Quantitative Rates of Convergence to Non-Equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

TL;DR

This paper analyzes the convergence rates to non-equilibrium steady states in a weakly anharmonic oscillator chain, demonstrating polynomial decay of the spectral gap with system size using advanced probabilistic and analytical techniques.

Contribution

It adapts Bakry-Emery theory to hypoelliptic generators in oscillator chains, providing explicit convergence rates and bounds on the spectral gap decay.

Findings

Exponential convergence to NESS in Wasserstein and entropy metrics.

Spectral gap decays at most polynomially with system size, order > N^{-3}.

Derived explicit constants and rates via Lyapunov matrix equations.

Abstract

We study a $1$-dimensional chain of $N$ weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with $N$ dependent bounds) perturbations of the harmonic ones. We show how a generalized version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by F. Baudoin (2017). By that we prove exponential convergence to non-equilibrium steady state (NESS) in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $N^{-3}$. For the purely harmonic chain the order of the rate is in $ [N^{-3},N^{-1}]$. This shows that, in this set up, the spectral gap decays at most polynomially with $N$.