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 $n$, $mN\mathrm{/}M\gg 1$ and $m\mathrm{/}M\ll 1$, we find a good approximation of Fourier’s law.