Energy Transfer in a Fast-Slow Hamiltonian System

Dolgopyat, Dmitry; Liverani, Carlangelo

doi:10.1007/s00220-011-1317-7

Cited by 42 publications

(61 citation statements)

References 30 publications

(91 reference statements)

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“…Finally thanks to the symmetries of the system, all the coefficients can be calculated from K p,p (β) and K β,β (β) := K β,β (β, 0), computed at zero average velocity. Relations (10) and (12) were already noted in [19]. One of the main mathematical problems in dealing with the deterministic infinite dynamics, is in proving that the limits defining K p,p (β) and K β,β (β) exist and are finite.…”

Section: Linear Response and Onsager Matrixmentioning

confidence: 87%

Role of conserved quantities in Fourier's law for diffusive mechanical systems

Olla

2019

Comptes Rendus. Physique

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Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space-time scale. In these situations the Fourier law depends also from the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamiltonian dynamics is perturbed by conservative stochastic terms. The presence of the stochastic dynamics allows to define the transport coefficients (thermal conductivity) and in some cases to prove the local equilibrium and the linear response argument necessary to obtain the diffusive equations governing the macroscopic evolution of the conserved quantities. Temperature profiles and other conserved quantities profiles in the non-equilibrium stationary states can be then understood from the non-stationary diffusive behaviour. We also review some results and open problems on the two step approach (by weak coupling or kinetic limits) to the heat equation, starting from mechanical models with only energy conserved. To cite this article: S. Olla, C. R. Physique X (2019).

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Section: Linear Response and Onsager Matrixmentioning

confidence: 87%

Role of conserved quantities in Fourier's law for diffusive mechanical systems

Olla

2019

Comptes Rendus. Physique

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“…We denote with [L] the corresponding equivalence class, which therefore uniquely identifies a probability measure. We say that a probability 19 Recall that a probability space is a Lebesgue space if it is isomorphic to the disjoint union of an interval [0, a] with Lebesgue measure and (at most) countably many atoms. 20 The set Lc 1 ,c 2 of (c 1 , c 2 )-standard pairs is in fact a space of smooth functions; it is thus a measurable space with the Borel σ-algebra.…”

Section: Standard Familiesmentioning

confidence: 99%

“…Note however that this would require a time scale ε −2 to bring any two standard pairs close enough to couple them, [16]. This situation is of considerable interest in non-Equilibrium Statistical Mechanics when the dynamics is Hamiltonian and the slow variables are the energies of nearby, weakly interacting, systems, see [19]. In this case we conjecture that, generically, the system should be mixing and the correlations should decay exponentially with rate which would be, at best, ε 2 .…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Another argument of renewed interest is averaging theory, due to new powerful results [16,18] and, among others, new applications of clear relevance for non-equilibrium Statistical Mechanics [19]. Yet, averaging theory only provides information on a given time scale; a natural and very relevant question is what happens at longer, possibly infinitely long, time scales.…”

Section: Introductionmentioning

confidence: 99%

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Statistical properties of mostly contracting fast-slow partially hyperbolic systems

Simoi

Liverani

2016

Invent. math.

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Abstract. We consider a class of C 4 partially hyperbolic systems on T 2 described by maps Fε(x, θ) = (f (x, θ), θ + εω(x, θ)) where f (·, θ) are expanding maps of the circle. For sufficiently small ε and ω generic in an open set, we precisely classify the SRB measures for Fε and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.

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“…(see in particular [1,[12][13][14]). See also a study with stochastic thermostats [5,6], and a promising investigation by Dolgopyat and Liverani [11] of the macroscopic behavior of a coupled lattice of strongly chaotic microscopic subsystems.…”

Section: Introductionmentioning

confidence: 98%

A Mechanical Model for Fourier’s Law of Heat Conduction

Ruelle

2011

Commun. Math. Phys.

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Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier's law can proceed rigorously.

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Energy Transfer in a Fast-Slow Hamiltonian System

Cited by 42 publications

References 30 publications

Role of conserved quantities in Fourier's law for diffusive mechanical systems

Role of conserved quantities in Fourier's law for diffusive mechanical systems

Statistical properties of mostly contracting fast-slow partially hyperbolic systems

A Mechanical Model for Fourier’s Law of Heat Conduction

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