Chaotic distributions for relativistic particles

TL;DR

This paper investigates a modified kinetic model for particles with arbitrary energy functions, demonstrating that the uniform distribution is chaotic and extending the analysis to relativistic particles and higher dimensions.

Contribution

It introduces a generalized Kac model with arbitrary energy functions and proves chaos for the uniform distribution, including relativistic cases and higher-dimensional extensions.

Findings

Uniform density with respect to the microcanonical measure is chaotic.

Relativistic kinetic energy is a special case within this framework.

The model can be extended to higher dimensions with momentum conservation.

Abstract

We study a modified Kac model where the classical kinetic energy is replaced by an arbitrary energy function $\phi(v)$, $v \in \mathbb{R}$. The aim of this paper is to show that the uniform density with respect to the microcanonical measure is $Ce^{-z_0\phi(v)}$-chaotic, $C,z_0 \in \mathbb{R}_+$. The kinetic energy for relativistic particles is a special case. A generalization to the case $v\in \mathbb{R}^d$ which involves conservation momentum is also formally discussed.