Fourier's Law in a Generalized Piston Model

A simplified, but non trivial, mechanical model—gas of N particles of mass m in a box partitioned by n mobile adiabatic walls of mass M—interacting with two thermal baths at different temperatures, is discussed in the framework of kinetic theory. Following an approach due to Smoluchowski, from an analysis of the collisions particles/walls, we derive the values of the main thermodynamic quantities for the stationary non-equilibrium states. The results are compared with extensive numerical simulations; in the limit of large n, mN/M 1 and m/M 1, we find a good approximation of Fourier’s law.