Topics on chaotic dynamics

Gallavotti, Giovanni

doi:10.1007/3-540-59178-8_34

Cited by 13 publications

(8 citation statements)

References 24 publications

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“…Further computer simulations and rigorous results (under suitable hypotheses) strongly suggest that thermostated stationary measures are indeed generically singular with respect to the Lebesgue measure [4]. They correspond to the Sinai-Ruelle-Bowen (SRB) measures for these systems [5]. A question then arose of what significance the singular nature of such measures, so different from those obtained from the traditional stochastic modeling of these systems, has for the behavior of macroscopic nonequilibrium systems.…”

Section: Introductionmentioning

confidence: 99%

Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

Bonetto¹,

Kupiainen²,

Lebowitz³

2005

Ergod. Th. Dynam. Sys.

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Absrtact:We study a d-dimensional coupled map lattice consisting of hyperbolic toral automorphisms (Arnold cat maps) that are weakly coupled by an analytic coupling map. We construct the Sinai-Ruelle-Bowen measure for this system and study its marginals on the tori. We prove they are absolutely continuous with respect to the Lebesgue measure if and only if the coupling satisfies a nondegeneracy condition.

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Section: Introductionmentioning

confidence: 99%

Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

Bonetto¹,

Kupiainen²,

Lebowitz³

2005

Ergod. Th. Dynam. Sys.

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“…where ν is a constant tuned so that the average kinetic energy is eN k B T /2. This model was considered in the perspective of the conjectures of ensemble equivalence in [Ga95], [Ga96b].…”

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Nonequilibrium in statistical and fluid mechanics. Ensembles and their equivalence. Entropy driven intermittency

Gallavotti

2000

Journal of Mathematical Physics

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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. §1. Introduction.A general theory of non equilibrium stationary phenomena extending classical thermodynamics to stationary non equilibria is, perhaps surprisingly, still a major open problem more than a century past the work of Boltzmann (and Maxwell, Gibbs,...) which made the breakthrough towards an understanding of properties of matter based on microscopic Newton's equations and the atomic model. In the last thirty years, or so, some progress appears to have been achieved since the recognition that non equilibrium statistical mechanics and stationary turbulence in fluids are closely related problems and, in a sense, in spite of the apparently very different nature of the equations describing them they are essentially the same. The unifying principle, originally proposed for turbulent motions by Ruelle, [Ru78], in the early 970's, has been extended to statistical mechanics and eventually called the "chaotic hypothesis", [GC95]:Chaotic hypothesis: Asymptotic motions of a chaotic system, be it a multi particle system of microscopic particles or a turbulent macroscopic fluid, can be regarded as a transitive Anosov system for the purposes of computing time averages in stationary states.

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“…This happens in particular for generic Anosov systems; for other examples, see [1,2]. In addition there may exist a stationary measure absolutely continuous with respect to dx, i.e.…”

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confidence: 99%

“…S µ has been of much interest recently in connection with "thermostatted" nonequilibrium systems [1,2,5]. Under suitable conditions on T t , it can be shown that µ t (dx) −→ t→±∞ ν ± (dx)…”

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Remark on the (non)convergence of ensemble densities in dynamical systems

Lebowitz

Sinaĭ

1998

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.

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Topics on chaotic dynamics

Cited by 13 publications

References 24 publications

Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

Nonequilibrium in statistical and fluid mechanics. Ensembles and their equivalence. Entropy driven intermittency

Remark on the (non)convergence of ensemble densities in dynamical systems

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