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

Abstract

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.

Hydrodynamic limit for a chain with thermal and mechanical boundary forces

“…Besides the aforementioned entropy production estimates, in order to obtain the hydrodynamic limit, one needs to establish also the property of equipartition of the random fluctuations of the mechanical and thermal components of the microscopic energy density, which was postulated in [11, identity (A.46)]. As we have pointed out in [11] this property seems to be out of reach of the relative entropy method and some other approach to resolve the difficulty is needed. In the present work we employ the Wigner distribution method to give a rigorous prove of the hydrodynamic limit for an open system with a random flip of momenta, see Section 2 below for its precise formulation.…”

“…In fact the random flip of the velocity signs does not move the energy in the system and the energy transport is entirely due to the hamiltonian part of the dynamics, that is very hard to control. This difficulty forced us to consider a different energy conserving random dynamics, where kinetic energy is exchanged between nearest neighbor particles by a continuous random mechanism, see [11]. In this case the stochastic dynamics is also responsible for energy transport.…”

“…in mechanical (but not thermal) equilibrium. In [11] the validity of the Fourier law for the NESS is extended also to the situation when an external tension force is present (then the system is in both mechanical and thermal non-equilibrium). Furthermore, the existence of both stationary macroscopic profiles for the temperature and volume stretch, at least in some situations, are established in [11].…”

“…In [11] the validity of the Fourier law for the NESS is extended also to the situation when an external tension force is present (then the system is in both mechanical and thermal non-equilibrium). Furthermore, the existence of both stationary macroscopic profiles for the temperature and volume stretch, at least in some situations, are established in [11]. In particular, the temperature profile has the interesting feature that the stationary temperatures in the bulk can be higher than at the boundaries, a general behavior conjectured in the NESS for many systems [13].…”

“…Concerning the hydrodynamic limit, in the appendix section of [11] we have formulated a heuristic argument, based on entropy production estimates, that has not been proved there, and that substantiated the validity of the macroscopic equations governing the dynamics in the case of a random momentum exchange microscopic model, see Section 2.2 of [11]. Besides the aforementioned entropy production estimates, in order to obtain the hydrodynamic limit, one needs to establish also the property of equipartition of the random fluctuations of the mechanical and thermal components of the microscopic energy density, which was postulated in [11, identity (A.46)].…”