581 research outputs found
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 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 (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 -trajectory from renormalization invariance
We formulate a renormalized running coupling expansion for the
--function and the potential of the renormalized --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
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