Long-time correlations for a hard-sphere gas at equilibrium

TL;DR

This paper proves that the fluctuations of a hard-sphere gas at equilibrium are governed by the linearized Boltzmann equation over all times using a new duality and pruning method, extending previous short-time results.

Contribution

It introduces a robust duality and pruning approach to establish the long-time behavior of fluctuations in a hard-sphere gas at equilibrium.

Findings

Covariance of fluctuations follows the linearized Boltzmann equation globally in time

Method applies to diffusive regimes and is simpler than previous approaches

Extends short-time Boltzmann limit results to long-time dynamics

Abstract

It has been known since Lanford [19] that the dynamics of a hard sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a duality method coupled with a pruning argument to prove that the covariance of the fluctuations around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simple than the one devised in [4] which was specific to the 2D case.