On the behaviour of a periodically forced and thermostatted harmonic chain

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