Energy Growth of Infinite Harmonic Chain under Microscopic Random Influence

TL;DR

This paper investigates the long-term energy growth in an infinite harmonic chain with a single particle influenced by white noise, revealing linear macroscopic growth, microscopic logarithmic growth, and equipartition properties.

Contribution

It provides a rigorous analysis of energy growth rates and equipartition in an infinite harmonic chain with microscopic stochastic influence, under general interaction conditions.

Findings

Energy grows linearly with time macroscopically.

Energy grows as logarithm of time microscopically.

System exhibits equipartition properties in non-equilibrium.

Abstract

Infinite harmonic chains of point particles with finite range translation invariant interaction have considered. It is assumed that the only one particle influenced by the white noise. We studied microscopic and macroscopic behavior of the system's energies (potential, kinetic, total) when time goes to infinity. We proved that under quite general condition on interaction potential the energies grow linearly with time on macroscopic scale, and grow as $\ln(t)$ on microscopic scale. Moreover it is turned out that the system exhibit some equipartition properties in this non equilibrium settings.