**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

bonetto@math.gatech.edu

J.L. Lebowitz

Department of Mathematics and Physics,

Rutgers University

New Brunswick, NJ 08903

lebowitz@math.rutgers.edu

Jani Lukkarinen

Munich University of Technology

Centre for Mathematical Sciences, M5

Boltzmannstr. 3

D-85747 Garching b. Muenchen

Germany

E-mail: jlukkari@ma.tum.de