A local fluctuation theorem

A mechanism for the validity of a local version of the fluctuation theorem, uniform in the system size, is discussed for a reversible chain of weakly coupled Anosov systems.Comment: plain TeX, 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

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

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

Reversible viscosity and Navier--Stokes fluids

Exploring the possibility of describing a fluid flow via a time-reversible equation and its relevance for the fluctuations statistics in stationary turbulent (or laminar) incompressible Navier-Stokes flows.Comment: 7 pages 6 figures, v2: replaced Fig.6 and few changes. Last version: appendix cut shorter, because of a computational erro

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

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

Irreversibility time scale

Entropy creation rate is introduced for a system interacting with thermostats ({\it i.e.}, in the usual language, for a system subject to internal conservative forces interacting with ``external'' thermostats via conservative forces) and a fluctuation theorem for it is proved. As an application a time scale is introduced, to be interpreted as the time over which irreversibility becomes manifest in a process leading from an initial to a final stationary state of a mechanical system in a general nonequilibrium context. The time scale is evaluated in a few examples, including the classical Joule-Thompson process (gas expansion in a vacuum). The new version (n.2) contains several comments on references pointed out to me after posting the version n.1.Comment: 6 pages 1 figur

Equilibrium Statistical Mechanics

An introductory review of Classical Statistical MechanicsComment: 56 page