TY - JOUR

T1 - Heat conduction and entropy production in anharmonic crystals with self-consistent stochastic reservoirs

AU - Bonetto, F.

AU - Lebowitz, J. L.

AU - Lukkarinen, J.

AU - Olla, S.

N1 - Funding Information:
Acknowledgements We thank Jonathan Mattingly and S.R.S. Varadhan for the help in the proof of the existence of the self-consistent profile. We also thank Herbert Spohn for useful discussions. The work of F. Bonetto was supported in part by NSF Grant DMS-060-4518, the work of J.L. Lebowitz was supported in part by NSF Grant DMR-044-2066 and by AFOSR Grant AF-FA 9550-04-4-22910, the work of J. Lukkarinen by Deutsche Forschungsgemeinschaft (DFG) project SP 181/19-2 and by the Academy of Finland, the work of S. Olla by ANR LHMSHE no. BLAN07-2 184264 (France).

PY - 2009/3

Y1 - 2009/3

N2 - We investigate a class of anharmonic crystals in d dimensions, d ≥ 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 TL and TR. 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 that it minimizes the entropy production to leading order in (TL - TR). In the NESS the heat conductivity k is defined as the heat flux per unit area divided by the length of the system and (TL - T R). In the limit when the temperatures of the external reservoirs go to the same temperature T, k(T) is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This k(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.

AB - We investigate a class of anharmonic crystals in d dimensions, d ≥ 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 TL and TR. 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 that it minimizes the entropy production to leading order in (TL - TR). In the NESS the heat conductivity k is defined as the heat flux per unit area divided by the length of the system and (TL - T R). In the limit when the temperatures of the external reservoirs go to the same temperature T, k(T) is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This k(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.

KW - Entropy production

KW - Green-kubo formula

KW - Nonequilibrium stationary states

KW - Self-consistent thermostats

KW - Thermal conductivity

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U2 - 10.1007/s10955-008-9657-1

DO - 10.1007/s10955-008-9657-1

M3 - Article

AN - SCOPUS:67349260760

VL - 134

SP - 1097

EP - 1119

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -