Modulated logarithmic Sobolev inequalities and generation of chaos

TL;DR

This paper links the generation of chaos in particle systems to modulated logarithmic Sobolev inequalities, providing a new approach to prove exponential convergence to equilibrium in mean-field dynamics.

Contribution

It introduces the concept of modulated logarithmic Sobolev inequalities and demonstrates their role in establishing chaos generation for particle systems.

Findings

Uniform modulated logarithmic Sobolev inequalities can be established in one-dimensional cases.

The method applies to systems with singular interactions like Riesz kernels.

Generation of chaos can be proved via these inequalities under certain conditions.

Abstract

We consider mean-field limits for overdamped Langevin dynamics of $N$ particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential-in-time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-$N$ "modulated logarithmic Sobolev inequality" holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension.