On convergence to equilibrium for one-dimensional chain of harmonic oscillators in the half-line

TL;DR

This paper investigates the long-term behavior of an infinite one-dimensional harmonic oscillator chain on the half-line with random initial data, proving convergence to a Gaussian measure and identifying stationary states with energy currents.

Contribution

It establishes the convergence of the distribution of solutions to a Gaussian measure and characterizes stationary states with non-zero energy flow, advancing understanding of such systems' asymptotics.

Findings

Distribution $mbda_t$ converges to Gaussian as $t o \u221e$

Stationary states with non-zero energy current identified

Long-term behavior characterized for random initial data

Abstract

The initial-boundary value problem for an infinite one-dimensional chain of harmonic oscillators on the half-line is considered. The large time asymptotic behavior of solutions is studied. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution $\mu_t$ of the random solution at time moments $t\in\mathbb{R}$. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$. We find stationary states in which there is a non-zero energy current at origin.