Virial expansion of molecular Brownian motion versus tales of "statistical independency"

TL;DR

This paper derives an exact virial expansion for molecular Brownian motion, revealing that real paths exhibit long-range correlations and power-law tails, challenging the traditional assumption of statistical independence in molecular chaos.

Contribution

It introduces a virial expansion based solely on statistical mechanics principles, linking path distribution responses to fluid perturbations and molecular correlations, and challenges the Gaussian assumption.

Findings

Correlations have finite spatial extent.

Path distribution exhibits power-law long tails.

Brownian diffusivity fluctuates scalelessly over low frequencies.

Abstract

Basing on main principles of statistical mechanics only, an exact virial expansion for path probability distribution of molecular Brownian particle in a fluid is derived which connects response of the distribution to perturbations of the fluid and statistical correlations of its molecules with Brownian particle. The expansion implies that (i) spatial spread of these correlations is finite, (ii) this is inconsistent with Gaussian distribution involved by the ``molecular chaos'' hypothesis, and (iii) real path distribution possesses power-law long tails. This means that actual Brownian path never can be disjointed into statistically independent fragments, even in the Boltzmann-Grad gas, but behaves as if Brownian particle's diffusivity undergoes scaleless low-frequency fluctuations.