# Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

## TL;DR

This paper introduces a notion of entropic curvature for mean-field dynamics on discrete spaces, linking curvature bounds to convergence rates and establishing explicit bounds for classical models.

## Contribution

It extends the concept of curvature bounds from linear Markov chains to non-linear mean-field dynamics, providing new tools for analyzing convergence to equilibrium.

## Key findings

- Positive curvature bounds imply functional inequalities for convergence.
- Explicit curvature bounds are derived for classical statistical mechanics models.
- The framework generalizes existing curvature notions to non-linear mean-field systems.

## Abstract

We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on convexity of the free energy along interpolations in a discrete transportation distance related to the gradient flow structure of the dynamics. This notion extends the one for linear Markov chain dynamics studied by Erbar and Maas. We show that positive curvature bounds entail several functional inequalities controlling the convergence to equilibrium of the dynamics. We establish explicit curvature bounds for several examples of mean-field limits of various classical models from statistical mechanics.