Stationary non-equilibrium properties for a heat conduction model

Abstract

We consider a stochastic heat conduction model for solids composed by N interacting atoms. The system is in contact with two heat baths at different temperature $T_{\ell }$ and $T_r$. The bulk dynamics conserve two quantities: the energy and the deformation between atoms. If $T_{\ell }\ne T_r$, a heat flux takes place in the system. For large $N$, the system adopts a linear temperature profile between $T_{\ell }$ and $T_r$. We establish the hydrodynamic limit for the two conserved quantities. We introduce the fluctuations field of the energy and of the deformation in the non-equilibrium steady state. As $N$ goes to infinity, we show that this field converges to a Gaussian field and we compute the limiting covariance matrix. The main contribution of the paper is the study of large deviations for the temperature profile in the non-equilibrium stationary state. A variational formula for the rate function is derived following the recent macroscopic fluctuation theory of Bertini et al.

References in corpus (4)

• Macroscopic fluctuation theory for stationary non equilibrium states
• A momentum conserving model with anomalous thermal conductivity in low dimension
• Stochastic interacting particle systems out of equilibrium
• Large deviations for a stochastic model of heat flow

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• Long range correlations in stochastic transport with energy and momentum conservation
• Hydrodynamic limit for a boundary driven stochastic lattice gas model with many conserved quantities
• Totally asymmetric limit for models of heat conduction
• Green-Kubo formula for weakly coupled system with dynamical noise
• Graph-Dynamics correspondence in metallic glass-forming liquids
• Dynamical large deviations for a boundary driven stochastic lattice gas model with many conserved quantities
• Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps
• Microscopic and macroscopic perspectives on stationary nonequilibrium states
• Diffusion of energy in chains of oscillators with bulk noise