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

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Abstract

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 T l and T r . 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 (T l −T r ). In the NESS the heat conductivity κ is defined as the heat flux per unit area divided by the length of the system and (T l −T r ). In the limit when the temperatures of the external reservoirs go to the same temperature T, κ(T) is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This κ(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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References

  1. Basile, G., Bernardin, C., Olla, S.: A momentum conserving model with anomalous thermal conductivity in low dimension. Phys. Rev. Lett. 96, 204303 (2006)

    Article  ADS  Google Scholar 

  2. Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conserving model. Commun. Math. Phys. (2009). doi:10.1007/s10955-008-9657-1. arXiv.org:cond-mat/0601544v3

    Google Scholar 

  3. Benabou, G.: Homogenization of Ornstein-Uhlenbeck Process in Random Environment. Commun. Math. Phys. 266, 699–714 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Bergmann, P.G., Lebowitz, J.L.: New approach to nonequilibrium processes. Phys. Rev. 99, 578–587 (1955)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Bernardin, C., Olla, S.: Fourier’s law for a microscopic model of heat conduction. J. Stat. Phys. 121(3/4), 271–289 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Bodineau, T., Lefevere, R.: Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats. J. Stat. Phys. 133, 1–27 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bolsterli, M., Rich, M., Visscher, W.M.: Simulation of nonharmonic interactions in a crystal by self-consistent reservoirs. Phys. Rev. A 4, 1086–1088 (1970)

    Article  ADS  Google Scholar 

  8. Bonetto, F., Lebowitz, J.L., Lukkarinen, J.: Fourier’s law for a harmonic crystal with self-consistent stochastic reservoirs. J. Stat. Phys. 116, 783–813 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Fokas, A., et al. (eds.) Mathematical Physics, pp. 128–150. Imperial College Press, London (2000)

    Google Scholar 

  10. Fritz, J.: Stochastic dynamics of two-dimensional infinite-particle systems. J. Stat. Phys. 20(4), 351–379 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  11. Hairer, M.: A probabilistic argument for the controllability of conservative systems. Preprint arXiv.org:math-ph/0506064v2 (2005)

  12. Katz, S., Lebowitz, J.L., Spohn, H.: Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors. J. Stat. Phys. 34, 497–537 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  13. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, New York (1999)

    MATH  Google Scholar 

  14. Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65–74 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  15. Maes, C., Netočný, K., Verschuere, M.: Heat conduction networks. J. Stat. Phys. 111, 1219–1244 (2003)

    Article  MATH  Google Scholar 

  16. Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipshitz vector fields and degenerate noise. Stoch. Proc. Appl. 101, 185–232 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mattingly, J.C.: Private communication (2008)

  18. Olla, S., Tremoulet, C.: Equilibrium fluctuations for interacting Ornstein-Uhlenbeck particles. Commun. Math. Phys. 233, 463–491 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  19. Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)

    MATH  Google Scholar 

  20. Villani, C.: Hypercoercivity, preprint. URL http://www.umpa.ens-lyon.fr/~cvillani/Cedrif/pre.Hypoco.pdf (2006)

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    F. Bonetto

  2. Department of Mathematics and Physics, Rutgers University, Piscataway, NJ, USA

    J. L. Lebowitz

  3. Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747, Garching, Germany

    J. Lukkarinen

  4. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsingin yliopisto, Finland

    J. Lukkarinen

  5. Ceremade, UMR CNRS 7534, Université de Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    S. Olla

Authors
  1. F. Bonetto
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  2. J. L. Lebowitz
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  3. J. Lukkarinen
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  4. S. Olla
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Correspondence to S. Olla.

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Dedicated to Jürg Fröhlich and Tom Spencer with friendship and appreciation.

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Bonetto, F., Lebowitz, J.L., Lukkarinen, J. et al. Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs. J Stat Phys 134, 1097–1119 (2009). https://doi.org/10.1007/s10955-008-9657-1

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  • DOI: https://doi.org/10.1007/s10955-008-9657-1

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