Decay of Information for the Kac Evolution

TL;DR

This paper analyzes how information and entropy decay exponentially in a particle system interacting with a heat reservoir, using Kac's Master Equation, and simplifies previous proofs of related results.

Contribution

It demonstrates exponential decay of entropy and information in a particle-reservoir system, simplifying the proof of a key result in kinetic theory.

Findings

Entropy decays exponentially to a small value

Information decay follows a similar exponential pattern

Simplifies the proof of a main result in prior work

Abstract

We consider a system of $M$ particles in contact with a heat reservoir of $N\gg M$ particles. The evolution in the system and the reservoir, together with their interaction, are modeled via the Kac's Master Equation. We chose the initial distribution with total energy $N+M$ and show that if the reservoir is initially in equilibrium, that is if the initial distribution depends only on the energy of the particle in the reservoir, then the entropy of the system decay exponentially to a very small value. We base our proof on a similar property for the Information. A similar argument allows us to greatly simplify the proof of the main result in [2].