Cumulants of the Rayleigh gas mixture model: statistical results

TL;DR

This paper analyzes the statistical properties of the nonideal Rayleigh gas model, focusing on cumulants, fluctuations, and large deviations, and refines previous results with new convergence rate insights.

Contribution

It introduces a detailed cumulant analysis of the Rayleigh gas, providing refined asymptotic behavior and convergence rates for fluctuations and large deviations.

Findings

Cumulants exhibit trivial limit behavior in overdilute regimes.

Fluctuations converge to the linear Rayleigh-Boltzmann equation.

Established a full convergence rate for cumulants.

Abstract

In this paper, we explore the statistical subtleties of the nonideal Rayleigh gas, in a grand canonical mixture framework. This model allows to consider a large amount of tagged particles close to equilibrium, and their empirical measure, whose first-order convergence has been shown to converge to the solution of the linear Rayleigh-Boltzmann equation [5]. Thanks to the study of the cumulants of the system, we analyze the asymptotic behaviour of the fluctuations and large deviations of this empirical measure, hence refining the previous statistical results in the same vein as [7]. This way, we exhibit the trivial limit behaviour of the fluctuations in any overdilute regime, proving the exact relevance at any statistical scale of the low density limit. In the case of large deviations, we present the linear Boltzmann-Hamilton-Jacobi system driving their asymptotic behaviour. Eventually, we optimize the geometrical estimates on the billiards dynamics [6] to finally achieve a full convergence rate for the cumulants.