On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures

TL;DR

This paper rigorously analyzes the convergence of a two-temperature Langevin dynamics to a non-equilibrium steady state, providing explicit convergence rates and exploring conditions under which these rates are sharp, with applications to complex models like spin glasses.

Contribution

It offers a rigorous analysis of convergence rates for Langevin dynamics with multiple temperatures, including explicit estimates and applications to spin-glass models.

Findings

Convergence towards the non-equilibrium steady state is exponential under certain conditions.

Logarithmic Sobolev inequalities are key to estimating convergence rates.

The estimates are sharp in the quadratic potential case.

Abstract

We study the solution of the two-temperatures Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular we consider a spin-glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra's hierarchical construction appearing in Talagrand's proof of the Parisi formula.