Hydrodynamic limit for the velocity-flip model

Simon, Marielle

doi:10.1016/j.spa.2013.05.005

Cited by 13 publications

(36 citation statements)

References 18 publications

(37 reference statements)

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“…The structure of steady state correlations and energy fluctuations are discussed in [5] with supporting numerical evidence presented in [6]. The validity of the proposed hydrodynamic limit equations (Fourier's law) is proven rigorously in [3] (see the Remark after Theorem 1.2. for the changes needed in case the model has pinning).…”

Section: Introductionmentioning

confidence: 59%

“…The model dynamics consists of a classical Hamiltonian evolution of the particle positions and velocities, as determined by a quadratic Hamiltonian, intercepted with random flips of the particle velocities. This model was first considered in [1], and it is one of the simplest known particle chain models which has a finite thermal conductivity and satisfies the time-dependent Fourier's law [2,3]. The model is blessed with many simplifying features which make possible the usually intractable rigorous analysis of heat transport properties.…”

Section: Introductionmentioning

confidence: 99%

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Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

2016

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We consider the detailed structure of correlations in harmonic chains with pinning and a bulk velocity flip noise during the heat relaxation phase which occurs on diffusive timewhere L is the chain length. It has been shown earlier that for nondegenerate harmonic interactions these systems thermalize, and the dominant part of the correlations is given by local thermal equilibrium determined by a temperature profile which satisfies a linear heat equation. Here we are concerned with two new aspects about the thermalization process: the first order corrections in 1/L to the local equilibrium correlations and the applicability of kinetic theory to study the relaxation process. Employing previously derived explicit uniform estimates for the temperature profile, we first derive an explicit form for the first order corrections to the particle position-momentum correlations. By suitably revising the definition of the Wigner transform and the kinetic scaling limit we derive a phonon Boltzmann equation whose predictions agree with the explicit computation. Comparing the two results, the corrections can be understood as arising from two different sources: a currentrelated term and a correction to the position-position correlations related to spatial changes in the phonon eigenbasis.

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Section: Introductionmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

2016

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“…For more details related to these moments bounds, that are still conjectured but not proved (on the contrary to what is claimed in[12]), we refer the reader to an erratum which is available online at http://chercheurs.lille.inria.fr/masimon/erratum-v2.pdf.…”

mentioning

confidence: 99%

“…Harmonic chains with energy conserving random perturbations of the dynamics have recently received attention in the study of the macroscopic evolution of energy [1,2,5,8,10,12]. They provide models that have a non-trivial macroscopic behavior which can be explicitly computed.…”

mentioning

confidence: 99%

“…The presence of the non-linearity in the evolution of the energy makes the macroscopic limit non-trivial. Relative entropy methods (as introduced in [13]) identify correctly the limit equation (see [12]), but in order to make them rigorous one needs sharp bounds on higher moments than cannot be controlled by the relative entropy 1 . In this sense the proof in [12] is not complete.…”

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confidence: 99%

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Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

Komorowski¹,

Olla²,

Simon³

2018

Kinetic &Amp; Related Models

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We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.

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Diffusion of Energy in Chains of Oscillators with Conservative Noise

Bernardin¹

2015

Springer Proceedings in Mathematics &Amp; Statistics

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These notes are based on a mini-course given during the conference Particle systems and PDE's -II which held at the Center of Mathematics of the University of Minho in December 2013. We discuss the problem of normal and anomalous diffusion of energy in systems of coupled oscillators perturbed by a stochastic noise conserving energy.The goal of statistical mechanics is to elucidate the relation between the microscopic world and the macroscopic world. Equilibrium statistical mechanics assume the microscopic systems studied to be in equilibrium. In this course we will be concerned with non-equilibrium statistical mechanics where time evolution is taken into account: our interest will not only be in the relation between the microscopic and the macroscopic scales in space but also in time.By microscopic system we refer to molecules or atoms governed by the classical Newton's equations of motion. The question is then to understand how do these particles manage to organize themselves in such a way as to form a coherent structure on a large scale. The "structure" will be described by few variables (temperature, pressure . . . ) governed by autonomous equations (Euler's equations, Navier-Stokes's equation, heat equation . . . ). The microscopic specificities of the system will appear on this scale only through the thermodynamics (equation of state) and

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Hydrodynamic limit for the velocity-flip model

Abstract: International audienc

Cited by 13 publications

References 18 publications

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

Diffusion of Energy in Chains of Oscillators with Conservative Noise

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