Stationary nonequilibrium properties for a heat conduction model.

We consider a stochastic heat conduction model for solids composed of N interacting atoms. The system is in contact with two heat baths at different temperatures Tl and Tr. The bulk dynamics conserves two quantities: the energy and the deformation between atoms. If Tl not equal to Tr, a heat flux occurs in the system. For large N, the system adopts a linear temperature profile between Tl and Tr. We establish the hydrodynamic limit for the two conserved quantities. We introduce the fluctuation field of the energy and of the deformation in the nonequilibrium 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 nonequilibrium stationary state. A variational formula for the rate function is derived following the recent macroscopic fluctuation theory of Bertini [J. Stat. Phys. 107, 635 (2002); Math. Phys., Anal. Geom. 6, 231 (2003); J. Stat. Phys. 121, 843 (2005)].

• Publication year

  2008

• Venue

  Physical review. E, Statistical, nonlinear, and soft matter physics

• Publication date

  2008-08-05

• Fields of study

  Medicine, Physics, Mathematics

• Identifiers
  DOI 10.1103/PhysRevE.78.021134arXiv 0808.0662PMID 18850813
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar, PubMed

• No claims are published for this paper.

• No concepts are published for this paper.

• Stochastic Processes

  2018cited by this paper
• Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise

  2007cited by this paper
• Stochastic interacting particle systems out of equilibrium

  2007cited by this paper
• Equilibrium Fluctuations for a System of Harmonic Oscillators with Conservative Noise

  2006cited by this paper
• Large Deviations for a Stochastic Model of Heat Flow

  2005cited by this paper
• Fourier’s Law for a Microscopic Model of Heat Conduction

  2005cited by this paper
• Momentum conserving model with anomalous thermal conductivity in low dimensional systems.

  2005cited by this paper
• Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States

  2001cited by this paper
• Scaling Limits of Interacting Particle Systems

  1998influential reference
• Navier-Stokes equations for stochastic lattice gases.

  1996cited by this paper
• Ergodicity in infinite Hamiltonian systems with conservative noise

  1996cited by this paper
• Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals

  1995cited by this paper
• Stationary states of random Hamiltonian systems

  1994cited by this paper
• Hydrodynamical limit for a Hamiltonian system with weak noise

  1993cited by this paper
• Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state

  1991cited by this paper
• Continuous martingales and Brownian motion

  1990cited by this paper
• Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

  1990cited by this paper
• Long range correlations for stochastic lattice gases in a non-equilibrium steady state

  1983cited by this paper
• Heat flow in an exactly solvable model

  1982cited by this paper
• What is a Stochastic Process

  1942influential reference

• Graph-Dynamics correspondence in metallic glass-forming liquids

  2025cites this paper
• Large Deviations and Additivity Principle for the Open Harmonic Process

  2023cites this paper
• Non-equilibrium Stationary Properties of the Boundary Driven Zero-Range Process with Long Jumps

  2022cites this paper
• Microscopic and macroscopic perspectives on stationary nonequilibrium states

  2019cites this paper
• Totally Asymmetric Limit for Models of Heat Conduction

  2017cites this paper
• Totally Asymmetric Limit for Models of Heat Conduction

  2016cites this paper
• Correlations of the density and of the current in non-equilibrium diffusive systems

  2016cites this paper
• Long range correlations in stochastic transport with energy and momentum conservation

  2015cites this paper
• Diffusion of Energy in Chains of Oscillators with Conservative Noise

  2015cites this paper
• Macroscopic fluctuation theory

  2014cites this paper
• Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

  2014cites this paper
• Diffusion of energy in chains of oscillators with bulk noise

  2014cites this paper
• Green-Kubo Formula for Weakly Coupled Systems with Noise

  2014cites this paper
• Green-Kubo formula for weakly coupled system with dynamical noise

  2013cites this paper
• Harmonic Systems with Bulk Noises

  2011cites this paper
• Nonequilibrium stationary states of harmonic chains with bulk noises

  2011cites this paper
• Hydrodynamic behavior of boundary driven stochastic lattice gas models and interacting particle systems with conductances in random environments

  2010influential citation
• Dynamical Large Deviations for a Boundary Driven Stochastic Lattice Gas Model with Many Conserved Quantities

  2009cites this paper
• Thermal conductivity for a chain of anharmonic oscillators perturbed by a conservative noise

  2009cites this paper
• Hydrodynamic Limit for a Boundary Driven Stochastic Lattice Gas Model with Many Conserved Quantities

  2009cites this paper