Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation

Abstract

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.

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“…is asymptotically similar to k 2 as k → 0. The model was later modified in [17] by replacing Gaussian noise with a Ornstein-Uhlenbeck (OU) type random field, which is space-time stationary Gaussian, Markovian and self-correlated. In the Ornstein-Uhlenbeck process, the temporal and spatial correlations are determined by two functions, denoted by σ(k)…”

“…However in [17], this assumption is not possible for mathematical reasons: γ must be separated from zero because of the way the process is constructed. Our approach based on the innovative construction of the OU process via the transition probability function opens the possibility of meeting the assumption (13).…”

“…It exists in a Hilbert space, but has no transition probability function in the strict sense and no Markov semigroup on a space of functions defined pointwise. In [17], it is defined by the covariance function…”

“…and, for u ∈ l 2 (Z), it is defined as the extension of (17) to the isometry between l 2 (Z) and L e 2 (1), here 1 ≡ 1 on T. The inverse Fourier transform F −1 f of a function f integrable on T will be denoted by f : fz = T e 2πikz f (k)dk, z ∈ Z.…”

“…Subsequently, in [16], a direct transition from the microscopic model to the macroscopic superdiffusion equation was made. In [17], the same authors modified the stochastic perturbation, introduced in [12], by replacing the Gaussian noise with the Ornstein-Uhlenbeck process, then obtained the linear Boltzmann equation at the mesoscopic scale.…”