A diagrammatic formulation of the kinetic theory of fluctuations in equilibrium classical fluids. VI. Binary collision approximations for the memory function for self-correlation functions

Abstract

We use computer simulation results for a dense Lennard-Jones fluid for a range of temperatures to test the accuracy of various binary collision approximations for the memory function for density fluctuations in liquids. The approximations tested include the moderate density approximation of the generalized Boltzmann-Enskog memory function (MGBE) of Mazenko and Yip [Statistical Mechanics. Part B. Time-Dependent Processes, edited by B. J. Berne (Plenum, New York, 1977)], the binary collision approximation (BCA) and the short time approximation (STA) of Ranganathan and Andersen [J. Chem. Phys. 121, 1243 (2004); J. Phys. Chem. 109, 21437 (2005)] and various other approximations we derived by using diagrammatic methods. The tests are of two types. The first is a comparison of the correlation functions predicted by each approximate memory function with the simulation results, especially for the self-longitudinal current correlation (SLCC) function. The second is a direct comparison of each approximate memory function with a memory function numerically extracted from the correlation function data. The MGBE memory function is accurate at short times but decays to zero too slowly and gives a poor description of the correlation function at intermediate times. The BCA is exact at zero time, but it predicts a correlation function that diverges at long times. The STA gives a reasonable description of the SLCC but does not predict the correct temperature dependence of the negative dip in the function that is associated with caging at low temperatures. None of the other binary collision approximations is a systematic improvement on the STA. The extracted memory functions have a rapidly decaying short time part, much like the STA, and a much smaller, more slowly decaying part of the type predicted by a mode coupling theory. Theories that use mode coupling commonly include a binary collision term in the memory function but do not discuss in detail the nature of that term. It is clear from the present work that the short time part of the memory function has a behavior associated with brief binary repulsive collisions, such as those described by the STA. Collisions that include attractive as well as repulsive interactions, such as those of the MGBE, have a much longer duration, and theories that include them have memory functions that decay to zero much too slowly to provide a good first approximation of the correlation function. This leads us to speculate that the memory function for density fluctuations can be usefully regarded as a sum of at least three parts: a contribution from repulsive binary collisions (the STA or something similar to it), another short time part that is related to all the other interactions (but whose nature is not understood), and a longer time slowly decaying part that describes caging (of the type predicted by the mode coupling theory).

A diagrammatic kinetic theory of density fluctuations in simple liquids in the overdamped limit. I. A long time scale theory for high density

“…The method for calculating this matrix element is discussed by Noah-Vanhoucke and Andersen. 24 These calculations do not involve dynamical simulations of the liquid, merely trajectory calculations for collisions of two particles.) Choose the effective hard sphere diameter d associated with the repulsive part of the potential of mean force of the fluid of interest so that, at the density of the fluid of interest, the value of M H s (0, 0) ẑ ẑ of the hard sphere fluid is equal to M R s (0, 0) ẑ ẑ for the fluid of interest.…”

“…In previous papers, [19][20][21][22][23][24] we have developed a diagrammatic theory of a hierarchy of time correlation functions. The theory defines retarded propagators χ and χ s for the correlation functions C and C s , respectively, 29 and expresses the C function for positive values of t − t in terms of the t = t value in the following way:…”

“…41 The matrix elements M H (k) λλ and M H s (k) λλ in the present theory correspond closely to (c) (α|β) and (s) (α|β) of Furtado et al 38 48 The fact that all the graphical series have these properties was not discussed explicitly in Andersen 21 and these properties were not needed for the subsequent development by Andersen and co-workers. [22][23][24] (In fact, diagrams that violate these conditions have zero value, so even if they are included in theoretical developments this introduces no error.) However, for the present analysis it is worthwhile to acknowledge and utilize these conditions explicitly.…”

“…The diagrammatic theory had the advantage of being able to represent both correlation functions and their memory functions in a unified way in terms of diagrams that could be ascribed straightforward physical interpretations thanks to the individual elements of the diagrams being relatively simple (even though the full diagrams in general were not). Andersen and co-workers [22][23][24] then used the physical intuition this theory afforded to derive several candidate graphical representations of the short time part of the memory function of the phase space density correlation function that could be evaluated numerically and used to compute the short time behavior of several correlation functions of interest. Each of these representations in turn implied an explicit diagrammatic series for the corresponding long time part of the memory function, but these series were always too complex to evaluate.…”

Temperature relaxation in binary hard-sphere mixture system: Molecular dynamics and kinetic theory study

“…In addition, a similar collision term including the product of effective and bare potentials was successfully applied to the theoretical description of shear viscosity in the dense one-component plasma [38][39][40][41] as well. It is known that the squared form of the effective potential in the collision integral does not satisfy the correct shorttime or high-frequency limit 35,37,[42][43][44] because the system should see the bare interaction in this limiting regime. This is a cost the present kinetic theory should pay instead of obtaining the merits of the description in the wavenumber space and the positive-definite collision integral.…”