Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model   with Gauss Heat Bath
  Large entropy fluctuations in a nonequilibrium steady state of classical mechanics were studied in extensive numerical experiments on a simple 2-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 thermostat were found to be non-Gaussian. Approximately, the fluctuations can be discribed 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 turned out to be 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 Poincar\'e recurrences to the initial state and beyond. This is a new interesting phenomenon to be farther studied together with some other open questions. Relation of this particular example of nonequilibrium steady state to a long-standing persistent controversy over statistical 'irreversibility', or the notorious 'time arrow', is also discussed. In conclusion, an unsolved problem of the origin of the causality 'principle' is touched upon. 
Boris Chirikov (Budker Institute of Nuclear Physics, Novosibirsk,   Russia)