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 in, 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 >> 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)root 8 pi mk(B)T, 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 = root 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. Copyright (C) EPLA, 2019