Fourier’s Law for a Microscopic Model of Heat Conduction

Abstract

We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostat at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove for the stationary state, in the limit as N → ∞, the Fourier's law and the linear profile for the energy average.

Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

“…Related works in systems defined by dynamical local rules include [4,5,6,7,17]. For stochastic models, the collection of results is much larger, and we mention only some that are closer to this paper in spirit: [1,2,8,9,10,16,18,19,23,24,26,29].…”

Duality and exact correlations for a model of heat conduction

“…21,1 They later appeared as the high-energy limit of a chain with deterministic dynamics 9 and were also used as a stochastic perturbation ͑mimicking nonlinearities͒ of an oscillator chain. 3,8 In the present paper we construct for our energy diffusion model a dual process that expresses the evolution of the K-point correlation functions of the kinetic energies in terms of a process of K interacting random walkers. We also give a closed expression for the stationary covariance ͗x i 2 ; x j 2 ͘ which confirms the presence of long-range correlations in the nonequilibrium stationary state ͓as already found before in the SEP Ref.…”

Nonequilibrium Invariant Measure under Heat Flow

“…Later, purely stochastic models, where energy is assumed to diffuse between neighboring boxes (the oscillators), have been considered [4,5]. More recently, systems of harmonic oscillators exchanging energy with ''conservative'' noise have been proven to admit a unique stationary state with a constant heat flux and a linear temperature profile [6]. Admittedly, the leap from such class of models to even the simplest deterministic, nonlinear ones is still a challenge for the theory [7].…”