Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators
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Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators Angeliki Menegaki1 Received: 29 October 2019 / Accepted: 10 May 2020 © The Author(s) 2020
Abstract 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
1 Introduction 1.1 Description of the Model We consider a model for heat conduction consisting of a one-dimensional chain of N coupled oscillators. The evolution is a Hamiltonian dynamics with Hamiltonian
H ( p, q) =
1≤i≤N
N pi2 Uint (qi+1 − qi ), + Upin (qi ) + 2 i=0
Communicated by Stefano Olla.
B 1
Angeliki Menegaki [email protected] DPMMS, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
123
A. Menegaki
where ( p, q) belong in the phase space R2N and q0 , q N +1 describe the boundaries which here are considered to be fixed: q0 = q N +1 = 0. We denote by q = (q1 , . . . , q N ) ∈ R N the displacements of the atoms from their equilibrium positions and by p = ( p1 , . . . , p N ) ∈ R N the momenta. Each particle has its own pinning potential Upin and it also interacts with its nearest neighbors through an interaction potential Uint . Notice that here all the masses are equal and we take them m i = 1. So we consider a homogeneous chain, where both the masses and the potentials that act on each oscillator, are the same. The classical Hamiltonian dynamics is perturbed by noise and friction in the following way: the two ends of the chain are in contact with heat Langevin baths at two different temperatures TL , TR > 0. So our dynamics is described by the following system of SDEs: dqi (t) = pi (t)dt for i = 1, . . . , N , d pi (t) = (−∂qi H )dt for i = 2, . . . , N − 1, d p1 (t) = (−∂q1 H − γ1 p1 )dt + 2γ1 TL dW1 (t), d p N (t) = (−∂q N H − γ N p N )dt + 2γ N TR dW N (t)
(1.1)
where γi are the friction constants, Ti are the two temperatures and W1 , W N are two independent
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