Exact Classical Nonequilibrium Statistical-Mechanical Analysis of the Finite Ideal Gas

A system consisting of $N$ noninteracting point particles bouncing elastically from the walls of a rectangular box is studied. The macroscopic observables are assumed to be the center of mass, total energy, and total momentum. The initial $N$-particle probability distribution $P\left(0\right)\mathrm{}$ corresponding to these observables is set up. Liouville's equation is solved exactly and analytically for $P\left(t\right)\mathrm{}$, and exact expressions are obtained for various reduced distributions and moments and for the time dependence of the macroscopic observables. It is shown that the expected value of any analytic phase function relaxes to equilibrium. The evolution of the nonequilibrium entropy $S\left(t\right)\mathrm{}$ is investigated. It is found that $S\left(t\right)\mathrm{}$ undergoes a nonmonotonic increase from a minimum at $t=0\mathrm{}$ to a maximum at $t\to \infty$, and that $S(t\to \infty )$ is the usual canonical entropy. It is shown that statistical irreversibility occurs for arbitrary $N$, but that predictability occurs only for large (but not necessarily infinite) $N$.