Dynamical ensembles in stationary states

Abstract

We propose as a generalization of an idea of Ruelle to describe turbulent fluid flow a chaotic hypothesis for reversible dissipative many particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to non equilibrium states and it leads to the identification of a unique distribution $\m$ describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution $\m$: it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a non trivial, parameter--free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov--Obuchov theory for turbulent flow.Comment: 31 pages, 3 figures, compile with dvips (otherwise no pictures

Extremes and Recurrence in Dynamical Systems

“…See discussions in, e.g., [263] and references therein. This is closely related to the chaos hypothesis [266].…”

“…As discussed in Chapter 8, on general grounds we may expect that in Axiom Alike systems (in the sense of the chaotic hypothesis [266]) physical observables have bounded fluctuations and that their extremes follow Weibull distributions [81,44]. The closer we are to a crisis, the more likely is for the system to explore regions of the phase space close to the saddle, so that there is an increasing probability that the physical observable will have anomalous values and feature (rare) very large fluctuations, much larger than the extreme fluctuations observed in the system far away from the crisis.…”

The McLennan–Zubarev steady state distribution and fluctuation theorems

“…This relation is called the fluctuation theorem, which was first found in numerical data of the simulations of the Nosé-Hoover thermostat systems [1]. After that, a careful examination using dynamical systems theory and ergodic theory, has elucidated the phenomena observed [2,3]. As a result, it is shown that the fluctuation of the entropy production is governed by time reversal symmetry of the system.…”

“…To elucidate the nature of these fluctuations is an issue of nonequilibrium statistical physics in last two decades, for instance, in refs. [1,2,3,4,5,6,7,8,9,10,11,12,13].…”

The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields

“…In 1993, a breakthrough occurred in nonequilibrium statistical mechanics, when Evans, Cohen and Morriss [12] found in computer simulations that the sample EPR of a steady flow has a highly nontrivial symmetry, which is called the fluctuation theorem in the mathematical theory put forward by Gallavotti and Cohen [15]. The fluctuation theorem gives a general formula valid in nonequilibrium systems, for the logarithm of the probability ratio of observing trajectories that satisfy or "violate" the second law of thermodynamics [37].…”