Infinite Chain of Harmonic Oscillators Under the Action of the Stationary Stochastic Force

TL;DR

This paper analyzes an infinite chain of harmonic oscillators influenced by a stationary stochastic force, deriving explicit representations of deviations and energy limits under specific conditions.

Contribution

It provides a novel explicit representation of particle deviations and energy limits for an infinite oscillator chain with stochastic forcing.

Findings

Deviations of particles are sums of stationary processes and vanishing terms.

The time limit of the mean energy of the system is explicitly determined.

Representation holds under positive definite potential and initial conditions in l2 space.

Abstract

We consider countable system of harmonic oscillators on the real line with quadratic interaction potential with finite support and local external force (stationary stochastic process) acting only on one fixed particle. In the case of positive definite potential and initial conditions lying in $l_2(\mathbb{Z})$-space the perpesentation of the deviations of the particles from their equilibrium points are found. Precisely, deviation of each particle could be represented as the sum of some stationary process (it is also time limiting process in distribution for that function) and the process which converges to zero as $t\rightarrow+\infty$ with probability one. The time limit for the mean energy of the whole system is found as well.