Chaos and Entropic Chaos in Kac's Model Without High Moments

TL;DR

This paper introduces a new local Lévy CLT for convergence to non-Gaussian stable states, and explores entropic chaos with moments of order 2α, providing insights into stability and entropy behavior.

Contribution

It presents a novel local Lévy CLT applicable to non-Gaussian stable states and analyzes entropic chaos with specific moment conditions.

Findings

Established convergence to non-Gaussian stable states.

Identified entropic chaotic families with moments of order 2α.

Proved lower semi-continuity of relative entropy in this context.

Abstract

In this paper we present a new local L\'evy Central Limit Theorem, showing convergence to stable states that are not necessarily the Gaussian, and use it to find new and intuitive entropically chaotic families with underlying one-particle function that has moments of order $2\alpha$, with $1<\alpha<2$. We also discuss a lower semi continuity result for the relative entropy with respect to our specific family of functions, and use it to show a form of stability property for entropic chaos in our settings.