Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs
Bonetto, Federico; Olla, Stefano; Lukkarinen, Jani; Lebowitz, Joel L. (2009), Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs, Journal of Statistical Physics, 134, 5, p. 1097. http://dx.doi.org/10.1007/s10955-008-9657-1
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00318755/en/
Journal nameJournal of Statistical Physics
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School of Mathematics - Georgia Institute of Technology
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lebowitz, Joel L.
Center for Mathematical Sciences Research
Abstract (EN)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.
Subjects / Keywordsnon-equilibrium stationary state; entropy production; self-consistent thermostats; Green-Kubo formula; Thermal condutivity
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