Molecular Brownian motion and invariance group of the Bogolyubov equation

TL;DR

This paper investigates the statistical properties of molecular Brownian motion using the Bogolyubov equation, revealing invariance groups that impose strict constraints on path distributions and exclude Gaussian asymptotics.

Contribution

It identifies an invariance group of the Bogolyubov equation, deriving exact relations and showing Gaussian long-range behavior is impossible for molecular paths.

Findings

Gaussian asymptotic is forbidden for molecular paths

Allowed asymptotic has power-law long tail

Invariance group leads to exact relations between path probabilities

Abstract

Statistics of molecular random walks in a fluid is considered with the help of Bogolyubov equation for generating functional of distribution functions. An invariance group of this equation is found. It results in many exact relations between path probability distribution of a test particle and its correlations with the fluid. As the consequence, significant restrictions on possible shape of the path distribution do arise. In particular, the hypothetical Gaussian form of long-range asymptotic proves to be forbidden, even (and first of all) under the Boltzmann-Grad limit. An allowed diffusive asymptotic possesses power-law long tail (cut off by free flight length).