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

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

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

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

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

A fluctuation theorem in a random environment

A simple class of chaotic systems in a random environment is considered and the fluctuation theorem is extended under the assumption of reversibility.Comment: 9 page

Note on nonequilibrium stationary states and entropy

In transformations between nonequilibrium stationary states, entropy might be a not well defined concept. It might be analogous to the ``heat content'' in transformations in equilibrium which is not well defined either, if they are not isochoric ({\it i.e.} do involve mechanical work). Hence we conjecture that un a nonequilbrium stationary state the entropy is just a quantity that can be transferred or created, like heat in equilibrium, but has no physical meaning as ``entropy content'' as a property of the system.Comment: 4 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