arXiv subjects
Publications and source records attributed to Ya. Sinai.
The purpose of this paper is to give a complete derivation of the limiting distribution of large Frobenius numbers outlined in earlier work of J. Bourgain and Ya. Sinai and fill some gaps formulated there as hypotheses.
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 $m=1$. We find that, 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), so that the mechanical trajectory of the piston converges, in probability, to the solution of the hydrodynamical equations for a certain period of time. We also discuss heuristically the dynamics of the system on longer intervals of time.
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 $M\sim L^2$ and undergoes elastic collisions with $N\sim L^3$ gas particles of mass $m$. In a previous paper, Lebowitz, Piasecki and Sinai 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.