DIFFUSION LIMIT FOR A KINETIC EQUATION WITH A THERMOSTATTED INTERFACE

Giada Basile (1) , Tomasz Komorowski (2) , Stefano Olla (3)

1 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]
2 PAN - Institut of Mathematics - Polish Academy of Sciences
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature T in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution ρ(t, y) of a heat equation with the boundary condition ρ(t, 0) ≡ T .

Mots clés: Diffusion limit; thermostatted boundary conditions

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

Giada Basile, Tomasz Komorowski, Stefano Olla. DIFFUSION LIMIT FOR A KINETIC EQUATION WITH A THERMOSTATTED INTERFACE. 2019. ⟨hal-02059831v1⟩