On the behaviour of a periodically forced and thermostatted harmonic chain

TL;DR

This paper analyzes a harmonic chain with periodic forcing and thermostats, showing long-term Gaussian behavior with force-independent covariance and explicit calculations of work and energy for all chain sizes.

Contribution

It provides an explicit analysis of the long-term statistical behavior of a thermostatted harmonic chain under periodic forcing, including formulas for work and energy for any chain size.

Findings

System reaches a Gaussian measure with force-independent covariance.

Means of the system are periodic in the long-time limit.

Explicit formulas for work and energy for all chain sizes, including infinite chain.

Abstract

We consider a chain consisting of $n+1$ pinned harmonic oscillators subjected on the right to a time dependent periodic force $\cF(t)$ while Langevin thermostats are attached at both endpoints of the chain. We show that for long times the system is described by a Gaussian measure whose covariance function is independent of the force, while the means are periodic. We compute explicitly the work and energy due to the periodic force for all $n$ including $n\to\infty$.