An open microscopic model of heat conduction: evolution and non-equilibrium stationary states

Tomasz Komorowski (1) , Stefano Olla (2) , Marielle Simon (3)

1 PAN - Institut of Mathematics - Polish Academy of Sciences
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique

We consider a one-dimensional chain of coupled oscillators in contact at both ends with heat baths at different temperatures, and subject to an external force at one end. The Hamiltonian dynamics in the bulk is perturbed by random exchanges of the neighbouring momenta such that the energy is locally conserved. We prove that in the stationary state the energy and the volume stretch profiles, in large scale limit, converge to the solutions of a diffusive system with Dirichlet boundary conditions. As a consequence the macroscopic temperature stationary profile presents a maximum inside the chain higher than the thermostats temperatures, as well as the possibility of uphill diffusion (energy current against the temperature gradient). Finally, we are also able to derive the non-stationary macroscopic coupled diffusive equations followed by the energy and volume stretch profiles.

Domaines: Mathématiques [math]; Probabilités [math.PR]

Tomasz Komorowski, Stefano Olla, Marielle Simon. An open microscopic model of heat conduction: evolution and non-equilibrium stationary states. Communications in Mathematical Sciences, 2020, 18 (3), pp.751-780. ⟨10.4310/CMS.2020.v18.n3.a8⟩. ⟨hal-02081200v2⟩