Ergodicity in Hamiltonian Systems

Liverani, Carlangelo; Wojtkowski, Maciej P.

doi:10.1007/978-3-642-61215-2_3

Cited by 93 publications

(183 citation statements)

References 22 publications

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“…Similar metric-type quantities have been used before, see e.g. the quadratic form Q used in [17], or the pseudo-metric called "p-metric" in [5,15,19]. These are known to increase on the unstable cone, but are not well-behaved on general tangent vectors.…”

Section: Riemannian Structure and Regularitymentioning

confidence: 98%

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

Bálint

Tóth

2008

Ann. Henri Poincaré

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Section: Riemannian Structure and Regularitymentioning

confidence: 98%

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

Bálint

Tóth

2008

Ann. Henri Poincaré

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“…32 30 The square root is for later convenience, see the proof of Lemmata 6.3 and 6.8. 31 For example, using the function p introduced to partition in time, one can define, in the charts κ ζ , p(kr θ /2 + 2r −θ τ − η) p(jr θ /2 + 2r −θ τ − ξ) p(īr θ /2 + 2r θ /2τ − s). 32 This can be achieved thanks to the fact that the Wα belong to the stable cone.…”

Section: The Dolgopyat Estimatementioning

confidence: 99%

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

Baladi

Liverani

2012

Commun. Math. Phys.

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Abstract. We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author's version [30] of Dolgopyat's estimates for contact flows and the first author's work with Gouëzel [6] on piecewise hyperbolic discrete-time dynamics.

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“…This scenario is easily ruled out by choosing Γ 0 to have concave (or scattering) walls. Such a choice of Γ 0 implies that ∂∆ N also has concave boundaries, and the free motion of a particle in a domain with concave boundaries is well known to be hyperbolic and ergodic [18,13]. We do not know if the absence of trapped tracers in the sense above implies ergodicity.…”

Section: Hyperbolicity Of Billiard Dynamics: a Necessary Conditionmentioning

confidence: 99%

Nonequilibrium Energy Profiles for a Class of 1-D Models

Eckmann

Young

2005

Commun. Math. Phys.

181

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As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which containing an energy storage device called a "tank". Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously -under physically natural assumptions -macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed.

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Ergodicity in Hamiltonian Systems

Abstract: Abstract. We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (non-uniform) hyperbolic behavior.

Cited by 93 publications

References 22 publications

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

Nonequilibrium Energy Profiles for a Class of 1-D Models

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