Dynamics of a massive piston in an ideal gas

Nikolai Chernov,
University of Alabama at Birmingham

We study a dynamical system consisting of a massive piston in a cubical container of large size $L$ filled with an ideal gas. The piston has mass $M\sim L^2$ and undergoes elastic collisions with $N\sim L^3$ non-interacting gas particles of mass $m=1$. Under suitable initial conditions, there is, in the limit $L \to \infty$, a scaling regime with time and space scaled by $L$, in which the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled equations (hydrodynamical equations). We prove that the mechanical trajectory of the piston converges, in probability, to the solution of the hydrodynamical equations for a certain period of time. On the other hand, we have observed experimentally that during longer periods of time the mechanical trajectory of the piston can, under certain conditions, diverge far from the corresponding solutions of the hydrodynamical equations with probability close to 1. In that case the piston experiences large oscillations that damp very slowly. This phenomenon is traced to the instability of the hydrodynamical equations.