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 $$\gamma $$γ . 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 $$n\rightarrow +\infty $$n → + ∞ limit, the heat equation for the macroscopic temperature profile T(t, u), $$t>0$$t > 0 , $$u \in [0,1]$$u ∈ [ 0 , 1 ] . It is to be solved for initial conditions T(0, u) and specified $$T(t,0)=T_-$$T ( t , 0 ) = T - , the temperature of the left heat reservoir and a fixed heat flux J, entering the system at $$u=1$$u = 1 . |J| equals the work done by the periodic force which is computed explicitly for each n.