On the Application of the Gallavotti-Cohen Fluctuation Relation to Thermostatted Steady States Near Equilibrium

The fluctuation relation of the Gallavotti-Cohen Fluctuation Theorem (GCFT) concerns fluctuations in the phase space compression rate of dissipative, reversible dynamical systems. It has been proven for Anosov systems, but it is expected to apply more generally. This raises the question of which non-Anosov systems satisfy the fluctuation relation. We analyze time dependent fluctuations in the phase space compression rate of a class of N-particle systems that are at equilibrium or in near equilibrium steady states. This class does not include Anosov systems or isoenergetic systems, however, it includes most steady state systems considered in molecular dynamics simulations of realistic systems. We argue that the fluctuations of the phase space compression rate of these systems at or near equilibrium do not satisfy the fluctuation relation of the GCFT, although the discrepancies become somewhat smaller as the systems move further from equilibrium. In contrast, similar fluctuation relations for an appropriately defined dissipation function appear to hold both near and far from equilibrium.Comment: 46 pages, for publication in PR

Effective statistical physics of Anosov systems

We present evidence indicating that Anosov systems can be endowed with a unique physically reasonable effective temperature. Results for the two paradigmatic Anosov systems (i.e., the cat map and the geodesic flow on a surface of constant negative curvature) are used to justify a proposal for extending Ruelle's thermodynamical formalism into a comprehensive theory of statistical physics for nonequilibrium steady states satisfying the Gallavotti-Cohen chaotic hypothesis.Comment: 38 pages, 17 figures. Substantially more details in sections 4 and 6; new and revised figures also added. Typos and minor errors (esp. in section 6) corrected along with minor notational changes. MATLAB code for calculations in section 16 also included as inline comment in TeX source now. The thrust of the paper is unaffecte

Banach spaces adapted to Anosov systems

We study the spectral properties of the Ruelle-Perron-Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the C^\infty case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the SRB measure, the variance for the CLT, the rates of decay for smooth observable, etc.).Comment: 26 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

Fluctuations relation and external thermostats: an application to granular materials

In this note we discuss a paradigmatic example of interacting particles subject to non conservative external forces and to the action of thermostats consisting of external (finite) reservoirs of particles. We then consider a model of granular materials of interest for experimental tests that had recently attracted lot of attentions. This model can be reduced to the previously discussed example under a number of assumptions, in particular that inelasticity due to internal collisions can be neglected for the purpose of measuring the large deviation functional for entropy production rate. We show that if the restitution coefficient in the granular material model is close to one, then the required assuptions are verified on a specific time scale and we predict a fluctuation relation for the entropy production rate measured on the same time scale.Comment: 7 pages; updated to take into account comments received on the first version; to appear on J.Stat.Mech.(2006

Equilibrium states of almost Anosov diffeomorphisms

We develop a thermodynamic formalism for a class of diffeomorphisms of a torus that are "almost-Anosov". In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium states for a collection of non-H\"older continuous geometric potentials over almost Anosov systems with an indifferent fixed point, as well as prove exponential decay of correlations and the central limit theorem for these equilibrium measures