Big entropy fluctuations in statistical equilibrium: The macroscopic kinetics

Large entropy fluctuations in the equilibrium steady state of classical mechanics are studied in extensive numerical experiments in a simple strongly chaotic Hamiltonian model with two degrees of freedom described by the modified Arnold cat map. The rise and fall of a large separated fluctuation is shown to be described by the (regular and stable) “macroscopic” kinetics, both fast (ballistic) and slow (diffusive). We abandon a vague problem of the “appropriate” initial conditions by observing (in a long run) a spontaneous birth and death of arbitrarily big fluctuations for any initial state of our dynamical model. Statistics of the infinite chain of fluctuations similar to the Poincaré recurrences is shown to be Poissonian. A simple empirical relationship for the mean period between the fluctuations (the Poincaré “cycle”) is found and confirmed in numerical experiments. We propose a new representation of the entropy via the variance of only a few trajectories (“particles”) that greatly facilitates the computation and at the same time is sufficiently accurate for big fluctuations. The relation of our results to long-standing debates over the statistical “irreversibility” and the “time arrow” is briefly discussed.