Scaling dynamics of a massive piston in a cube filled with ideal gas: Exact results

We continue the study of the time evolution of a system consisting of a piston in a cubical container of large size L filled with an ideal gas. The piston has mass M ∼ L² and undergoes elastic collisions with N ∼ L³ gas particles of mass m. In a previous paper, Lebowitz et al. considered a scaling regime, with time and space scaled by L, in which they argued heuristically that the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled differential equations. Here we state exact results and sketch proofs for this behavior.