Entropy, Thermostats and Chaotic Hypothesis

The chaotic hypothesis is proposed as a basis for a general theory of nonequilibrium stationary states. Version 2: new comments added after presenting this talk at the Meeting mentioned in the Acknowledgement. One typo corrected.Comment: 6 page

Chaotic Hypothesis, Fluctuation Theorem and singularities

The chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them. However its application to Physics problems requires attention and can lead to apparent inconsistencies. In particular there are several cases that have been considered in the literature in which singularities are built in the models: for instance when among the forces there are Lennard-Jones potentials (which are infinite in the origin) and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded. The situation is well understood in certain special cases in which the system is subject to Gaussian noise; here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived. The main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider, i.e. deterministic systems with thermostat forces acting according to Gauss' principle for the constraint of constant total kinetic energy (``isokinetic Gaussian thermostats''), close and far from equilibrium. Near equilibrium it even predicts a fluctuation relation which, in deterministic cases with more general thermostat forces (i.e. not necessarily of Gaussian isokinetic nature), extends recent relations obtained in situations in which the thermostatting forces satisfy Gauss' principle. This relation agrees, where expected, with the fluctuation theorem for perfectly chaotic systems. The results are compared with some recent works in the literature.Comment: 7 pages, 1 figure; updated to take into account comments received on the first versio

Chaotic Hypothesis and Universal Large Deviations Properties

Chaotic systems arise naturally in Statistical Mechanics and in Fluid Dynamics. A paradigm for their modelization are smooth hyperbolic systems. Are there consequences that can be drawn simply by assuming that a system is hyperbolic? here we present a few model independent general consequences which may have some relevance for the Physics of chaotic systems. Expanded version of a talk at ICM98, Berlin.Comment: 29 pages: Plain-TeX, 1 figur

Dynamical ensembles in stationary states

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

Chaotic hypothesis: Extension of Onsager reciprocity to large fields and the chaotic hypothesis

The fluctuation theorem (FT), the first derived consequence of the {\it Chaotic Hypothesis} (CH) of ref. [GC1], 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.Comment: Revises previous paper with the same title and extends the result

Gallavotti-Cohen theorem, Chaotic Hypothesis and the zero-noise limit

The Fluctuation Relation for a stationary state, kept at constant energy by a deterministic thermostat - the Gallavotti-Cohen Theorem -- relies on the ergodic properties of the system considered. We show that when perturbed by an energy-conserving random noise, the relation follows trivially for any system at finite noise amplitude. The time needed to achieve stationarity may stay finite as the noise tends to zero, or it may diverge. In the former case the Gallavotti-Cohen result is recovered, while in the latter case, the crossover time may be computed from the action of `instanton' orbits that bridge attractors and repellors. We suggest that the `Chaotic Hypothesis' of Gallavotti can thus be reformulated as a matter of stochastic stability of the measure in trajectory space. In this form this hypothesis may be directly tested

Non equilibrium in statistical and fluid mechanics. Ensembles and their equivalence. Entropy driven intermittency

We present a review of the chaotic hypothesis and discuss its applications to intermittency in statistical mechanics and fluid mechanics proposing a quantitative definition. Entropy creation rate is interpreted in terms of certain intermittency phenomena. An attempt to a theory of the experiment of Ciliberto-Laroche on the fluctuation law is presented.Comment: 22 page

Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem

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 fluctuation dissipation theorem in various models for coexisting transport phenomena.Comment: 16 pages, postscrip

Heat and Fluctuations from Order to Chaos

The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as $10^{23}$ degrees of freedom systems, {\it i.e.} for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems ({\it i.e.} the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.Comment: 1) added comment at the end of Sec. 1 to explain the meaning of the title (referee request) 2) added comment at the end of Sec. 17 (i.e. appendix A4) to refer to papers related to the ones already quoted (referee request