0n the Fluctuation Law(s) for Hamiltonian Systems (with Equilibrium Steady State): a Comment on Cond–mat/0008421

A generalization of the fluctuation law (FL) (" theorem "), formulated in 1993 by Evans, Cohen and Morriss for a nonequilibrium steady state, on the chaotic Hamiltonian systems with equilibrium steady state in recent publication by Evans, Searles and Mittag (cond–mat/0008421) is briefly discussed. We argue that the physical meaning of this law, as presented in the latter publication, is qualitatively different from the original one. Namely, the original FL concerns the local (in time) fluctuations with an intriguing result: a high probability for the " violation " of the Second Law. Instead, the new law describes the global fluctuations for which this remarkable unexpected phenomenon is absent or hidden. We compare both types of fluctuations in both classes of Hamiltonian systems, and discuss remarkable similarities as well as the interesting distinctions. The " Fluctuation Theorem " has been first obtained by Evans, Cohen and Morriss [1] for a particular example of the nonequilibrium steady state, using both the theory as well as numerics. For our purposes it can be represented in the form: ln p(∆S) p(−∆S) = F · ∆S , F = 2∆S σ 2 (1) Here p(∆S) is the probability of entropy (or entropy–like quantity as in [1, 2]) change ∆S in the ensemble of trajectory segments of a fixed (appropriately scaled) duration τ for the mean change ∆S > 0 and variance σ 2 , and F the fluctuation parameter usually taken to be unity (F = 1). We call this type of fluctuations the local fluctuations. By itself, the relation (1) is but a specific reduced representation of the normal probabilistic law, the Gaussian distribution, in a suitable random variable (∆S): p(∆S) = 1 √ 2πσ 2 · exp − (∆S − ∆S) 2 2σ 2 (2) shifted with respect to ∆S = 0 due to the permanent entropy production at a constant rate in the nonequilibrium steady state. The FL (1) immediately follows from the normal law (2) but not vice versa. Notice also that this distribution is not universal, yet it is rather typical indeed. However, the surprise (to many) was in that the probability of negative (" abnormal ") entropy change ∆S < 0 (without time reversal!) is generally not small at all reaching 50% for sufficiently short τ. That is every second change may be abnormal !? Implicitly, all that is contained in the well developed statistical theory (see, e.g., [3], …