On the Behaviour of a Periodically Forced and Thermostatted Harmonic Chain

We consider a chain consisting of \(n+1\) $n+1$ harmonic oscillators subjected on the right to a time dependent periodic force \({{\mathcal {F}}}(t)\) $\mathcal{F}(t)$ 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 \(n\rightarrow \infty \) $n\to \infty$ .