Trend to equilibrium for granular media equations under non-convex potential and application to log gases

TL;DR

This paper develops new inequalities for granular media equations with non-convex potentials, demonstrating convergence and stability of log gases under various non-convex external potentials.

Contribution

It introduces novel HWI inequalities applicable to non-convex potentials and applies them to establish convergence rates and stability results for log gases.

Findings

Log gases converge to equilibrium at a square root rate in Wasserstein distance.

Exponential stability of log gases under double-well and non-confining potentials.

Handling of singular logarithmic interaction potential W.

Abstract

We derive new HWI inequalities for the granular media equation, which external potential $V$ and interaction potential $W$ are only strictly convex on complementary parts of the space. Particularly, potentials are not assumed convex. After solving technicalities related to the singularity of a logarithmic $W$, we apply our result to obtain stability rates of log gases under non-strictly convex or quartic external potentials. We prove that the distribution of a log gas converges towards an equilibrium with respect to the Wasserstein distance at a square root rate. Finally, we establish exponential stability of log gases under the double-well potential $V(x) = \frac{x^4}{4} + c\frac{x^2}{2}$, $c < 0$ and the non-confining potential $V(x) = g\frac{x^4}{4} + \frac{x^2}{2}$, $g<0$ for $|c|$ and $|g|$ small enough.