Approach of One-Dimensional Systems to Equilibrium

This paper defines an equilibrium coarse-grained probability density $F(\mathrm{X},\mathrm{P}, \infty )$ in terms of the Liouville fine-grained density $f(\mathrm{X},\mathrm{P}, t)$ through the equation $F(\mathrm{X},\mathrm{P}, \infty )=limit\; of\left(\frac{1}{\Delta \Omega }\right)\text{as}\Delta \Omega \to 0limit\; of\int \stackrel{}{\Delta \Omega }d\mathrm{X}d\mathrm{P}f(\mathrm{X},\mathrm{P}, t)\text{as}t\to \infty ,$ and points out that $F(\mathrm{X},\mathrm{P}, \infty )$ is the density which determines the observable properties of a system at equilibrium. Given a sufficiently smooth initial density $f_0({\mathrm{X}}_0, {\mathrm{P}}_0, 0)$, $F(\mathrm{X},\mathrm{P}, \infty )$ is shown to exist and describe the equilibrium behavior of two one-dimensional systems. This equilibrium behavior is occasioned by the presence of nonlinear forces. Because of the nonlinearity, the Poincaré recurrence time, for a given accuracy of return, depends on initial conditions. It is shown that this dependence causes a Gibbs-type stirring of phase space which leads to equilibrium.