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On the Fluctuation Relation for Nose-Hoover Boundary Thermostated Systems.pdf

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On the Fluctuation Relation for Nose-Hoover Boundary Thermostated Systems

a r X i v : 0 7 1 0 .3 6 7 3 v 2 [ c o n d - m a t .s t a t - m e c h ] 4 S e p 2 0 0 8 On the Fluctuation Relation for Nose?-Hoover Boundary Thermostated Systems Carlos Mej??a-Monasterio, Lamberto Rondoni Dipartimento de Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino Italy Abstract We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nose?- Hoover thermostats applied at the boundaries of the system. The transient fluc- tuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in [D.J. Searles, et al., J. Stat. Phys. 128, 1337 (2007)], for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate Λ and of the dissipation function ?, a similar relaxation regime at shorter averaging times is found. The quantity ? satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity Λ appears to begin a monotonic convergence after such times. This is consistent with the fact that ? and Λ differ by a total time derivative, and that the tails of the probability distribution function of Λ are Gaussian. 1 Introduction In the last decades it has been recognized the central role that the fluctuations around stationary states play for a microscopic theory of nonequilibrium phenom- INTRODUCTION 2 ena