On two-temperature problem for harmonic crystals

TL;DR

This paper studies the long-term behavior of a harmonic crystal with random initial data, showing convergence to a Gaussian measure and analyzing the energy flow between regions at different temperatures.

Contribution

It establishes the convergence of the distribution of the harmonic crystal's solution to a Gaussian measure and analyzes the energy current for different temperature regions.

Findings

Convergence of the solution's distribution to a Gaussian measure as time goes to infinity.

Explicit expression for the limiting mean energy current density.

Application to Gibbs measures with different temperatures demonstrating the Second Law.

Abstract

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n \ge 1$. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_d\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution $\mu_t$ of the solution at time $t\in\R$. The main result is the convergence of $\mu_t$ to a Gaussian translation-invariant measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green function and on Bernstein's `room-corridor' argument. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density is $- (0,...,0,C(T_+ - T_-))$ with some positive constant $C>0$ what corresponds to Second Law.