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# Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs

Abstract : We investigate a class of anharmonic crystals in $d$ dimensions, $d\ge 1$, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths, applied at the boundaries in the $1$-direction, are at specified, unequal, temperatures $\tlb$ and $\trb$. The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). We prove the existence of such a stationary self-consistent profile of temperatures for a finite system and show it minimizes the entropy production to leading order in $(\tlb -\trb)$. In the NESS the heat conductivity $\kappa$ is defined as the heat flux per unit area divided by the length of the system and $(\tlb -\trb)$. In the limit when the temperatures of the external reservoirs goes to the same temperature $T$, $\kappa(T)$ is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature $T$. This $\kappa(T)$ remains bounded as the size of the system goes to infinity. We also show that the corresponding infinite system Green-Kubo formula yields a finite result. Stronger results are obtained under the assumption that the self-consistent profile remains bounded.
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https://hal.archives-ouvertes.fr/hal-00318755
Contributor : Stefano Olla Connect in order to contact the contributor
Submitted on : Friday, November 21, 2008 - 4:44:15 PM
Last modification on : Tuesday, January 18, 2022 - 3:24:29 PM
Long-term archiving on: : Wednesday, September 22, 2010 - 10:49:45 AM

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### Citation

Federico Bonetto, Joel L. Lebowitz, Jani Lukkarinen, Stefano Olla. Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs. Journal of Statistical Physics, 2009, 134 (5 (2009)), pp.1097. ⟨10.1007/s10955-008-9657-1⟩. ⟨hal-00318755v2⟩

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