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 $n$.
