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 ≫ m), can be described effectively by a Langevin equation. Various numerical evidences suggest that the frictional coefficient in the Langevin equation is given by γ = (1/X) √ 8πmkBT , 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 ǫ = 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.

Universal framework for the long-time position distribution of free active particles

“…For example, starting from the Kramers equation, the Fokker-Planck equation of a Brownian particle in the presence of a potential can be obtained in the overdamped limit (see chapter VIII, section 7 of [43]). Another wellknown example is the derivation of the Fokker-Planck equation for slow degrees of freedom in an interacting many particle system, by integrating out the fast degrees [44,45]. Recently, similar procedures were used in the context of active particles, for obtaining a perturbative expansion of the stationary states of AOUP [29] and ABP [37] in external potentials about the respective passive Boltzmann distributions.…”

One-particle engine with a porous piston

“…The demon consists of an engine working with a single particle gas, and since then several models for single particle engines and processes have been studied (see for example [2][3][4] ). Though some variants of this demon, known as Szilard's engine, have been proposed [5][6][7][8][9][10] , the basic design is composed of the following parts: A particle inside a container along with a removable piston, a heat bath at temperature T in contact with the container walls, a hanging mass which is attached to the piston in such a way that work is performed on it as the piston moves, and an external agent in charge of resetting the system once the piston has reached one of the ends of the container.…”