On the relaxation rate of short chains of rotors interacting with Langevin thermostats

Cuneo, Noé; Poquet, Christophe

doi:10.1214/17-ecp62

Cited by 11 publications

(10 citation statements)

References 17 publications

(41 reference statements)

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“…One respect in which our problem differs from this preceding work is that we do not deal with small coupling, but with high frequencies. It seems this problem is not readily transformed into one with small coupling, but is rather close to ideas having to do with breathers, isolated high-energy states in extended systems [35,36,21,22,12,11,13].…”

Section: Introductionmentioning

confidence: 94%

Energy dissipation in Hamiltonian chains of rotators

2017

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We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.

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Section: Introductionmentioning

confidence: 94%

Energy dissipation in Hamiltonian chains of rotators

2017

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“…If T R = T L or τ L = τ R , the stationary probability measure cannot be computed explicitly. In fact even the existence of an invariant probability measure is an open problem for chains of lengths greater than 4 (see [4][5][6]). In what follows, we assume the existence and uniqueness of the stationary state and the expectation with respect to the stationary probability measure is denoted by • N ,ss .…”

Section: The Stationary Statementioning

confidence: 99%

Thermo-mechanical Transport in Rotor Chains

2021

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We study the macroscopic profiles of temperature and angular momentum in the stationary state of chains of rotors under a thermo-mechanical forcing applied at the boundaries. These profiles are solutions of a system of diffusive partial differential equations with boundary conditions determined by the thermo-mechanical forcing. Instead of expensive Monte Carlo simulations of the underlying microscopic dynamics, we perform extensive numerical computations based on a finite difference method for the system of partial differential equations describing the macroscopic steady state. We first present a formal derivation of these stationary equations based on a linear response argument and local equilibrium assumptions. We then study various properties of the solutions of these equations. This allows to characterize the regime of parameters leading to uphill energy diffusion-a situation in which the energy flows in the direction of the gradient of temperature-and to identify regions of parameters corresponding to a negative energy conductivity (i.e. a positive linear response of the energy current to a gradient of temperature). The macroscopic equations we derive are consistent with some previous results obtained by numerical simulation of the microscopic physical system, which confirms their validity. KeywordsThermal transport • Rotor chain • Thermo-mechanical response • Macroscopic diffusion equations • Uphill diffusion B Alessandra Iacobucci

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“…The difference with the Langevin heat baths is that the dissipation and the noise act on the momenta only indirectly through some auxiliary variables. Finally let us mention that the relaxation rates have been studied for short chains of rotors with Langevin thermostats in [11,13].…”

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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On the relaxation rate of short chains of rotors interacting with Langevin thermostats

Cited by 11 publications

References 17 publications

Energy dissipation in Hamiltonian chains of rotators

Energy dissipation in Hamiltonian chains of rotators

Thermo-mechanical Transport in Rotor Chains

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

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