Role of conserved quantities in Fourier's law for diffusive mechanical systems

Abstract

Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space-time scale. In these situations the Fourier law depends also from the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamiltonian dynamics is perturbed by conservative stochastic terms. The presence of the stochastic dynamics allows to define the transport coefficients (thermal conductivity) and in some cases to prove the local equilibrium and the linear response argument necessary to obtain the diffusive equations governing the macroscopic evolution of the conserved quantities. Temperature profiles and other conserved quantities profiles in the non-equilibrium stationary states can be then understood from the non-stationary diffusive behaviour. We also review some results and open problems on the two step approach (by weak coupling or kinetic limits) to the heat equation, starting from mechanical models with only energy conserved. To cite this article: S. Olla, C. R. Physique X (2019).

Hydrodynamic limit for a chain with thermal and mechanical boundary forces

“…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]. Furthermore, because of the presence of other conservation laws, the stationary energy current can have the same sign as the gradient of the temperature -a phenomenon called an uphill diffusion in the literature.…”

Thermo-mechanical Transport in Rotor Chains

“…However, in many realistic systems, there are other conserved quantities besides energy, and the interplay among them has a deep impact on the system thermal properties, in particular when all these conserved quantities evolve on the macroscopic diffusive scale [21]. In the specific case of one dimensional rotor chains, the extra conserved quantity is angular momentum.…”

“…In order to derive this macroscopic description, we consider the stationary case (see [21] for the time-dependent problem under a space-time diffusive scaling), and rely on a local equilibrium assumption, that has been numerically verified in [12]. It is then possible to associate stationary profiles of temperature to stationary profiles of the conserved quantities (energy and angular momentum).…”

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

“…a non-acoustic harmonic chain with a random exchange of momentum as considered in [15], where the non-stationary hydrodynamic limit is proven. An attempt to describe more generally the systems for which the phenomena of an uphill energy diffusion and heating inside the system occur is made in [21].…”