Hydrodynamic Limit for the Velocity-Flip Model

Simon, Marielle

doi:10.1007/978-3-642-54271-8_13

Cited by 8 publications

(25 citation statements)

References 13 publications

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“…In the present work we improve on the result proven in [12] in two ways. We prove that it is not necessary to assume that the initial state is close to an LTE state; indeed, our main theorem is applicable for arbitrary deterministic initial data, including those in which just one of the sites carries energy.…”

Section: Introductionmentioning

confidence: 75%

“…This was postulated in [8,9], based on earlier mathematical work on similar models by Bernardin and Olla (see e.g. [10,11]), and it was later proven by Simon in [12]. (Although the details are only given for the case without pinning, it is mentioned in the Remark after Theorem 1.2 that the proofs can be adapted to include interactions with pinning.)…”

Section: Introductionmentioning

confidence: 91%

“…The strategy for proving the hydrodynamic limit in [12] was based on relative entropy methods introduced by Yau [13] and Varadhan [14]; see also [15] and [16] for more references and details. There one begins by assuming that the initial state is close to a local thermal equilibrium (LTE) state, which allows for unique definition of the initial profiles of the hydrodynamic fields.…”

Section: Introductionmentioning

confidence: 99%

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Thermalization in Harmonic Particle Chains with Velocity Flips

Lukkarinen

2014

J Stat Phys

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We propose a new mathematical tool for the study of transport properties of models for lattice vibrations in crystalline solids. By replication of dynamical degrees of freedom, we aim at a new dynamical system where the "local" dynamics can be isolated and solved independently from the "global" evolution. The replication procedure is very generic but not unique as it depends on how the original dynamics are split between the local and global dynamics. As an explicit example, we apply the scheme to study thermalization of the pinned harmonic chain with velocity flips. We improve on the previous results about this system by showing that after a relatively short time period the average kinetic temperature profile satisfies the dynamic Fourier's law in a local microscopic sense without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the above thermalization period is at most of the order L 2/3 , where L denotes the number of particles in the chain. In particular, even before the diffusive time scale Fourier's law becomes a valid approximation of the evolution of the kinetic temperature profile. As a second application of the dynamic replica method, we also briefly discuss replacing the velocity flips by an anharmonic onsite potential.

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Section: Introductionmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Thermalization in Harmonic Particle Chains with Velocity Flips

Lukkarinen

2014

J Stat Phys

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“…We also remark here that another two-component system under the recent attention is the chain of harmonic oscillators [21,28]. For these models, two conserved quantities are the energy and the volume of the system.…”

Section: Introductionmentioning

confidence: 92%

Scaling Limit of Two-component Interacting Brownian Motions

Seo¹

2015

Preprint

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This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain form of singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and the interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a partial differential equation known as the Maxwell-Stefan equation.

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“…This is not a consequence of Theorem 2.1, because the relative entropy does not control the convergence of the energy. In the harmonic case it can be proven by using similar argument as in [6] (in fact in this case f N (t) is a gaussian distribution where we have control of any moments). Assuming (4.9), we have that Q N (t) converges, as N → ∞, to the deterministic…”

mentioning

confidence: 89%

From Particle Systems to Partial Differential Equations

Simon

2014

Springer Proceedings in Mathematics &Amp; Statistics

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We obtain macroscopic isothermal thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics in contact with a heat bath. The microscopic dynamics is given by a chain of anharmonic oscillators subject to a varying tension (external force) and the contact with the heat bath is modeled by independent Langevin dynamics acting on each particle. After a diffusive space-time scaling and cross-graining, the profile of volume converges to the solution of a deterministic diffusive equation with boundary conditions given by the applied tension. This defines an irreversible thermo-dynamic transformation from an initial equilibrium to a new equilibrium given by the final tension applied. Quasistatic reversible isothermal transformations are then obtained by a further time scaling. Heat is defined as the total flux of energy exchanged between the system and the heat bath. Then we prove that the relation between the limit heat, work, free energy and thermodynamic entropy agree with the first and second principle of thermodynamics.

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Hydrodynamic Limit for the Velocity-Flip Model

Cited by 8 publications

References 13 publications

Thermalization in Harmonic Particle Chains with Velocity Flips

Thermalization in Harmonic Particle Chains with Velocity Flips

Scaling Limit of Two-component Interacting Brownian Motions

From Particle Systems to Partial Differential Equations

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