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Large entropy fluctuations in a nonequilibrium steady state of classicalmechanics were studied in extensive numerical experiments on a simple 2-freedommodel with the so-called Gauss time-reversible thermostat. The localfluctuations (on a set of fixed trajectory segments) from the average heatentropy absorbed in thermostat were found to be non-Gaussian. Approximately,the fluctuations can be discribed by a two-Gaussian distribution with acrossover independent of the segment length and the number of trajectories('particles'). The distribution itself does depend on both, approaching thesingle standard Gaussian distribution as any of those parameters increases. Theglobal time-dependent fluctuations turned out to be qualitatively different inthat they have a strict upper bound much less than the average entropyproduction. Thus, unlike the equilibrium steady state, the recovery of theinitial low entropy becomes impossible, after a sufficiently long time, even inthe largest fluctuations. However, preliminary numerical experiments and thetheoretical estimates in the special case of the critical dynamics withsuperdiffusion suggest the existence of infinitely many Poincar\'e recurrencesto the initial state and beyond. This is a new interesting phenomenon to befarther studied together with some other open questions. Relation of thisparticular example of nonequilibrium steady state to a long-standing persistentcontroversy over statistical 'irreversibility', or the notorious 'time arrow',is also discussed. In conclusion, an unsolved problem of the origin of thecausality 'principle' is touched upon.Comment: 21 pages, 7 figure

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