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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

### Abstract

Starting 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

Received 02 December 2013
Accepted 28 March 2014
Published online 21 April 2014

Acknowledgments:
This work was supported by the National Science Foundation through Grant No. CHE-0716047.

Article outline:

I. INTRODUCTION
II. CORRELATION FUNCTIONS AND DIAGRAMS
A. Definitions
B. Diagrammatic theory of time correlation functions
III. REPRESENTATION OF SHORT RANGED REPULSIVE FORCES
IV. HERMITE POLYNOMIAL REPRESENTATION
V. GRAPHICAL KINETIC THEORY IN THE HERMITE REPRESENTATION
A. The propagator
1. Relationship of the propagator to observables
2. Another expression for the propagator
3. The relationship between the propagator and the coherent intermediate scattering function
B. The projected propagator
C. The propagators associated with the longitudinal and transverse current correlation functions
D. Summary of this section
VI. TOPOLOGICAL ANALYSIS OF THE SERIES FOR THE PROJECTED PROPAGATOR
A. The generalized Enskog projected propagator
B. Topological reduction of the series for χ_{ P }
VII. THE OVERDAMPED LIMIT OF THE PROJECTED PROPAGATOR
A. *M* ^{ H }, , and the hard sphere collision frequency ν
B. Properties of
C. The evaluation of diagrams for large ν
1. Time integrations in the evaluation of a diagram
2. Objects and subgraphs
3. Quasi-simultaneous objects and subgraphs
4. The *t* and ν dependence of a diagram for large ν
D. The topological structure of diagrams to be retained for large ν
1. -connectivity
2. A theorem about maximal qs subgraphs
3. Properties of overdamped subgraphs
4. Characterization of diagrams to be retained for large ν
E. Overdamped theory and overdamped limit
F. Analysis of the overdamped χ_{ P } and the irreducible memory kernel
G. Comment
VIII. DISCUSSION

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2014-04-21

2016-09-30

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