Molecular random walks in a fluid and an invariance group of the Bogolyubov generating functional equation

TL;DR

This paper develops an exact theoretical framework for analyzing molecular random walks in fluids using the Bogolyubov generating functional, revealing invariance properties that impose constraints on the path distribution of a Brownian particle.

Contribution

It introduces a new set of correlation functions and derives an invariance group of the generating functional, leading to exact relations in molecular random walk statistics.

Findings

Invariance group leads to infinitely many exact relations.

Path probability distribution is significantly restricted by these relations.

Framework applies to both dynamic and equilibrium correlations.

Abstract

The problem of statistics of molecular random walks in a classical fluid is analyzed by means of the BBGKY hierarchy of equations reformulated in terms of the Bogolyubov evolution equation for generating functional of many-particle distribution functions. A proper equivalent set of correlation functions is introduced so that all they are integrable, vanish in statistical equilibrium, otherwise accumulate statistical information about history of collisions of a ``molecular Brownian particle'' (test molecule) with other molecules of the fluid. An exact evolution equation for generating functional of such correlation functions is derived. Then it is shown that time-dependent solution to this equation, as well as a properly defined generating functional of static thermodynamically equilibrium correlations, possesses invariance with respect to a definite group of transformations of independent variables of the functional, if density of the fluid (number of molecules per unit volume) is treated as one of the independent variables. Such invariance results in infinitely many exact relations between the correlation functions and probability distribution of path of the molecular Brownian particle. Even simplest of these relations suggest significant restrictions on a profile of the path probability distribution, even without literal solving the BBGKY hierarchy.