Hard-sphere Brownian motion in ideal gas: inter-particle correlations, Boltzmann-Grad limit, and destroying the myth of molecular chaos propagation

TL;DR

This paper critically examines the BBGKY hierarchy for a particle in an ideal gas, demonstrating that the Boltzmann-Grad limit does not produce the classical Boltzmann hierarchy or molecular chaos, due to persistent correlations.

Contribution

It introduces a new perspective on the BBGKY hierarchy and shows that correlations remain significant in the Boltzmann-Grad limit, challenging traditional assumptions.

Findings

Correlations of all orders remain significant in the Boltzmann-Grad limit.

The Boltzmann hierarchy and molecular chaos are not valid in this limit.

Exact relations connect correlations with the particle's probability distribution.

Abstract

The BBGKY hierarchy of equations for a particle interacting with ideal gas is analyzed in terms of irreducible many-particle correlations between gas atoms and the particle's motion. The transition to the hard-sphere interaction is formulated from viewpoint of the recently discovered exact relations connecting the correlations with the particle's probability distribution. Then the Boltzmann-Grad limit is considered and shown not to lead to the Bolzmann hierarchy and the molecular chaos, since correlations of all orders keep significant.in this limit, merely taking a singular form.