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Thermal Conductivity for a Momentum Conservative Model

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Abstract

We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t −d/2 in the unpinned case and like t −d/2–1 if an on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.

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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., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. http://arxiv.org/abs/0805.3012, 2008

  3. 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 

  4. Bodineau, T., Helffer, B.: Correlations, Spectral gap and Log-Sobolev inequalities for unbounded spins systems. In: Differential equations and mathematical physics (Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., 16, Providence, RI: Amer. Math. Soc., 2000, pp 51–66

  5. 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 

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

  7. 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 

  8. Bricmont J., Kupianen A.: Towards a Derivation of Fouriers Law for Coupled Anharmonic Oscillators. Comm. Math. Phys. 274(3), 555–626 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Deuschel J.D., Delmotte T.: On estimating the derivative of symmetric diffusions in stationary random environment, with pplications to \({\nabla \phi}\) interface model. Prob. Theory Relat. Fields 133, 358–390 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Diaconis P., Freedman D.: A dozen de Finetti style results in search of a theory. Ann. Inst. H. Poincaré (Probabilités et Statistiques) 23, 397–423 (1987)

    MathSciNet  Google Scholar 

  11. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24. Amsterdam: North-Holland Publishing Co., 1989

  12. Jara, M., Komorowski, T., Olla, S.: Limit theorems for additive functionals of a Markov chain. http://arxiv.org/abs/0809.0177, 2008

  13. Lepri S., Livi R., Politi A.: Thermal Conduction in classical low-dimensional lattices. Phys Rep. 377, 1–80 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  14. Mangad M.: Asymptotic expansions of Fourier transforms and discrete polyharmonic Green’s functions. Pacific J. Math. 20, 85–98 (1967)

    MATH  MathSciNet  Google Scholar 

  15. Lebowitz J.L., Lieb E., Rieder Z.: Properties of harmonic crystal in a stationary non-equilibrium state. J. Math. Phys. 8, 1073–1078 (1967)

    Article  ADS  Google Scholar 

  16. Sethuraman S.: Central limit theorems for additive functionals of the simple exclusion process. Ann. Probab. 28(1), 277–302 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Spohn H.: Large Scale Dynamics of interacting Particles. Springer, Berlin-Heidelberg-New York (1991)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. WIAS, Mohrenstr. 39, 10117, Berlin, Germany

    Giada Basile

  2. Université de Lyon, CNRS (UMPA), Ecole Normale Supérieure De Lyon, 46, Allée D’Italie, 69364, Lyon Cedex 07, France

    Cédric Bernardin

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

    Stefano Olla

Authors
  1. Giada Basile
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  2. Cédric Bernardin
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  3. Stefano Olla
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Correspondence to Stefano Olla.

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Communicated by A. Kupiainen

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Basile, G., Bernardin, C. & Olla, S. Thermal Conductivity for a Momentum Conservative Model. Commun. Math. Phys. 287, 67–98 (2009). https://doi.org/10.1007/s00220-008-0662-7

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  • DOI: https://doi.org/10.1007/s00220-008-0662-7

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