Poincaré Recurrences

In connection with the tracing of the origin of the apparent irreversibility exhibited by a class of simple mechanical systems, namely all multiply or conditionally periodic Hamilton-Jacobi systems, estimates are obtained for the Poincaré recurrence time of such a system in terms of the preassigned limits of error of the mechanical recurrence, $\epsilon$. By applying the theory of diophantine approximations, the asymptotic fraction of the time a system spends in such recurrences is found exactly. These results allow further deductions concerning the fraction of time a given system obeys a strict version of the second law of thermodynamics, as well as the existence and order of magnitude of the average Poincaré recurrence time of a Gibbsian ensemble of such systems whose degrees of freedom are indistinguishable.