Ruelle's principle for turbulence leading to what is usually called the Sinai-Ruelle-Bowen distribution (SRB) is applied to the statistical mechanics of many particle systems in nonequilibrium stationary states. A specific prediction, obtained without the need to construct explicitly the SRB itself, is shown to be in agreement with a recent computer experiment on a strongly sheared fluid. This presents the first test of the principle on a many particle system far from equilibrium. A possible application to fluid mechanics is also discussed.Comment: Postscript, 12 pages, 132K, 1 uncompressed file Keywords: chaos, nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov exponents, ga

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

The fluctuation theorem (FT), the first derived consequence of the Chaotic Hypothesis (CH) of [1], can be considered as an extension to arbitrary forcing fields of the fluctuation dissipation theorem (FD) and the corresponding Onsager reciprocity (OR), in a class of reversible nonequilibrium statistical mechanical systems. 47.52.+j, 05.45.+b, 05.70.Ln, A typical system studied here will be N point particles subject to (a) mutual and external conservative forces with potential V ( q 1 , . . . , q N ), (b) external (non conservative) forces, forcing agents, { F j }, j = 1, . . . , N , whose strength is measured by parameters {G j }, j = 1, . . . , s, and (c) also to forces { ϕ j }, j = 1, . . . N , generating constraints that provides a model for the thermostatting mechanism that keeps the energy of the system from growing indefinitely (because of the continuing action of the forcing agents).An observable O({ q,˙ q}) evolves in time under the time evolution S t solving the equations of motion:with m = particles mass, and ϕ j "thermostatting" forces assuring that the system reaches a (non equilibrium) stationary state. The time evolution of the observable O on the motion with initial data x = ( q,˙ q) is the function t → O(S t x) so that the motion statistics is the probability distribution µ + on the phase space C such that:for all data x ∈ C except a set of zero measure with respect to the volume µ 0 on C. The distribution µ + is assumed to exist: a property called zero-th law, [2,3]. The thermostatting mechanism will be described by force laws ϕ j which enforce the constraint that the kinetic energy (or the total energy) of the particles, or of subgroups of the particles, remains constant, [4]. It is convenient also to imagine that the constraints keep the total kinetic energy bounded (hence phase space is bounded). The constraint forces { ϕ j } will be supposed such that the system is reversible: this means that there will be a map i, defined on phase space, anticommuting with time evolution: i.e. S t i ≡ i S −t .Reversibility is a key assumption, [1,5].In [2] the divergence of the r.h.s. of Eq. (1) is a quantity −σ(x) defined on phase space that has been identified with the entropy production rate.The chaotic hypothesis, (CH), of [1] implies a fluctuation theorem or (FT) which, [2], is a property of the fluctuations of the entropy production rate. Namely if we denote σ + = C σ(y)µ + (dy) the time average, over an infinite time interval by Eq. (2), then the dimensionless finite time average p = p(x):has a statistical distribution π τ (p) with respect to the stationary state distribution µ + such that:provided (of course) σ + > 0. Following [1] a reversible system for which σ + > 0 will be called dissipative. Ruelle's H-theorem, [5], states that σ + > 0 unless the stationary distribution µ + has the form ρ(x)µ 0 (dx).Hence we shall suppose that the system is dissipative when the forcing G does not vanish and, without real loss of generality, that σ(ix) = −σ(x) and that σ(x) ≡ 0 when the external forcing vanishes...

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The chaotic hypothesis discussed in Gallavotti and Cohen (1995) is tested experimentally in a simple conduction model. Besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which, as the forcing strength grows, the attractor ceases to be an Anosov system and becomes an Axiom A attractor. A first text of the new predictions is also attempted

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A self-contained analysis is given of the simplest quantum fields from the renormalization group point of view: multiscale decomposition, general renormalization theory, resummations of renormalized series via equations of the Callan-Symanzik type, asymptotic freedom, and proof of ultraviolet stability for sin-Gordon fields in two dimensions and for other super-renormalizable scalar fields. Renormalization in four dimensions (Hepp's theorem and De Calan-Rivasseau n! bound) is presented and applications are made to the Coulomb gases in two dimensions and tot he convergence of the planar graph expansion in four dimensional field theories (t'Hooft-Rivasseau theorem). ÌÀËÁ Á È ÍÇAEÌ Á1

It is shown that the "chaoticity hypothesis", analogous to Ruelle's principle for turbulence and recently introduced in statistical mechanics, implies the Onsager reciprocity and the uctuation dissipation theorem in various reversible models for coexisting transport phenomena.x1: Introduction.In reference GC2] we introduced, as a principle holding when a system motions have an empirically chaotic nature, that:Chaotic hypothesis: A many particle system in a stationary state can be regarded as a smooth dynamical system with a transitive axiom A global attractor for the purpose of computing macroscopic properties. In the reversible case it can be regarded, for the same purposes, as a smooth transitive Anosov system.