I0 V/~ =3 8

kR = . 6 / 6

•

6

3

4

2

O0

"

"

t

i

I

2

5

oJ

Fig. 8. Frequency dependence of the hard sphere dynamic scattering function for V/Vo= 3, kR = 0.616, and S(k) = 0.149. See fig. 7. negligible at about the I% level if kvo1"m~ 1/10.

(99)

combining this with eq. (97) implies that a sufficient condition is 1)0Tm ~

or, letting

R/IO, ~m

be the mean free path,

h~ <~R/IO.

(100)

This condition is easily met 42) in the hard sphere fluid at V/Vo = 1.6. If both eqs. (97) and (100) are satisfied, the spatially nonlocal hydrodynamic expressions for F(k, t) should be valid for all significant times, If either of these conditions is violated, we expect the theory to be accurate only for short times. Indeed using eq. (98) again, we find that in the absence of conditions (97) and (100) the results for F(k, t) should be accurate only when 0<~ t ~ l]lOkvo.

(101)

However, for these times the theory should be accurate for all k, since it

332

J. KEIZER AND M. MEDINA-NOYOLA

V / V o = 1.6

kR = . 7 5 9

3 v

(/3

5

I0 O.I

Fig. 9. Frequency dependence of the hard sphere dynamic scattering function for V/Vo = !.6, kR = 0.759, and S(k) = 0.027. See fig. 7 and cf. fig. 2.

J

J JT I/R

I

.,,J

/J~R/IO

Fig. 10. A schematic diagram of the space of parameters time (t), wave vector (k), and mean free path (hm). For parameter values beneath the indicated surface, the nonlocal hydrodynamic theory is accurate for hard spheres.

SPATIALLYNONLOCALFLUCTUATIONTHEORIES

333

properly includes the correlations at t = 0. Thus, for example, even in dilute gases where the mean free path greatly exceeds the collision diameter (i.e. eq. (100) violated), the spatially nonlocal theory should be accurate for all times in the limit of small k-vectors. A schematic graph describing these estimates of the validity of the spatially nonlocal hydrodynamic approach is given in fig. 10. At all points (k, t, )tin) beneath the two-dimensional surface shown, F(k, t) for hard spheres is accurately described near equilibrium. The theory clearly is at its best for liquids and when kR <~ 1.

10. Prospects for far from equilibrium calculations Since the theory outlined in section 2 is not limited to equilibrium, it is natural to inquire about its application away from equilibrium. Indeed this is the purpose for which the mechanistic fluctuation theory was originally devised8). Calculations based on the Landau-Lifshitz fluctuating hydrodynamics- which is equivalent to the local approximation of the theory presented h e r e - have already been carried out for fluids with temperature or shear gradients44). Those calculations are not easy and only recently have the various groups reached general agreement about the results44'45). The difficulty stems from considerations that do not occur at equilibrium: (i) boundary conditions and (ii) the fact that fluctuations in k space satisfy integrodifferential equations due to the spatial nonuniformities. Because these calculations are based on the local theory, they are valid only in the limit k ~ 0. In earlier work we have calculated spatial correlations for solute molecules in nonideal solutions based on a limited version of the present theory'7). There we neglected the center of mass momentum density, the energy density, and the solvent density. Thus our calculations included only diffusive effects, and we found that local equilibrium always develops at steady states in such systems. The calculations*~) using the spatially local hydrodynamic theory have made it clear that nonlocal equilibrium effects arise when other hydrodynamic elementary processes are important. Indeed, this has been known for a number of years for chemical reactions and for chemical reactions coupled to diffusion46). Recently we have carried out calculations using our spatially nonlocal fluctuating diffusion theory coupled to chemical reactions47). We have obtained corrections to the Debye-Hiickel radical distribution function in the presence of rapid chemical reactions and used it to obtain the rate constant for rapid reactions in solution48). These calculations involve the nonlocal

334

J. K E I Z E R A N D M. M E D I N A - N O Y O L A

effects discussed in this paper, although since gradients in the average variables do not occur, neither of the complications discussed above appear. In general the present theory has the advantage that it can be used systematically to calculate the static correlations in a far from equilibrium stationary state. This cannot be done using the local (Landau-Lifshitz) theory. Our present calculations for hard spheres suggest that one will be able to examine correlations lengths down to the order of molecular diameters. The technique of calculation is straightforward, being based on the fluctuation-dissipation theorem H=7) as generalized to stationary states. Indeed, this has been our approach for reaction-diffusion problems where many analytical calculations are tractable. Based on our experience with chemical reactions, and recent nonequilibrium computer dynamics calculationsS°), we do not expect to see significant modifications in spatial correlations on the molecular scale for the thermal gradient problem unless gradients of molecular dimensions are applied. Nonetheless, the present theory should have experimental application. We are thinking primarily about the onset of hydrodynamic instabilities, sometimes referred to as nonequilibrium phase transitions. It is known that the mechanistic nonequilibrium thermodynamic theory predicts a great increase in both the magnitude N) and range of fluctuations near a point of instability~6). Indeed, both experiment s~) and theory s2) have borne this out for the Gunn instability. The present spatially nonlocal theory can be used, in principle, to calculate the entire range of static fluctuations near a hydrodynamic instability.

Appendix A

The scalar Ornstein-Zernicke equation To obtain the scalar Ornstein-Zernicke equation we change variables from 80, ~e to 8n (the number density) and ST. Symbolically

8n'(r) = f A(r, r')Sn(r') dr'.

(A.1)

Using eq. (16) in the form 8T =

- Te28(l/T)

= - re2 f E2i(r, r')Sni(r') dr',

we find that

A(r,

r t)

( \ - T 2E2t(r, r')

o

-Te2E22(r, r')/'

(A.2)

SPATIALLY NONLOCAL FLUCTUATION THEORIES

335

where the momentum variables are suppressed as they are not effected by the variable change. Eq. (16) and the rule for transformation of matrices under a linear change of variables then gives rye'(r, r ' ) = -kB(El-~'(r'or')/m2

0

Te4EEE(r, r'))'

(A.3)

the covariance matrix of 8n and ST, with E~t! the 1,1 element of the inverse of E. Consequently by eq. (34)

E~(r, r,) = (E~l(or, r')

O) E'22(r, r') '

(A.4)

where we have defined

-kafm-2E?d(r,r'OE~l(r", r') d r " = 8(r - r'), -kB f E22(r, r'~E'22(r", r') d r " = 8(r - r')/T ¢4.

(A.5)

As in eqs. (39) and (42) we next introduce the matrices h' and c' by the equations

o'e'(r,

r')

=

or~'8(r

-

r') +

cri~'h ' c r ~' i,

E'(r, r') = E~8(r - r') + ksc'(r, r'),

(A.6)

(A.7)

with 0T

0 0T / ' 2mT¢2/3p ¢ 0 peksT~Im/ / - kBm/p ° 0 0T ~ E~ = ~ 0 -3p ~ks/2T¢2m 0T , 0 0 -Ira/peT \

o'~'=(P°! m t

(A.8)

as follows from eqs. (40), (41) and (A.3). The matrix Ornstein-Zernicke equation then follows by direct substitution of (A.6) and (A.7) into eq. (34)

h'(r, r') = c'(r, r') + f c'(r, r'Oor['h'(r", r') dr".

(A.9)

Because E' and tr' are diagonal (cf. (A.3) and (A.4)), so are c' and h'. Thus the 1,1 term solves the equation

h~l(r, r') = c~l(r, r') +

f

rO(p~/m)h~l(r ", r') dr".

(A.10)

However, since the new variable n~ is the number density, it follows from eq.

336

J. KEIZER AND M. MEDINA-NOYOLA

(A.6) and the usual definitions that h~l(r, r') = g~2)(r, r ' ) - 1 = h~2)(r, r'),

(A.1 1)

where g~2)is the radial distribution function. Eqs. (A.10) and (A.1 1) then give h<2)(r, r ' ) = c~l(r, r') +

(pelm)f c h ( r ,

r")h~2)(r ", r') dr",

This is the usual scalar Ornstein-Zernicke

ch(r,

r') =

(A.12)

equation and implies that

c~2)(r, r ' ) .

Appendix B D e r i v a t i o n o f the c o m p r e s s i b i l i t y e q u a t i o n

We use wave dependent thermodynamics to obtain the compressibility equation. We begin by obtaining (alS/al~)~ from the Gibbs-Duhem relationship in wave vector space dO = - T e h e d(I/T) - Tep ~ d(-/2/T)

(B.1)

and eq. (16) which gives d(-t2/T) =/~t2 d~ +/~1~ diS,

(B.2)

d(i/T) =/~22 d~ +/~2, d~3.

(B.3)

Solving (B.3) for d& substituting the result into (B.2), and then substituting the resulting expression for d(-tMT) into (B.1) yields (3#/3D)eT = -- T e p e det/~2]/~22,

(B.4)

where det is the determinant and /~2 is the 2 x 2 matrix. Elementary algebra gives ET~ = /~2Jdet/~2 and eq. (34) implies that / ~ = -kB-~cru. ^e Consequently (B.4) can be rewritten (Of9/O#)~ = 6"~(k)/p~kBTL

(B.5)

Recalling that ~,(k) = f+_~ eik (r-r'> < 8p(r)Sp(r') > ~d(r - r') and the definition of h(E)(r- r') in eq. (78), then gives k a T e (Of9/O~)~ = 1 + (p~/m)f~(2)(k). m

(B.6)

Eq. (B.6) is the wave vector dependent generalization of the compressibility equation (57) which follows from (B.6) in the limit that k -> 0.

SPATIALLY NONLOCAL FLUCTUATION THEORIES

337

Acknowledgement W e w o u l d like to t h a n k Drs. B. A l d e r a n d N . E . A l l e y for g e n e r o u s l y s u p p l y i n g the r e s u l t s of t h e i r c o m p u t e r d y n a m i c s c a l c u l a t i o n s prior to p u b lication. T h i s w o r k was s u p p o r t e d b y N S F G r a n t C H E 8 0 - 0 9 8 1 6 a n d a C O N A C Y T ( M e x i c o ) f e l l o w s h i p to M. M - N .

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J. KEIZER AND M. MEDINA-NOYOLA

33) D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (W.A. Benjamin, New York, 1975), chapt. 4. 34) R. Fox, Phys. Rep. 48 (1978) 179, section 1.7. 35) Ref. 20, pp. 307-317. 36) T.L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1953), chapt. 6. 37) B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956) pp. 119-121. 38) P. Resibois and M. DeLeener, Classical Kinetic Theory of Fluids (Wiley-lnterscience, New York, 1977) pp. 124-129. 39) Ref. 30, pp. 279-289. 40) Ref. 38, pp. 156-170. 41) B.J. Alder, D.M. Gass and T.E. Wainwright, J. Chem. Phys. 53 (1970) 3813. 42) J.A. Barker and D. Henderson, Rev. Mod. Phys. 48 (1976) 587. 43) Ref. 33, p. 15. 44) A.-M. Tremblay, E.D. Siggia and M. Arai, Phys. Lett. 76A (1980) 57. G. van der Zwan and P. Mazur, Phys. Lett. 75A (1980) 370. D. Ronis and S. Putterman, Phys. Rev. A 22 (1980) 773. R. Fox, to be published in J. Phys. Chem. 45) A.-M. Tremblay, M. Arai and E.D. Siggia, Phys. Rev. A23 (1981) 1451 has a complete bibliography of w o r k - published and unpublished - on these problems. 46) For a recent review see D. McQuarrie and J. Keizer, in Theoretical Chemistry: Advances and Perspectives, D. Henderson, ed. (Academic Press, New York, 1981). 47) J. Keizer, to be published in J. Phys. Chem. 48) J. Keizer, J. Phys. Chem. 85 (1981) 940. 49) Ref. 30, chapt. 14. 50) D.J. Evans, Phys. Rev. A23 (1981) 1988. 51) S. Kabashima, H. Yamazaki and T. Kawakubo, J. Phys. Soc. Japan 40 (1976) 921. 52) J. Keizer, J. Chem. Phys. 74 (1981) 1350.