Non-equilibrium steady states for networks of oscillators

Cuneo, Noé; Eckmann, Jean‐Pierre; Hairer, Martin; Rey-Bellet, Luc

doi:10.1214/18-ejp177

Cited by 28 publications

(31 citation statements)

References 28 publications

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“…Note that the MZ equation ( 4) and its projected form (5) for stochastic systems have the same structure as the classical MZ equations for deterministic (autonomous) systems [45,42,44]. However, the Liouville operator L is now replaced by the Kolmogorov operator K. Let us consider the weighted Hilbert space…”

Section: Mori-type Generalized Langevin Equations For Sdesmentioning

confidence: 99%

“…where K * is the L 2 (M )-adjoint operator of K. Throughout the paper, we further assume that ρ is a smooth function which decays to 0 when approaching boundary ∂M . For frequently used statistical physics models, these two assumptions are generally hard to prove since it involves the analysis of the degenerate elliptic operator K. Some recent studies on hypoellipticity [5,7] have shown the possibility to obtain affirmative answers using the rather complicated Hörmander analysis. These theoretical results, together with numerical studies such as [22,23], suggest that the uniqueness, smoothness and the decaying properties of ρ at ∂M are rather technical assumptions.…”

Section: Generalized Fluctuation-dissipation Theoremmentioning

confidence: 99%

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Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

Zhu,

Lei,

Kim

2021

Preprint

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In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and only uses the properties of the Kolmogorov operator. The new second FDT expresses the memory kernel of the generalized Langevin equation as the correlation function of the fluctuation force plus an additional term. In particular, we show that for nonequilibrium states such as heat transport between two thermostats with different temperatures, the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The obtained theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that the new model of heat transfer yields better predictions than the Green-Kubo formula for systems far from the equilibrium.

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Section: Mori-type Generalized Langevin Equations For Sdesmentioning

confidence: 99%

Section: Generalized Fluctuation-dissipation Theoremmentioning

confidence: 99%

Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

Zhu,

Lei,

Kim

2021

Preprint

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“…/ norm was constructed in which the dynamics contracts. Although structurally different but similar in spirit, we refer the reader to the papers [4,11,13] and the references therein for scenarios where Langevin dynamics produces subgeometric rates of convergence either due to absence of friction on certain particles or weak Poincaré inequalities being satisfied.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

Herzog

Mattingly

2019

Comm Pure Appl Math

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We study Langevin dynamics of N particles on ℝd interacting through a singular repulsive potential, e.g., the well‐known Lennard‐Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. © 2019 Wiley Periodicals, Inc.

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“…Regarding the existence, uniqueness of a non-equilibrium stationary state and exponential convergence towards it in more complicated networks of oscillators (multi-dimensional cases) see [12]. The proofs there are inspired by the above-mentioned works in the 1-dimensional chains.…”

Section: State Of the Artmentioning

confidence: 99%

Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

Menegaki

2020

J Stat Phys

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We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N −3. For the purely harmonic chain the order of the rate is in [N −3 , N −1 ]. This shows that, in this set up, the spectral gap decays at most polynomially with N. Keywords Chain of oscillators • Spectral gap • Hypocoercivity • Hypoellipticity • Functional inequalities • Nonequilibrium steady states • Heat bath • Exponential convergence N i=0 U int (q i+1 − q i), Communicated by Stefano Olla.

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Non-equilibrium steady states for networks of oscillators

Cited by 28 publications

References 28 publications

Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

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