Heat flow in a periodically forced, thermostatted chain II

TL;DR

This paper derives a macroscopic heat equation for a thermostatted harmonic chain under periodic forcing, connecting microscopic dynamics with macroscopic heat conduction and explicitly computing the heat flux.

Contribution

It introduces a new derivation of the heat equation for a periodically forced, thermostatted harmonic chain with explicit calculation of heat flux.

Findings

Finite heat conductivity established.

Macroscopic temperature profile converges to a heat equation.

Explicit expression for heat flux under periodic forcing.

Abstract

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 $\gamma$. The latter produces a finite heat conductivity. Starting with an initial probability distribution for a chain of $n$ particles we compute the local temperature given by the expected value of the local energy and current. Scaling space and time diffusively yields, in the $n\to+\infty$ limit, the heat equation for the macroscopic temperature profile $T(t,u),$ $t>0$, $u \in [0,1]$. 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 $u=1$. $J$ is the work done by the periodic force which is computed explicitly for each $n$.