Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat

Tomasz Komorowski (1) , Stefano Olla (2)

1 PAN - Institut of Mathematics - Polish Academy of Sciences
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat attached at the origin and energy, momentum and volume conserving noise that models the collisions between atoms. The noise is rarefied in the limit, that corresponds to the hypothesis that in the macroscopic unit time only a finite number of collisions takes place (Boltzmann-Grad limit). We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function W (t, y, k) that evolves according to a linear kinetic equation, with the interface condition at y = 0 that corresponds to reflection, transmission and absorption of phonons. The paper extends the results of [3], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.

Mots clés: Harmonic chains with stochastic noise; Wigner functions; Linear kinetic equation with interface; Duhamel formula

Domaines: Mécanique statistique [cond-mat.stat-mech]; Physique mathématique [math-ph]; Probabilités [math.PR]; Physique [physics]

Tomasz Komorowski, Stefano Olla. Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat. Journal of Functional Analysis, 2020, 279, pp.108764. ⟨10.1016/j.jfa.2020.108764⟩. ⟨hal-02305797⟩