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A Mechanical Model for Fourier’s Law of Heat Conduction

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Abstract

Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier’s law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier’s law can proceed rigorously.

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References

  1. Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Introduction to nonequilibrium quantum statistical mechanics. In: Open Quantum Systems III Recent developments. Lecture Notes in Mathematics 1882, Berlin: Springer, 2006, pp. 1–66

  2. Baladi, V., Smania, D.: Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps. To be published, available at http://www.mitlag-letter.se/preprints/files/IML-0910503.pdf, 2010

  3. Bonatti C., Diaz L., Viana M.: Dynamics beyond uniform hyperbolicity. Springer, Berlin (2005)

    MATH  Google Scholar 

  4. Bonetto, F., Lebowitz, J., Rey-Bellet, L.: Fourier’s law: a challenge for theorists. In: Mathematical Physics 2000, Fokas A., Grigoryan A., Kibble T., Zegarlinsky B. (eds), Imperial College, London, 2000, pp. 128–150

  5. 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  MathSciNet  ADS  MATH  Google Scholar 

  6. Bonetto F., Lebowitz J.L., Lukkarinen J., Olla S.: The heat conduction and entropy production in anharmonic crystals with self-consistent stochastic reservoirs. J. Stat. Phys. 134, 1097–1119 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Butterley O., Liverani C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dolgopyat D.: Decay of correlations in Anosov flows. Ann. of Math. 147, 357–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolgopyat, D.: Prevalence of rapid mixing in hyperbolic flows. Ergod. Th. and Dynam. Syst. 18, 1097-1114 (1998); Prevalence of rapid mixing-II: topological prevalence. Ergod. Th. and Dynam. Syst. 20, 1045–1059 (2000)

    Google Scholar 

  10. Dolgopyat D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Dolgopyat, D., Liverani, C.: Energy transfer in a fast-slow Hamiltonian system. Preprint, available at http://arxiv.org/abs/1010.3972v1 [math-ph], 2010

  12. Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two baths at different temperatures. Commun. Math. Phys. 201, 657–697 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Eckmann J.-P., Young L.-S.: Temperature profiles in Hamiltonian heat conduction. Europhys. Lett. 68, 790–796 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Eckmann J.-P., Young L.-S.: Nonequilibrium energy profiles in Hamiltonian for a class of 1-D models. Commun. Math. Phys. 262, 237–267 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Evans D.J., Morriss G.P.: Statistical mechanics of nonequilibrium fluids. Academic Press, New York (1990)

    Google Scholar 

  16. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Letters 74, 2694–2697 (1995); Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995)

    Google Scholar 

  17. Hirsch, M., Pugh, C.C., Shub, M.: Invariant manifolds. Lect. Notes in Math. 583, Berlin: Springer, 1977

  18. Hoover, W.G.: Molecular dynamics. Lecture Notes in Physics 258. Heidelberg: Springer, 1986

  19. Katok A., Knieper G., Pollicott M., Weiss H.: Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98, 581–597 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Pesin Ya.B., Sinai Ya.G.: Gibbs measures for partially hyperbolic attractors. Ergod. Th. and Dynam. Syst. 2, 417–438 (1982)

    MathSciNet  MATH  Google Scholar 

  21. Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997); Correction and complements. Commun. Math. Phys. 234, 185–190 (2003)

    Google Scholar 

  22. Ruelle D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95, 393–468 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Ruelle D.: Differentiation of SRB states for hyperbolic flows. Ergod. Theor. Dynam. Syst. 28, 613–631 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ruelle D.: Singularities of the susceptibility of an SRB measure in the presence of stable-unstable tangencies. Phil. Trans. R. Soc. A 369, 482–493 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Kampen N.G.: The case against linear response theory. Phys. Norv. 5, 279–284 (1971)

    Google Scholar 

  26. Young L.-S.: What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 108, 733–754 (2002)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    David Ruelle

  2. IHES, 91440, Bures sur Yvette, France

    David Ruelle

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  1. David Ruelle
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Correspondence to David Ruelle.

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Communicated by G. Gallavotti

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Ruelle, D. A Mechanical Model for Fourier’s Law of Heat Conduction. Commun. Math. Phys. 311, 755–768 (2012). https://doi.org/10.1007/s00220-011-1304-z

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  • DOI: https://doi.org/10.1007/s00220-011-1304-z

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