Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

By Luc Rey-Bellet, L. Thomas. 2001

137 citations recorded by the source corpus.

Abstract: We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators.

Subfields: Computer Networks and Communications, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c884d4e289deef330ad89d08d9a3fbb68c847984

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In Statistical and Nonlinear Physics

Citations in the corpus

Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential (2004, background): “Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.”
Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations (2009, background): “There are however many examples (see for example [ RBT02 , Hai09]”
Couplings and quantitative contraction rates for Langevin dynamics (2017, methodology): “me explorations around the Gaussian case and the eect of hypoellipticity. There are only few articles which study the ergodic properties of Langevin processes using more probabilistic arguments, cf. [41, 46, 33, 39, 1, 5, 20]. Most of these results ultimately rely on arguments used in Harris’ type t”
Hypocoercive Diffusion Operators (2006, background): “Hypoellipticity and hypocoercivity are often found together, and have been actually studied together, by refined hypoelliptic techniques [6], [7], [15], [16], [19], and sometimes by probabilistic methods [8], [22], [23].”