Aspects of Ergodic, Qualitative and Statistical Theory of Motion

Gallavotti, Giovanni; Bonetto, Federico; Gentile, Guido

doi:10.1007/978-3-662-05853-4

Cited by 96 publications

(181 citation statements)

References 129 publications

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“…Analyticity and convexity of large deviation rates are general properties, established by Sinai and valid for the SRB-averages of smooth observables (in Anosov systems), [6,9,10]. In fact more can be said for the specific case of the large deviation rate of the observable p, and one can prove the following fluctuation theorem:…”

Section: The Fluctuation Relationmentioning

confidence: 93%

“…If S is an Anosov maps, existence of a unique invariant probability distribution µ, called the SRB distribution and describing the long-time statistics of the motions whose initial data are chosen randomly with respect to the volume measure, is established, [5,6]. It has the property that, with the exception of points x ∈ Ω in a set of 0-volume, we have…”

Section: The Fluctuation Relationmentioning

confidence: 99%

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Chaotic Hypothesis, Fluctuation Theorem and Singularities

et al. 2006

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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.

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Section: The Fluctuation Relationmentioning

confidence: 93%

Section: The Fluctuation Relationmentioning

confidence: 99%

Chaotic Hypothesis, Fluctuation Theorem and Singularities

et al. 2006

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“…. , R − 1), and F (ε ′ ) is an arbitrary smooth observable defined on phase space regarded as a function on the symbolic sequences and evaluated at a sequence ε ′ which extends (rather arbitrarily) ε to an infinite compatible sequence by continuing ε to the right with a sequence ε > and to the left with a sequence ε < into ε ′ = (ε < , ε, ε > ) so that ε < depends only on the symbol ε −R and ε > depends only on the symbol ε R : see [18,20,21].…”

Section: Finite Time Corrections To the Fluctuation Relationmentioning

confidence: 99%

“…The one dimensional systems are very well understood and the above is a well studied problem in statistical mechanics, known as a finite size corrections calculation. For instance it can be attacked by cluster expansion, [21]; this is a technique to deal with the average of the exponential of a spin Hamiltonian which is defined in terms of potentials φ X exponentially decaying with rate κ, such as those appearing in the numerator and in the denominator of Eq. 16.…”

Section: B Finite Time Corrections To the Characteristic Functionmentioning

confidence: 99%

Fluctuation Relation beyond Linear Response Theory

2005

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The Fluctuation Relation (FR) is an asymptotic result on the distribution of certain observables averaged over time intervals τ as τ → ∞ 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 τ . 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.

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“…It follows, by standard arguments on one-dimensional Gibbs distributions, see Chap.6 in [8], that the "short range" Gibbs distribution ν with formal energy function…”

Section: Construction Of the Srb Distributionmentioning

confidence: 99%