Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

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

Pages: 751 – 780



Tomasz Komorowski (Institute of Mathematics, Polish Academy Of Sciences, Warsaw, Poland)

Stefano Olla (Centre de Recherche en Mathématiques de la Décision (CEREMADE), Université Paris-Dauphine, Paris, France)

Marielle Simon (Laboratoire Paul Painlevé, Institut national de recherche en sciences et technologies du numérique (Inria), Université Lille, France)


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.


open chain of oscillators, heat conduction, non-equilibrium stationary state, uphill heat diffusion

2010 Mathematics Subject Classification

60K35, 82C70

Received 28 March 2019

Accepted 19 November 2019

Published 30 June 2020