Non-equilibrium dynamics of the piston in the Szilard engine

Abstract

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.