On the local space-time structure of non-equilibrium steady states

Abstract

We consider the Gibbs representation over space-time of non-equilibrium dynamics of Hamiltonian systems defined on a lattice with local interactions. We first write the corresponding action functional as a sum of local terms, defining a local action functional. We replace the local system by a translation-invariant system whose dynamics has an identical space-time characterization. We study in details the irreversible properties of the new dynamics, define the local conductivity and show its equivalence with the Green-Kubo formula. Given the definition of the local heat conductivity and using conservation of energy, we derive the shape of the temperature profile. Next, we find an explicit formula for the non-equilibrium stationary measure of harmonic systems. Finally, we apply our scheme to various approximations of anharmonic Hamiltonian models, show how to compute their thermal conductivity and recover results confirmed in numerical simulations.

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References in corpus (14)

• A Gallavotti-Cohen Type Symmetry in the Large Deviation Functional for Stochastic Dynamics
• Thermal conduction in classical low-dimensional lattices
• Fluctuation theorem for stochastic dynamics
• The Fluctuation Theorem as a Gibbs Property
• Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
• A momentum conserving model with anomalous thermal conductivity in low dimension
• Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators
• Entropy Production in Non-Linear, Thermally Driven Hamiltonian Systems
• Fermi-Pasta-Ulam $\beta$ lattice: Peierls equation and anomalous heat conductivity
• Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics
• Energy Transport in Weakly Anharmonic Chains
• Fluctuations of the Entropy Production in Anharmonic Chains
• Normal Heat Conductivity in a strongly pinned chain of anharmonic oscillators
• On the derivation of Fourier's law for coupled anharmonic oscillators

Cited by in corpus (1)

• Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats