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 Poincar\'e recurrence time of such a system in terms of the preassigned limits of error of the mechanical recurrence,. 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 Poincar\'e recurrence time of a Gibbsian ensemble of such systems whose degrees of freedom are indistinguishable. The relation of the results obtained for this important class of mechanical systems and the resolution of the paradoxes of heat theory propounded by Zermelo, Loschmidt, etc. , due to Boltzmann and von Smoluchowski, is discussed. An especially easily visualized model, the one-dimensional gas of hard spheres, is treated, in particular, in some detail.