Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

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 TLand TR, 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 (ldd)−1where ldis 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.

• Publication year

  2003

• Venue

  Journal of Statistical Physics

• Publication date

  2003-07-17

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1023/B:JOSS.0000037232.14365.10arXiv math-ph/0307035
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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  1991influential reference
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  1989cited by this paper
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  1987cited by this paper
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  1975cited by this paper
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  1970cited by this paper
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