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Towards a derivation of fourier's law for coupled anharmonic oscillators

Bibliographic reference Bricmont, Jean ; Kupiainen, Antti. Towards a derivation of fourier's law for coupled anharmonic oscillators. In: Communications in Mathematical Physics, Vol. 274, no. 3, p. 555-626 (2007)
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  1. Aoki K., Kusnezov D. (2002). Nonequilibrium statistical mechanics of classical lattice φ 4 field theory. Ann. Phys. 295: 50–80
  2. Aoki K., Lukkarinen J., Spohn H. (2006). Energy Transport in Weakly Anharmonic Chains. J. Stat. Phys. 124: 1105–1129
  3. Bernardin C., Olla S. (2005). Fourier’s law for a microscopic model of heat conduction. J. Stat. Phys. 121: 271–289
  4. Basile G., Bernardin C., Olla S. (2006). A momentum conserving model with anomalous thermal conductivity in low dimension. Phys. Rev. Lett. 96: 204303
  5. Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conserving model., 2006
  6. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier Law: A challenge to Theorists. In: Mathematical Physics 2000, London: Imp. Coll. Press, 2000, pp. 128–150
  7. Bonetto F., Lebowitz J.L., Lukkarinen J. (2004). Fourier’s Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs. J. Stat. Phys. 116: 783–813
  8. Cercignani C., Kremer G.M. (1999). On relativistic collisional invariants. J. Stat. Phys. 96: 439–445
  9. Eckmann J.-P., Pillet C.-A., Rey-Bellet L. (1999). Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201: 657–697
  10. Eckmann J.-P., Pillet C.-A., Rey-Bellet L. (1999). Entropy production in non-linear, thermally driven Hamiltonian systems. J. Stat. Phys. 95: 305–331
  11. Eckmann J.-P., Hairer M. (2000). Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212: 105–164
  12. Eckmann J.-P., Hairer M. (2003). Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235: 233–253
  13. Eckmann, J.-P.: Non-equilibrium steady states. In: Proceedings of the International Congress of Mathematicians, Beijing, Vol. III, Beijing: Higher Education Press, 2002, pp. 409–418
  14. Eckmann J.-P., Young L.-S. (2004). Temperature profiles in Hamiltonian heat conduction. Europhys. Lett. 68: 790–796
  15. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. To appear in Commun. Math Phys
  16. Esposito, R., Pulvirenti, M.: From particles to fluids. In: Friedlander S., Serre D. (eds) Handbook of Mathematical Fluid Dynamics, Vol. III, Amsterdam: Elsevier Science, 2004
  17. Galves A., Kipnis C., Marchioro C., Presutti E. (1981). Nonequilibrium measures which exhibit a temperature gradient; study of a model. Commun. Math. Phys. 81: 127–147
  18. Golse F., Sant-Raymond L. (2004). The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155: 81–161
  19. Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with high degree potential. Arch. Rat. Mech. Anal. 171, 151, 218 (2004)
  20. Kipnis C., Marchioro C., Presutti E. (1982). Heat flow in an exactly solvable model. J. Stat. Phys. 27: 65–74
  21. Lefevere, R., Schenkel, A.: Normal heat conductivity in a strongly pinned chain of anharmonic oscillators. J. Stat. Mech., L02001 (2006), available on:, 2006
  22. Lepri S., Livi R., Politi A. (2003). Thermal conductivity in classical low-dimensional lattices. Phys. Reports 377: 1–80
  23. Lepri S., Livi R., Politi A. (1998). On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43: 271
  24. Narayan O., Ramaswamy S. (2002). Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89: 200601
  25. Pereverzev A. (2003). Fermi-Pasta-Ulam β lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E. 68: 056124
  26. Rey-Bellet L., Thomas L.E. (2000). Asymptotic behavior of thermal non-equilibrium steady states for a driven chain of anharmonic oscillators. Commun. Math. Phys. 215: 1–24
  27. Rey-Bellet L., Thomas L.E. (2002). Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225: 305–329
  28. Rey-Bellet, L.: Nonequilibrium statistical mechanics of open classical systems. In: XIVTH International Congress on Mathematical Physics, edited by Jean-Claude Zambrini, Singapore: World Scientific, 2006
  29. Rieder Z., Lebowitz J.L., Lieb E. (1967). Properties of a harmonic crystal in a stationary non-equilibrium state. J. Math. Phys. 8: 1073–1085
  30. Spohn H., Lebowitz J.L. (1977). Stationary non-equilibrium states of infinite harmonic systems. Commun. Math. Phys. 54: 97–120
  31. Spohn H. (2006). The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124: 1041–1104
  32. Spohn H. (2006). Collisional invariants for the phonon Boltzmann equation. J. Stat. Phys. 124: 1131–1135
  33. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Oxford: Clarendon Press, 1948
  34. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D., (eds.) Handbook of Mathematical Fluid Dynamics, Vol. I. Amsterdam: Elsevier Science, 2002