FM 2003-12; arXiv math.DS/0310009;

Author: Federico Bonetto, Joel Lebowitz and Jani Lukkarinen

Title :Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs

Abstract: We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ``exterior'' left and right heat baths are at specified values T_L and T_R, respectively, while the temperatures of the ``interior'' baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich and Visscher in 1970 who first studied this model. In the other limit, d >> 1, the stationary infinite volume heat conductivity behaves as 1/(l d) where l is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

Keywords: Fourier's law; harmonic crystal; non-equilibrium systems; thermodynamic limit; Green-Kubo formula.

F. Bonetto
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332

J.L. Lebowitz
Department of Mathematics and Physics,
Rutgers University
New Brunswick, NJ 08903

Jani Lukkarinen
Munich University of Technology
Centre for Mathematical Sciences, M5
Boltzmannstr. 3
D-85747 Garching b. Muenchen