Advertisement

Energy Transfer in a Fast-Slow Hamiltonian System

  • Dmitry Dolgopyat
  • Carlangelo LiveraniEmail author
Article

Abstract

We consider a finite region of a lattice of weakly interacting geodesic flows on manifolds of negative curvature and we show that, when rescaling the interactions and the time appropriately, the energies of the flows evolve according to a nonlinear diffusion equation. This is a first step toward the derivation of macroscopic equations from a Hamiltonian microscopic dynamics in the case of weakly coupled systems.

Keywords

Manifold Invariant Measure Negative Curvature Limit Equation Martingale Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Anosov D.V., Sinai Ja.G.: Certain smooth ergodic systems. Uspehi Mat. Nauk 22(5), 107–172 (1967)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aoki K., Lukkarinen J., Spohn H.: Energy transport in weakly anharmonic chains. J. Stat. Phys. 124(5), 1105–1129 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Basile G., Bernardin C., Olla S.: Thermal conductivity for a momentum conservative model. Commun. Math. Phys. 287(1), 67–98 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Basile G., Olla S., Spohn H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Rat. Mech. Anal. 195(1), 171–203 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bernardin C.: Thermal conductivity for a noisy disordered harmonic chain. J. Stat. Phys. 133(3), 417–433 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Mathematical Physics 2000, Imp. Coll. Press, London, 2000, pp. 128–150Google Scholar
  7. 7.
    Bricmont J., Kupiainen A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274(3), 555–626 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Bricmont J., Kupiainen A.: Random walks in space time mixing environments. J. Stat. Phys. 134(5-6), 979–1004 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Collet P., Eckmann J.-P.: A model of heat conduction. Commun. Math. Phys. 287(3), 1015–1038 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Croke C.B., Sharafutdinov V.A.: Spectral rigidity of a compact negatively curved manifold. Topology 37(6), 1265–1273 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dolgopyat D.: On decay of correlations in Anosov flows. Ann. Math. 147, 357–390 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dolgopyat D.: Averaging and Invariant measures. Moscow Math. J. 5, 537–576 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dolgopyat D., Keller G., Liverani C.: Random walk in Markovian environment. Ann. Probab. 36, 1676–1710 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dolgopyat D., Liverani C.: Random walk in deterministically changing environment. Lat. Am. J. Probab. Math. Stat. 4, 89–116 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dolgopyat D., Liverani C.: Non-perturbative approach to random walk in Markovian environment. Elect. Commun. Prob. 14, 245–251 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Eckmann J.-P., Young L.-S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262(1), 237–267 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Journé J.-L.: A regularity lemma for functions of several variables. Rev. Mat. Iberoamericana 4(2), 187–193 (1988)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Freidlin, M.I., Wentzell, A.D. Random perturbations of dynamical systems. Translated from the 1979 Russian original by Joseph Szcs. 2nd edition. Fundamental Principles of Mathematical Sciences, 260. New York: Springer-Verlag, 1998Google Scholar
  19. 19.
    Gaspard P., Gilbert T.: Heat conduction and Fourier’s law in a class of many particle dispersing billiards. New J. Phys. 10, 103004 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    Gaspard P., Gilbert T.: Heat conduction and Fourier’s law by consecutive local mixing and thermalization. Phys. Rev. Lett. 101, 020601 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved n-manifolds. Proc. Sympos. Pure Math., XXXVI. Providence, RI: Amer. Math. Soc., 1980, pp. 153–180Google Scholar
  22. 22.
    Guo M.Z., Papanicolaou G.C., Varadhan S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118(1), 31–59 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Hunt T.J., MacKay R.S.: Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity 16(4), 1499–1510 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Liverani C.: On Contact Anosov flows. Ann. of Math. 159(3), 1275–1312 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Liverani, C., Olla, S.: Toward the Fourier law for a weakly interacting anharmonic crystal. preprint http://arXiv.org/ans/1006.2900v1 [math.PR], 2010
  26. 26.
    Lukkarinen J., Spohn H.: Anomalous energy transport in the FPU-beta chain. Commun. Pure Appl. Math. 61(12), 1753–1786 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Paternain G.P.: Geodesic flows. Progress in Mathematics, 180. Boston, MA, Birkhäuser Boston, Inc. (1999)CrossRefGoogle Scholar
  28. 28.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edition. Fundamental Principles of Mathematical Sciences, 293. Springer-Verlag, Berlin (1999)Google Scholar
  29. 29.
    Ruelle, D.: A mechanical model for Fourier’s Law of the heat conduction http://arXiv.org/abs/1102.5488 [nlin.CD], 2011
  30. 30.
    Spohn H.: Large scale dynamics of interacting particles. Springer-Verlag, Berlin, New York (1991)zbMATHGoogle Scholar
  31. 31.
    Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions-II. Pitman Res. Notes Math. Ser. 283. Harlow: Longman Sci. Tech., 1993, pp. 75–128Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Dipartimento di MatematicaII Università di Roma (Tor Vergata), Via della Ricerca ScientificaRomaItaly

Personalised recommendations