Heat Flow in a Periodically Forced, Thermostatted Chain II

We derive a macroscopic heat equation for the temperature of a pinned harmonic chain subject to a periodic force at its right side and in contact with a heat bath at its left side. The microscopic dynamics in the bulk is given by the Hamiltonian equation of motion plus a reversal of the velocity of a particle occurring independently for each particle at exponential times, with rate ? . The latter produces a finite heat conductivity. Starting with an initial probability distribution for a chain of n particles we compute the current and the local temperature given by the expected value of the local energy. Scaling space and time diffusively yields, in the n ? +8 limit, the heat equation for the macroscopic temperature profile T (t, u), t > 0, u ? [0, 1]. It is to be solved for initial conditions T (0, u) and specified T(t, 0) = T-, the temperature of the left heat reservoir and a fixed heat flux J, entering the system at u = 1. |J | equals the work done by the periodic force which is computed explicitly for each n.