Modified logarithmic Sobolev inequalities for canonical ensembles

TL;DR

This paper establishes modified logarithmic Sobolev inequalities for certain statistical physics models, enabling analysis of convergence properties in Kawasaki dynamics within the Ginzburg-Landau framework.

Contribution

It extends the iterated two-scale approach to prove these inequalities for superquadratic potentials, linking them to concentration and transport-entropy inequalities.

Findings

Proved modified logarithmic Sobolev inequalities for canonical ensembles.

Derived convergence results in Wasserstein distance for Kawasaki dynamics.

Connected inequalities to concentration of measure and transport-entropy concepts.

Abstract

In this paper, we prove modified logarithmic Sobolev inequalities for canonical ensembles with superquadratic single-site potential. These inequalities were introduced by Bobkov and Ledoux, and are closely related to concentration of measure and transport-entropy inequalities. Our method is an adaptation of the iterated two-scale approach that was developed by Menz and Otto to prove the usual logarithmic Sobolev inequality in this context. As a consequence, we obtain convergence in Wasserstein distance $W_p$ for Kawasaki dynamics on the Ginzburg-Landau model.