A gradient flow approach to the Boltzmann equation

TL;DR

This paper demonstrates that the spatially homogeneous Boltzmann equation can be viewed as a gradient flow of entropy, providing a new proof of convergence from Kac's random walk by leveraging the stability of gradient flows.

Contribution

It introduces a novel gradient flow framework for the Boltzmann equation that incorporates collision dynamics, offering new insights and proof techniques.

Findings

The Boltzmann equation is shown to be a gradient flow of entropy.

A new proof of convergence from Kac's random walk is provided.

The stability of the gradient flow structure is exploited for convergence analysis.

Abstract

We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which takes the collision process into account. This gradient flow structure allows to give a new proof for the convergence of Kac's random walk to the homogeneous Boltzmann equation, exploiting the stability of gradient flows.