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.

Automated summary

This article investigates the macroscopic heat equation for a pinned harmonic chain under periodic force and in contact with a heat bath. The microscopic dynamics involve a Hamiltonian equation of motion and a velocity reversal for each particle at exponential times, resulting in a finite heat conductivity. By computing the current and local temperature, we derive the heat equation for the macroscopic temperature profile and solve it with specified initial conditions and fixed heat flux.