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The fluctuation theorem as a Gibbs property. (English) Zbl 0941.60099

Let \(\xi\to\varphi(\xi)\), \(\xi\in\Omega\), be a smooth discrete-time, reversible dynamical system on a compact, connected manifold \(\Omega\) and consider the quantity \(N^{-1}\sum^{N/2}_{-N/2} \dot S(\varphi^n(\xi))\) where \(\dot S\) is the phase space contraction rate (entropy production rate), i.e. \(\dot S= \log J\), \(J\) being the Jacobian corresponding to the dynamics. The fluctuation theorem of Gallavotti and Cohen [Phys. Rev. Lett. 74, 2694-2697 and 80, 931-970] establishes an asymptotic symmetry possessed by the distribution of this quantity under the stationary state of the system. The present paper shows how this property can be understood as a property of Gibbs states once the Gibbs formalism is applied at the level of space-time histories. The connection with the dynamical system comes from the fact that the space-time measure of the system is akin to a Gibbs measure for an interaction determined by the transition probabilities.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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