Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. V. The Short Time Approximation for the Memory Function

Ranganathan, Madhav; Andersen, Hans Christian

doi:10.1021/jp0523017

Cited by 5 publications

(16 citation statements)

References 15 publications

(33 reference statements)

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“…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:…”

Section: B Diagrammatic Theory Of Time Correlation Functionsmentioning

confidence: 99%

“…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.…”

Section: Appendix E: the Calculation Of Correlation Functionsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Pilkiewicz

Andersen

2014

The Journal of Chemical Physics

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Articles you may be interested inA diagrammatic kinetic theory of density fluctuations in simple liquids in the overdamped limit. II. The one-loop approximation 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 A diagrammatic formulation of the kinetic theory of fluctuations in equilibrium classical fluids. IV. The short time behavior of the memory functionStarting with a formally exact diagrammatic kinetic theory for the equilibrium correlation functions of particle density and current fluctuations for a monatomic liquid, we develop a theory for high density liquids whose interatomic potential is continuous and has a strongly repulsive short ranged part. We assume that interparticle collisions via this short ranged part of the potential are sufficient to randomize the velocities of the particles on a very small time scale compared with the fundamental time scale defined as the particle diameter divided by the mean thermal velocity. When this is the case, the graphical theory suggests that both the particle current correlation functions and the memory function of the particle density correlation function evolve on two distinct time scales, the very short time scale just mentioned and another that is much longer than the fundamental time scale. The diagrams that describe the motion on each of these time scales are identified. When the two time scales are very different, a dramatic simplification of the diagrammatic theory at long times takes place. We identify an irreducible memory function and a more basic function, which we call the irreducible memory kernel. This latter function evolves on the longer time scale only and determines the time dependence of the density and current correlation functions of interest at long times. In Paper II, a simple one-loop approximation for the irreducible memory kernel is used to calculate correlation functions for a Lennard-Jones fluid at high density and a variety of temperatures. © 2014 AIP Publishing LLC. [http://dx.

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Section: B Diagrammatic Theory Of Time Correlation Functionsmentioning

confidence: 99%

Section: Appendix E: the Calculation Of Correlation Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Pilkiewicz

Andersen

2014

The Journal of Chemical Physics

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“…43 In this method, the equations for successively larger finite sets of coupled equations are solved. We solve this infinite dimensional equation numerically using the method of Ranganathan and Andersen.…”

Section: B Solution Of the Kinetic Equationsmentioning

confidence: 99%

“…43 The accuracy of the theory is different for different correlation functions but is surprisingly good at high temperatures for the viscosity and diffusion constant. 43 The accuracy of the theory is different for different correlation functions but is surprisingly good at high temperatures for the viscosity and diffusion constant.…”

Section: Introductionmentioning

confidence: 99%

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

Noah-Vanhoucke

Andersen

2007

The Journal of Chemical Physics

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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).

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The second entropy: a general theory for non-equilibrium thermodynamics and statistical mechanics

Attard

2009

Annu. Rep. Prog. Chem., Sect. C: Phys. Chem.

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The second entropy is introduced, which is a new type of entropy that provides a basis for the non-equilibrium thermodynamics of time-dependent systems. Whereas the first or ordinary entropy counts the molecular configurations associated with a given structure, the second entropy counts the molecular configurations associated with a transition between two given structures in a specified time. Maximization of the second entropy gives the optimum rate of change or flux, and as such it provides a quantitative principle for non-equilibrium systems. In contrast, the second law of thermodynamics only provides a direction for change, not a rate of change. The probability distribution function for time-dependent systems is also given, which is the focal point of non-equilibrium statistical mechanics. This and the second entropy are used, for example, to derive the Langevin equation, the Green-Kubo relations, the transition and path probability, the fluctuation and work theorems, and a generalised fluctuation-dissipation theorem. They are also used to develop computer simulation algorithms suited for time-dependent systems, specifically non-equilibrium Monte Carlo and stochastic Molecular Dynamics. The analysis is illustrated and quantitatively tested for the case of steady heat flow, and for the case of time-varying, driven, Brownian motion.

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Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. V. The Short Time Approximation for the Memory Function

Cited by 5 publications

References 15 publications

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

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

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

The second entropy: a general theory for non-equilibrium thermodynamics and statistical mechanics

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