 03344 Federico Bonetto, Joel L. Lebowitz, Jani Lukkarinen
 Fourier's Law for a Harmonic Crystal with Selfconsistent Stochastic Reservoirs
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Jul 24, 03

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Abstract. We consider a ddimensional 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 selfconsistently 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 GreenKubo 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*d) where l_d is
the coupling to the intermediate reservoirs. We also analyze the effect of
having a nonuniform distribution of the heat bath couplings. These results are
proven rigorously by controlling the behavior of the correlations in the
thermodynamic limit.
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