Big entropy fluctuations in the nonequilibrium steady state: A simple model with the gauss heat bath

Large entropy fluctuations in a nonequilibrium steady state of classical mechanics are studied in extensive numerical experiments on a simple two-freedom model with the so-called Gauss time-reversible thermostat. The local fluctuations (on a set of fixed trajectory segments) from the average heat entropy absorbed in the thermostat are found to be non-Gaussian. The fluctuations can be approximately described by a two-Gaussian distribution with a crossover independent of the segment length and the number of trajectories (“particles”). The distribution itself does depend on both, approaching the single standard Gaussian distribution as any of those parameters increases. The global time-dependent fluctuations are qualitatively different in that they have a strict upper bound much less than the average entropy production. Thus, unlike the equilibrium steady state, the recovery of the initial low entropy becomes impossible after a sufficiently long time, even in the largest fluctuations. However, preliminary numerical experiments and the theoretical estimates in the special case of the critical dynamics with superdiffusion suggest the existence of infinitely many Poincaré recurrences to the initial state and beyond. This is a new interesting phenomenon to be further studied together with some other open questions. The relation of this particular example of a nonequilibrium steady state to the long-standing persistent controversy over statistical “irreversibility”, or the notorious “time arrow”, is also discussed. In conclusion, the unsolved problem of the origin of the causality “principle” is considered.