Space-time statistical solutions for an inhomogeneous chain of harmonic oscillators

TL;DR

This paper investigates the long-term statistical behavior of an inhomogeneous harmonic oscillator chain, showing convergence to Gaussian measures under certain conditions, thus advancing understanding of inhomogeneous systems in statistical mechanics.

Contribution

It introduces a framework for analyzing space-time statistical solutions in inhomogeneous harmonic chains and proves their convergence to Gaussian measures.

Findings

Convergence of statistical solutions to Gaussian measures.

Conditions under which the convergence holds.

Extension of statistical mechanics to inhomogeneous systems.

Abstract

We consider an one-dimensional inhomogeneous harmonic chain consisting of two different semi-infinite chains of harmonic oscillators. We study the Cauchy problem with random initial data. Under some restrictions on the interaction between the oscillators of the chain and on the distribution of the initial data, we prove the convergence of space-time statistical solutions to a Gaussian measure.