Borel summability and Lindstedt series

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining $C^\infty$ functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in the case of two-dimensional rotation vectors with Diophantine exponent $\tau=1$ (e. g. with ratio of the two independent frequencies equal to the golden mean).Comment: 17 pages, 1 figur

Fluctuation Theorem and Chaos

The heat theorem (i.e. the second law of thermodynamics or the existence of entropy) is a manifestation of a general property of hamiltonian mechanics and of the ergodic Hypothesis. In nonequilibrium thermodynamics of stationary states the chaotic hypothesis plays a similar role: it allows a unique determination of the probability distribution (called {\rm SRB} distribution on phase space providing the time averages of the observables. It also implies an expression for a few averages concrete enough to derive consequences of symmetry properties like the fluctuation theorem or to formulate a theory of coarse graining unifying the foundations of equilibrium and of nonequilibrium.Comment: Basis for the plenary talk at StatPhys23 (Genova July 2007

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

Fluctuation Relation beyond Linear Response Theory

The Fluctuation Relation (FR) is an asymptotic result on the distribution of certain observables averaged over time intervals T as T goes to infinity and it is a generalization of the fluctuation--dissipation theorem to far from equilibrium systems in a steady state which reduces to the usual Green-Kubo (GK) relation in the limit of small external non conservative forces. FR is a theorem for smooth uniformly hyperbolic systems, and it is assumed to be true in all dissipative ``chaotic enough'' systems in a steady state. In this paper we develop a theory of finite time corrections to FR, needed to compare the asymptotic prediction of FR with numerical observations, which necessarily involve fluctuations of observables averaged over finite time intervals T. We perform a numerical test of FR in two cases in which non Gaussian fluctuations are observable while GK does not apply and we get a non trivial verification of FR that is independent of and different from linear response theory. Our results are compatible with the theory of finite time corrections to FR, while FR would be observably violated, well within the precision of our experiments, if such corrections were neglected.Comment: Version accepted for publication on the Journal of Statistical Physics; minor changes; two references adde

Nonequilibrium and irreversibility

The booklet contain an overview on selected recent developments in nonequilibrium statistical mechanics and chaos theory: SRB distributions, chaotic hypothesis, fluctuation theorem, proposals for tests and applications to granular materials, fluidodynamics, irreversibility of quasi static processes. In appendices examples of the kind of technical work necessary for actual construction of nonequilibrium stationary states.Comment: XI+247 pages, latex, V.2 with new references and typos eliminated V.3 references added, further typos eliminated, style adjustment

Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus

We consider the "thermodynamic limit"of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual "cluster expansion" techniques.Comment: 28 pages, 2 figure

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

Twistless KAM tori

A selfcontained proof of the KAM theorem in the Thirring model is discussed.Comment: 7 pages, 50 K, Plain Tex, generates one figure named gvnn.p

Running coupling expansion for the renormalized ϕ44\phi^4_4-trajectory from renormalization invariance

We formulate a renormalized running coupling expansion for the $\beta$--function and the potential of the renormalized $\phi^4$--trajectory on four dimensional Euclidean space-time. Renormalization invariance is used as a first principle. No reference is made to bare quantities. The expansion is proved to be finite to all orders of perturbation theory. The proof includes a large momentum bound on the connected free propagator amputated vertices.Comment: 14 pages LaTeX2e, typos and references correcte

Is it possible to experimentally verify the fluctuation relation? A review of theoretical motivations and numerical evidence

The theoretical motivations to perform experimental tests of the stationary state fluctuation relation are reviewed. The difficulties involved in such tests, evidenced by numerical simulations, are also discussed.Comment: 36 pages, 4 figures. Extended version of a presentation to the discussion "Is it possible to experimentally verify the fluctuation theorem?", IHP, Paris, December 1, 2006. Comments are very welcom