Molecular random walks and invariance group of the Bogolyubov equation

TL;DR

This paper investigates molecular random walks in fluids using the Bogolyubov equation, discovering invariance properties that impose strict constraints on path distributions, ruling out Gaussian asymptotics and supporting power-law tails.

Contribution

It uncovers an invariance group of solutions to the Bogolyubov equation, deriving exact relations that restrict the shape of the path distribution in molecular random walks.

Findings

Gaussian long-range asymptotics are forbidden

Diffusive asymptotics with power-law tails are supported

Exact relations between path distribution and fluid correlations are established

Abstract

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