Non-equilibrium dynamics of the piston in the Szilard engine

We consider a Szilard engine in one dimension, consisting of a single particle of mass $m$, moving between a piston of mass $M$, and a heat reservoir at temperature $T$. In addition to an external force, the piston experiences repeated elastic collisions with the particle. We find that the motion of a heavy piston ($M \gg m$), can be described effectively by a Langevin equation. Various numerical evidences suggest that the frictional coefficient in the Langevin equation is given by $\gamma = (1/X)\sqrt{8 \pi m k_BT}$, where $X$ is the position of the piston measured from the thermal wall. Starting from the exact master equation for the full system and using a perturbation expansion in $\epsilon= \sqrt{m/M}$, we integrate out the degrees of freedom of the particle to obtain the effective Fokker-Planck equation for the piston albeit with a different frictional coefficient. Our microscopic study shows that the piston is never in equilibrium during the expansion step, contrary to the assumption made in the usual Szilard engine analysis --- nevertheless the conclusions of Szilard remain valid.