Nonequilibrium stationary states of harmonic chains with bulk noises

Bernardin, Cédric; Kannan, Venkateshan; Lebowitz, Joel L.; Lukkarinen, Jani

doi:10.1140/epjb/e2011-20746-0

Cited by 11 publications

(15 citation statements)

References 12 publications

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“…2. The results indicate that the limit L → ∞ exists and agrees with κ ∞ = γ −1 /(2 + ω 2 0 + ω 0 ω 2 0 + 4) which is the value obtained in previous works on this model [8,9,12].…”

Section: Assumption 41supporting

confidence: 89%

“…(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.) Also the structure of steady state correlations and energy fluctuations are discussed in [9] with supporting numerical evidence presented in [8]. For a more general explanatory discussion about hydrodynamic fluctuation theory, we refer to a recent preprint by Spohn [2].…”

Section: Introductionmentioning

confidence: 55%

“…It was later proven that, unlike the self-consistent heat bath model, the velocity flip model satisfies also the dynamic Fourier's law. 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].…”

Section: Introductionmentioning

confidence: 90%

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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: Assumption 41supporting

confidence: 89%

Section: Introductionmentioning

confidence: 55%

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Thermalization in Harmonic Particle Chains with Velocity Flips

Lukkarinen

2014

J Stat Phys

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“…This guarantees that on the diffusive scale I (5) + I (6) = O(L −2 ). Putting together all the terms, we finally get the anticipated equation (3.13): is analytic on a neighbourhood of the real axis and, consequently, its inverse Fourier-transform is an exponentially decreasing function on Z.…”

Section: Derivation Of Equation (313)mentioning

confidence: 88%

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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“…For instance, if T L = 1 and T R = 8, the LTE contribution to the N → ∞ limit of s N := N H; H / H 2 can be rounded to 1.20, while simulations produce 1.40(2) consistently for both N = 200 and N = 400 (more results are given in [2]). A possible mechanism, yielding an approximate value 1.40 in this case, is explained in Section 5.…”

Section: Comparison Of Energy Fluctuations At the Nessmentioning

confidence: 99%

Energy fluctuations, hydrodynamics and local correlations in harmonic systems with bulk noises

Lukkarinen¹

2012

Phys. Scr.

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In this note, I summarise and comment on joint work with C. Bernardin, V. Kannan, and J. L. Lebowitz concerning two harmonic systems with bulk noises whose nonequilibrium steady states (NESS) are nearly identical (they share the same thermal conductivity and two-point function), but whose hydrodynamic properties (convergence towards the NESS) are very different. The goal is to discuss the results in the general context of nonequilibrium properties of dynamical systems, in particular, what they tell us about possible effective models, or predictive approximations, for such systems.

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Nonequilibrium stationary states of harmonic chains with bulk noises

Cited by 11 publications

References 12 publications

Thermalization in Harmonic Particle Chains with Velocity Flips

Thermalization in Harmonic Particle Chains with Velocity Flips

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

Energy fluctuations, hydrodynamics and local correlations in harmonic systems with bulk noises

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