Faraday Discuss. Chem. SOC., 1957, 83, 21-31 The Generalized Langevin Equation as a Contraction of the Description An Approach to Tracer Diffusion Magdaleno Medina-Noyola Departamento de Fisica, Centro de Investigacion y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Mkxico, D. F., Mexico An effective Langevin equation for a tracer Brownian particle immersed in a macrofluid of other diffusing particles is derived as a contraction of the description involving the stochastic equations for the local concentration and the local current of the macrofluid particles. The resulting Langevin equation contains the effects of the interactions with the other diffusing particles in a temporally non-local friction term plus a fluctuating force representing the random, diffusion-driven departures from spherical sym- metry of the distribution of macrofluid particles around the tracer.This fluctuating force satisfies a fluctuation-dissipation relation with the effective time-dependent friction. This program is fully developed here only in the absence of hydrodynamic interactions, although the formal aspects of its extension are also suggested. The results derived here, however, are found to provide a unifying framework to describe, for example, self-friction and electrolyte friction in suspensions of charged colloidal particles within the same theoretical scheme. 1. Introduction The theory of Brownian motion consists, to a large extent, of the many efforts to understand, derive, generalize and apply the well known Langevin equation: d v ( t ) dt M - - ( v ( t ) +f( t ) .This equation constitutes the simplest phenomenological model of the random motion of an individual Brownian particle, whose average behaviour is summarized by quantities such as the velocity autocorrelation function and the mean-squared displacement. In the classical theory’ f( t ) is postulated to be a purely random Gaussian stochastic vector. The stationarity of the equilibrium state leads then to the fluctuation-dissipation theorem (FDT), (J;( t ) J ( O ) ) = kT52S( t ) S , J , and to the conclusion that u( t ) is a stationary, Gaussian, Markov process. [ T above is the temperature, k , Boltzmann’s constant, M and 5 the mass and friction coefficient of the particle. S ( t ) is Dirac’s delta function and a,, is Yuronccker’s delta.] As a phenomenological model of equilibrium fluctuations, this theory has been extended by Onsager and Machlup.“‘ It has been e~tablished,”~ for example, that the Landau- Lifshitz stochastic version of the hydrodynamic equations,s and the linearized Boltzmann equation with added fluctuation^,^-^ are two examples of fluctuation theories that fit in such general scheme, i.e.they can be cast as multivariate, stationary, Gaussian, Markov processes. These two equations also constitute a good example of two phenomenological descriptions of the same fluctuating phenomenon, differing only in 2122 Generalized Langevin Equation and Tracer Difusion the degree of detail. Thus one would expect that projecting the latter into the subspace spanned by the hydrodynamic variables one should be able to derive the former.Such derivation has been carried and constitutes an illustrative example of the general idea of contracting the de~cription,~ in which the aim is to establish a connection between two phenomenological levels of description of the same fluctuation phenomenon. Another important realization of the same idea was suggested and developed by Fox and Uhlenbeck,’ who derived eqn (1,l) from the stochastic hydrodynamic equations plus the equation of motion of a suspended particle: This equation couples the motion of the particle, through the pressure tensor P integrated on its surface, with the stochastic hydrodynamic fields. In general, the contraction procedure introduces a memory effect in the contracted description.Latter of the program of Fox and Uhlenbeck have led to the derivation of a Langevin equation with memory, along with a corresponding FDT, and to the explicit determination of the memory function, from a purely hydrodynamic level of description. This program has been extended by Mazurgc to many Brownian particles in an incompressible fluid. A microscopic derivation exists, however, of the N-particle Langevin equation,” which reads In this equation, M,, v,( t ) and rr( t ) are the mass, velocity and position, respectively, of the ith Brownian particle, which is subjected to a random forcef;. GI, is the hydrody- namic friction tensor, and 4,J( I r, - 5 I) the pair potential, between particles i and j . Eqn (1.3), or its equivalent many-body Fokker-Planck equation,12 may be regarded as the fundamental ‘microscopic’ description of the dynamic properties of the macrofluid constituted by the N Brownian particles.Starting from this level of description, a formal projection to a hydrodynamic level has been carried out by Hess and Klein.13 The variables describing the state of the system at this level are the local concentration n ( r, t ) and the local current j ( r, t ) of Brownian particles. An essential feature of the macrofluid analogue of the Navier-Stokes equation is its temporal and spatial non-locality in the typical time- and length-scales probed in dynamic light-scattering experiments. l 4 Unfortunately no simple phenomenological laws, analogous to Newton’s law in the Navier-Stokes equations, are known for the macrofluid’s hydrodynamic description, except in certain limiting condition^.'^ Assuming, however, that such law could be derived, or approximated, one could in principle set up an analogous program, at the macrofluid level, of the program suggested by Fox and Uhlenbeck3 at the simple fluid level, i.e.one could derive an effective Langevin equation for a tracer Brownain particle immersed in a macrofluid of other diffusing particles as the contraction of a description based on the stochastic version of the hydrodynamic equations of the macrofluid. It is the purpose of the present work to discuss some of the formal aspects involved in such a program, and to present the results of its explicit application in the simple case in which hydrodynamic interactions are neglected.Thus, in sections 2 and 3 we discuss the general aspects of the program proposed. We then carry out this program explicitly in section 4, in the absence of hydrodynamic interactions. The result of its application to specific systems is then reviewed in section 5. 2. Contractions of Non-Markov Processes One of the main concerns in the derivation of the Langevin equation from fluctuating hydrodynamic^'^^-'^ has been the demonstration of the FDT for the former, given thatM. Medina-Noyola 23 a FDT holds at the hydrodynamic level. As Berman” has pointed out, this particular task may be carried out in a more formal and general fashion, by verifying that certain conditions, the FDT included, hold at the non-contracted description. In our derivation of the generalized Langevin equation of a tracer Brownain particle from the stochastic hydrodynamic equations of the surrounding macrofluid, we adopt a similar attitude.Thus in this section we develop the formal elements which will allow us to carry out such derivation in a fairly straightforward manner. Since the stochastic hydrodynamic equations of the macrofluid are known to exhibit memory, we must first establish the conditions under which a fluctuation-dissipation theorem holds for projections of non-Markov processes. These conditions are stated in the following two theorems, whose demonstration will be provided elsewhere.’’ Theorem A (the ‘stationarity’ theorem). Consider a general N-dimensional stochastic vector a ( t ) defined as the solution of the generalized Langevin equation: *= -lor G( t - t’)a( t’) dt’+f( t ) dt with random initial condition a ( 0 ) of zero mean, and driven by the stochastic vector f( t ) of vanishing average and statistically independent of a(O), i.e. such that ( a(0)fT( t ) ) = 0.Then the following three statements are equivalent. (The superindex T indicates transpose.) (i) The correlation function c( t, t’) = ( a ( t)aT( t’)) is time-translational invariant, i.e. c( t + T, t‘+ T) = c( t, t’) for all t, t’ and 7. [For simplicity, we shall generally refer to this property of a ( t ) as its stationarity condition.] (ii) The correlation function off( t ) is related with the memory function matrix G( t ) by the following fluctuation-dissipation theorem (f( t)f’( t‘)) = 6 ( t - t‘)G( t - t ’ ) ~ + 6 ( t’ - t)uGT( t’- t ) , ( t # t‘) (2.2) where 6 ( t ) is Heaviside’s step function and u = (a(O)a’(O)) is the covariance matrix of (iii) f( t ) is stationary [in the sense defined in (i) for the process a( t)], and G( t ) is a(0).such that the generalized Langevin equation, eqn (2.1), has the following structure: -- dt‘ L( t - t‘)u-’a( t’) +f( t ) d a ( t ) dt general (2.3) where o is an antisymmetric, time-independent matrix and L( t ) = (f( t)f’r(0)). Theorem B (the ‘contraction’ theorem). Let a( t ) be the N-dimensional stochastic vector defined by the conditions of theorem A. Let us assume, in addition, that a ( t ) is indeed stationary. Consider now a projection of a ( t ) defined by the n-dimensional stochastic vector a( t ) with components ai( t ) = ai( t ) , i = 1,2, .. , n < N. Assume, in addition, that (ai(0)aj(O)) = 0 for 1 s i d n < j s N. Then, the vector a( t ) also satisfies all the conditions of theorem A, and is also stationary. Furthermore, the generalized Langevin equation for a(t) may be written as dcllo= -lor r(t- t’)a(t’) dt’+cp(t) d t where q ( t ) is the new vector of fluctuating ‘forces’, and r(t) is given in terms of the memory matrix G ( t ) of a ( t ) by f ( z ) = e, * ( 2 ) - 6 I*( z)[ 2122 + e*Z( z)] - l & ( z ) . (2.5)24 Generalized Langevin Equation and Tracer Diflusion In this equation, I,, is the ( N - n ) x ( N - n ) identity matrix and the circumflex indicates a Laplace transform. G , t ) , GIz( t ) , . . . , are the submatrices of dimension n x n, n x ( N - n ) , .. . , of G ( t ) , defined by the following partition: The demonstration of theorem A follows essentially Kubo’s derivation16 of the (‘second’) Auctuation-dissipation theorem. The demonstration of theorem B is based on straightforward projection operator algebra to identify r(t) as given in the theorem, and on the explicit demonstration that the resulting vector cp( t ) satisfies the fluctuation- dissipation relation in statement (ii) of theorem A. We should mention that these theorems also hold for non-Gaussian processes, for variables a , ( t ) of arbitrary sym- metries (time-reversal etc.) and for stationary states other than the thermodynamic equilibrium state. Let us notice, however, that if a, ( t ) -+ y,a, ( t ) under time-reversal, with y, = f 1, then w = - I ‘ d , and LT( t ) = rL( t)r, with rj, = S,,y,.The latter equation is a generalization of Onsager’s reciprocity relations, which result if r = * I and L( t ) = 2 S ( t ) 9 . The previous theorems constitute the formal scheme in which we shall carry out the contraction procedure involved in the program we proposed before. 3. Fluctuating Hydrodynamic Equations for a Macrofluid As a simple illustration of the use of the theorems above, we formally derive the fluctuating diffusion equation for the local macroparticle concentration N n(r, t ) - C S [ r - r i ( t ) ] , = I (3.1) of a macrofluid in the presence of an external static field $(I-), from the macrofluid analogue of the stochastic Navier-Stokes equation. This we do as an illustration of the use of the theorems above, and because in the absence of hydrodynamic interactions, the motion of a tracer only couples directly with n(r, t ) , as discussed in the following section, and this will be the only case which we shall consider explicitly.The macrofluid analogue of the fluctuating Navier-Stokes equation describes the departures S j ( r, t ) of the local current: N j ( r , t ) = 1 u i ( t ) S [ r - r , ( t ) ] 1 1 1 (3.2) from its vanishing equilibrium average. The variable an( r, t ) = n( r, t ) - neq( r ) satisfies the exact continuity equation, which reads: d S n ( r, t ) d t = -V Sj(r, t ) . (3 3) The analogue of the Navier-Stokes equation can be constr~cted’~ from the require- ment that the vector a( t ) , with components a,( r, t ) = an( r, t ) and a, ( r , t ) = @,( r, t ) ( i = 1,2,3) may be cast as a stationary non-Markov process, satisfying the conditions and consequences of theorem A above.As a result, Sj(r, t ) is found to satisfy the following stochastic equation -0 d’r’a,-,’(r, r’)Sn(r‘, t ) J d S j ( r, t ) kBTneq( r ) a t M - - (3.4) dt’ \ d3r’6( r, r’; t - t’) .Sj( r’, t’) +f( r, t )M. Medina-Noyola where the stochastic vector f( r, t ) satisfies the fluctuation-dissipation theorem (J;(r, t)$(r’, O > > = [ G ( r , r’; t)li,jk,Tn”‘(r’)lM 25 and the inverse function of uoo( r, r’) = ( S n ( r, 0) Sn ( r ‘ , 0)) is indicated by 0;: ( r, r’). The tensor G(r, r’; t ) embodies all the essential dynamic information of the macrofluid, and its exact calculation is only possible in certain limiting cases.” Hence, approximations or assumptions will eventually have to be provided to define this missing piece of information. Since owing to time-reversal symmetry the static correlation (an( r, t)&( r’, 0)) vanishes, then the additional condition of theorem B is also satisfied for the vector a( t ) with components a,( r, t ) = Sn( r, t ) . Thus, according to theorem B, a n ( r, t ) is also a stationary non-Markov process, which obeys the following generalized diffusion equation: t)=-[o‘dt’[ d3r’G(r, r’; t - t ’ ) S n ( r ’ , t ’ ) + f ( r , t ) d t where G ( r , r‘; t ) = - V ; d’r’’H(r, r”; t)neq(r”) * V r ~ o & , l ( d ’ , r’). kBT M I (3.5) ( 3 . 6 ) The matrix Hij(r, r’; t ) , syqbolically denoted by [ H ( t ) ] , ( r , r ’ ) , is given by the inverse Laplace transform of [ z 1 + GZ2( z ) ] - ’ , where [ 13 j j ( r, r’) = 6,s ( r - r’) and [ G22( t ) ] j j ( r, r’) = [G(r, r’; t > l i j - In the next section we shall refer to the ordinary Fick’s diffusion law: I aSn(r, t ) at -- - -V, - Doneq(r)V, d’r’a,-d( r, #)an( r’, t ) +f( r, t ) (3.7) in which Do= k , T / i o is the free diffusion coefficient of a macrofluid particle.This equation results from the Markov limit of eqn (3.5), with the approximation Gij(r, r’; t ) = ( i o / / M ) 6 ( r - r ’ ) S j j 2 S ( t ) , which is the strict hydrodynamic limit of Gij(r, r’; t ) in the absence of hydrodynamic interaction^.'^ 4. Effective Langevin Equation Going back to our original project, we now need an equation of motion for a Brownian tracer, analogous to eqn (1.2), which expresses the force on that tracer in terms of the hydrodynamic variables.Such an equation may be suggested by the many-particle Langevin equation, eqn (1.3), as applied to the tracer. Let us assume that under certain approximations, the tensors cij may be substituted by a pairwise effective friction tensor cif(ri - q), depending only on the relative positions of the particles i and j . Under this condition, eqn (1.3) may be written, for the tracer, as +I [ v r + ( l r - r - l - ( t ) ~ ) ~ n ( r , t> d ’ r + f ~ t ) (4.1) with n(r, t ) and j ( r , t ) being defined by eqn (3.1) and ( 3 . 2 ) , <TD being the effective friction tensor, and + ( I - ) the potential of the interaction between the tracer and any of the other diffusing particles. cT is the solvent friction on the tracer.This equation, along with the stochastic hydrodynamic equations discussed in section 3, might provide the26 Generalized Langevin Equation and Tracer Difusion non-contracted description leading to the effective Langevin equation for the tracer after the elimination of the hydrodynamic variables. In this paper we only carry out a simplified version of this program, in which hydrodynamic interactions are neglected, i.e. GTD = 0. Thus there is no need to consider explicitly the variable j ( r , t ) , and we may base our hydrodynamic description on the generalized diffusion equation of the previous section. The equation of motion for the tracer may then be written as where we have elimiated rT( t ) from the integral by shifting the origin of the integration variables r to the centre of the tracer.Thus, the new local concentration n ’ ( p , t ) is defined as n’(p, t ) = n[ rT( t ) + p, t ] . Its equilibrium average, neq((p), is the local concentra- tion of diffusing particles in the static field # ( p ) . Because of its radial symmetry, n e q ( p ) does not contribute to the integral in eqn (4.2). Hence n’(p, t ) must be substituted in that equation by its fluctuation around neq(p), Sn’(p, t ) = n’(p, t ) - neq((p). The equili- brium equal-time correlation of an’( p, t ) must also be time-independent, and is essentially the two-body correlation function of the macrofluid in the static field + ( p ) . Thus definining croo(p, p’) = (8n‘(p, O)Sn’(p’, 0)), we must have that” V,n‘“p> = -P I %(P, P’)V,,+(P’) d3P’ (4.3) where P = l/(kBT).equation: The fluctuations 6n (p, t ) are now expected to satisfy a stochastic generalized diffusion = [V,,neq(p)] - u( t ) - [‘ dt’ d’p’ G’(p, p’; t - t ’ ) S n ’ ( p ’ , t’) + f ’ ( p , t ) . (4.4) a w p , t ) a t - 0 The linearized streaming term originates from the change of coordinates to the reference frame of the tracer. G’(p, p’; t ) determines the diffusive relaxation of the concentration fluctuations as described from such a reference frame, which is in fact diffusing. For the moment, however, we do not specify G’(p, 9’; t ) . Eqn (4.2) and (4.4) constitute our non-contracted description, to which we now apply the protocol developed above. Thus, we cast these equations as a generalized Langevin equation for the vector a( t ) , defined as aT( t ) = [a,( t ) , a2( t ) ] = [ u( t ) , Sn’(p, t ) ] .The corresponding static correlation matrix u is given by and the matrix G ( t ) , with analogous notation, by Using eqn (4.3), (4.5) and (4.6), we may verify that Thus, according to theorem A of section 2, a ( t ) will be stationary, provided that the following ff uctuation-dissipation relations hold: and 0 1 2 = G 1 2 u 2 2 = - [ G ~ I U ~ ~ ] ~ = - m21. ( f T ( t )~T(o)) = k,T!%2~(t)l (4.7)M. Medina- Noyola 27 These FD relations are, of course, assumed. Hence, we may now apply the contraction theorem, which implies, given that uI2 = uTl = 0, that the subvector a,( t ) = u( t ) is also a stationary process, satisfying the generalized Langevin equation M , d o o = -&( t) + f T ( t ) - Gint( t - t ’ ) u( t ’ ) dt’+fin‘( t ) .d t I,‘ The additional fluctuating force, f i n ‘ ( t ) , satisfies the FDT ( f i n ‘ ( t)fint(0)) = MTkB TG’”‘( t ) with Gin‘(t) given by Gint( t ) = - d’p d’p’[V+(p)]x’(p, p’; t)[V’neq(p’)] I I where ~ ’ ( p , p’; t ) is the solution of (4.9) (4.10) (4.1 1) with initial condition ~ ’ ( p , p’; t = 0 ) = S(p - p‘). Eqn (4.9) is the generalized Langevin equation for the tracer that we set out to derive. From symmetry considerations, one expects that Gin‘(?) = I A l ( t ) . Using eqn (4.3) and (4.10) we may show that there are two additional, equivalent expressions for Gin‘( t ) which, in terms of A&( t ) , read (4.12) and where C’( p, p’; t ) = (Sn’( p, t)Sn’( p‘, 0)) is essentially the van Hove function of the macro- fluid in the reference frame of the tracer, i.e.it is the solution of eqn (4.11) with initial value aoo(p, p’). The function [ ‘ ~ ; d ~ ’ ( t ) ] ( p , p’) in eqn (4.13) is the convolution of u,-d(r, r’) with the propagator ~ ’ ( p , p ’ ; I ) . More explicit results require the specification of G’(p, p’; t ) in eqn (4.4). Let us notice, however, that a general simplification results if we assume that G’(p, p’; t ) and o,,(p, p’) depend on p and p’ only through I p - p’l. This amounts to ignoring the inhomogeneity produced by the field + ( p ) of the tracer. In this case eqn (4.13) may also be written as (4.14) where hTD( k ) is the Fourier transform of [ neq(p)/ n - 11, and 1 -I- n h D D ( k) = goo( k)/ n is the static structure factor of the homogeneous macrofluid.Similarly, x’(k, t ) is the Fourier transform of the propagator x’( I p - p’ I; t ) of the local concentration fluctuations as described from the reference frame of the diffusing tracer. The results above still require the determination of the function G’(p, p’; t ) , or equivalently, of C’ or x’. The exact relationship between these quantities and the corresponding objects, G( r, r’; t ) , C and x, which describe the concentration fluctuations of the macrofluid from a reference frame at rest, is unknown. With the help of reasonable assumptions, however, such a relationship could be determined in an approximate fashion; doing that would complete the program we meant to carry out, namely, writing the effective generalized Langevin equation for the tracer, eqn (4.9), in terms of the28 Generalized Langevin Equation and Tracer Diflusion quantities which describe the dynamics of the macrofluid at a hydrodynamic level.As an example, let us adopt the approximation in which C’(p, p’; t ) = C’( 1 p - p’l, t ) . Its Fourier transform C’( k, t ) is then given by N C ’ ( k , t ) = ( i , j = l C exp{ik.[p,(t)--pj(O)]})-n2(2li):’d‘(k) =({exp [-ik*ArT(t)]} i , j = l exp{ik* [ r j ( t ) - r j ( 0 ) ] } ) ) - n 2 ( 2 r ) : ’ 6 ( k ) (4.15) where ri( t ) = rT( t ) +pi( t ) is the position of particle i in a fixed reference frame and ArT( t ) = rT( t ) - rT(0) is the displacement of the tracer at time t. If we assume that the average of the product in this equation may be approximated by the product of the averages, then we have that C’(k, t ) = x T ( k , t ) C ( k , t ) - n 2 ( 2 n ) ’ 8 ( k ) where X T ( k t , (exp rik ArT( 11) is the tracer-diffusion propagator.The only change produced in eqn (4.14) when this relationship is used is the substitution of the collective propagator ~ ’ ( k , t ) referred to the tracer, by the product of the normal (i.e. referred to a fixed reference frame) collective propagator x ( k , t ) times the. tracer-diffusion propagator XT(k, t ) . We shall call the resulting expression for Al( t) the mode-mode coupling approximation, in reference to the fact that such an expression was first derived by Hess and JSlein13 via mode-mode coupling arguments, for the case in which the tracer is one of the macrofluid particles (self-diffusion).Writing A[( t ) in terms of xr( k, t ) reflects the intrinsic non-linearity of the phenomenon of tracer diffusion. In our derivations, we have disguised such non- linearity by the change of the integration variable in eqn (4.2). One hopes, however, that an additional relation between A l ( t ) and xT( k, t ) could be provided by additional, independent approximations. For example, we might suggest the Gaussian approxima- tion14 for xT( k, t ) . Alternatively, Hess and Klein have suggested another closure relation, derived from their systematic application of the mode-mode coupling method.” On the other hand, although we aimed at establishing a connection betweeen tracer diffusion properties and the hydrodynamic description of the macrofluid, the end result, namely, writing A l ( t ) in terms of G ( r , r’; t ) or x ( r , r’; t ) , does not constitute a self- contained theory of tracer diffusion.The determination of these quantities requires additional approximations. The hope, again, is that simple approximations for x ( r, r’; t ) will lead to useful results for A[( t ) . In the following section we shall consider the results of the simplest version of the mode-mode coupling approximation,” suggested by Hess and Klein and referred to as the mean-field approximation. This is defined by approximating and xT(k, t ) = exp (- k 2 ~ ; t ) x ( k, t ) = [ -k2D0t/( 1 + nh,,( k ) ] where D; = kBT/ [; and Do = k , T / 5’. The mode-mode coupling relationship between x’ and x is, however, not the only possible relationship between these quantities.Other approximations could also be suggested based on similar considerations, or on other phenomenological considerations. In recent work we have considered the approximation that results from a modification of Fick’s diffusion law in eqn (3.7) in which we approximate G’(p, p‘; t ) by G’(p, p‘; t ) = V , * D*neq(p)V,a,-,’(p, p’)2S( t ) (4.16)M. Medina-Noyola 29 with D" = D;+ Do. The rationale for this approximation is that Do in eqn (3.7) describes the free diffusion of a macrofluid particle. When this phenomenon is observed from the tracer, whose free diffusion is described by D:, the apparent free diffusion coefficient of the macrofluid particles is described by D:+ Do. In the following section we shall present specific results derived from this approximation.As a final note, let us mention that all the results in this section can be extended to the case of a polydisperse macrofluid in a rather straightforward manner, by including an additional index to denote macroparticle species. 5. Applications The results of the previous section have been applied to specific systems. Here we mention some of these applications. 5.1. Self-friction The simplest application corresponds to the case in which the tracer is identical to the other diffusing particles. Let us illustrate this situation by considering the system whose direct interactions are defined by P + ( r ) = K exp[-z(x-l)]/x, 1<x= r / o = 00, 1 > x. This model has been used 1371472*721 to represent the direct interactions between charged colloidal particles in suspension.The correlation function of this model is calculated using the mean spherical appro~imation'~ (MSA), with the understanding that the rescaling prescription of Hansen and Hayter2' is employed when it is necessary. The mode-mode coupling approximation, within the mean-field closure for xT( k, t ) , has been applied before to this model by Hess and Klein.13 Recently, an extensive comparison22 of such results with the Brownian dynamics simulations of Gaylor et aL2l indicates that the mode-mode coupling approximation provides an accurate description of the self-diffusion properties at short times even at values of the coupling constant K , and of the volume fraction 7, near the freezing transition. For long times, a similar agreement is observed only at states not so close to the freezing transition. Similar results may also be obtained by using the simple Fick's diffusion law in eqn (4.16) with neq( r ) = n, although the quantitative accuracy of the mode-mode coupling approxima- tion is in general superior.It is also interesting to consider the results of the two approximations for a system with attractive, rather than repulsive Yukawa interactions. This corresponds to a solution of non-ionic micelles, with the Yukawa tail representing, in a crude approximation, the van der Waals direct interactions, and in the idealized conditions of no hydrodynamic interactions. The results of the two approximations discussed above are now illustrated with the long-time limit of ((ArT( t)')/6?, i.e. the self-diffusion coefficient D,, which in fig.1 is plotted as a function of the volume fraction for a system with z = 5. The two isotherms considered correspond to K = -0.5 and K = -3.0. This system shows a gas-liquid phase transition, whose critical isotherm corresponds to K , = -3.32, as calculated from the MSA using the compressibility equation. The results in the figure illustrate the fact that for systems with attractive Yukawa interactions, the two approxima- tions yield quite similar results for D,, although as the critical point is approached still closer, important differences appear). Finally, let us mention that the two approximations considered above coincide in the low concentration limit. For hard spheres, their result is given by D,/ Do = 1 -$q, rather than by the exact value 1 -27.The origin of this discrepancy is the homogeneous30 - I I I 1 Generalized Langevin Equation and Tracer Difusion Fig. 1. Long-time self-diffusion coefficient D,, normalized with Do, as a function of volume fraction q for the hard-sphere plus Yukawa model with screening constant z = 5, calculated from the mode-mode coupling approximation (-) and from the modification of Fick’s diffusion equation (- - -). The upper curves correspond to the isotherm K = -0.5 and the lower curves to K = -3.0. macrofluid approximation introduced for simplicity. One can show” that if the inhomogeneous Fick’s diffusion approximation in eqn (4.16) is employed, the correct result should follow. 5.2. Electrolyte Friction As mentioned before, the results of the previous section may be extended to the case in which the macrofluid contains various species of diffusing particles.Imagine that such diffusing particles are in fact the ions of an electrolyte solution in which a charged tracer diffuses. Then, A c ( t ) describes the effects of the interactions of the tracer with the electrolyte (in the absence, however, of hydrodynamic interactions). Our first a p p l i ~ a t i o n ~ ~ of the method discussed in this paper was precisely to this problem. Such an application was based on the multi-species version of eqn (4.16). It was that when the interionic correlation functions ajj( r, r’) were approximated by their Debye- Hiickel limit, the result derived before by S ~ h u r r ~ ~ for the static electrolyte friction, i.e.the time integral of A l ( t ) , is obtained. In further we have also calculated A c ( t ) in the Debye-Huckel limit, and have also extended Schurr’s results to include the effects of a finite size of the small ions using the primitive model and the mean spherical approximation.28 So far, the theories of electrolyte friction and of self-friction have been developed without a clear connection with each other. We should mention that the results of the previous section provide a simple approach that unifies the treatment of these two effects, which are simultaneously present in the phenomenon of tracer diffusion of charged colloidal particles. 29*30 6. Conclusions In this paper we have derived a generalized Langevin equation [eqn (4.9)] for a tracer Brownian particle immersed in a macrofluid of other diffusing particles.This derivation (section 4) proceeded as a contraction of the description involving the stochastic hydrodynamic equations of the macrofluid, and it ignored hydrodynamic interactions. The brief summary of applications in section 5 illustrates how the results thus derivedM. Medina-Noyola 31 may be explicitly applied, within reasonable approximations, to the description of tracer-diff usion in important specific systems. Our derivation in section 4 made use of other formal results developed before in sections 2 and 3. The theorems in section 2 provided the formal basis for the process of contraction. In section 3 it was indicated that the stochastic hydrodynamic equations may be cast as a stationary non-Markov process satisfying a generalized Langevin equation.These results, together with the equation of motion for the tracer in eqn (4.1), already suggest the main ingredients for an extension of the results presented here to include hydrodynamic interactions. It is our intention to pursue this work in such direction in the near future. The collaboration of the author with A. Vizcarra-Rendon and J. L. Del Rio-Correa was essential to the early development of this work, and is gratefully acknowledged. It is also a pleasure to thank Prof. R. Klein and the members of his group for many fruitful discussions on the dynamics of Brownian systems, and for their kind hospitality at Universitat Konstanz, where this paper was written. 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