Advertisement

Communications in Mathematical Physics

, Volume 292, Issue 1, pp 131–177 | Cite as

How Hot Can a Heat Bath Get?

  • Martin Hairer
Article

Abstract

We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme’ non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence/non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state.

Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.

Keywords

Invariant Measure Heat Bath Integrability Property Exponential Convergence Invariant Probability Measure 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BCG08.
    Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254(3), 727–759 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bon69.
    Bony, J.-M.: Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19 , no. fasc. 1, 277–304 xii (1969)Google Scholar
  3. Car07.
    Carmona P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stoch. Process. Appl. 117(8), 1076–1092 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. CGGR08.
    Cattiaux, P., Gozlan, N., Guillin, A., Roberto, C.: Functional inequalities for heavy tails distributions and application to isoperimetry. http://arxiv.org/abs/0807.3112v1[math.PR], 2008
  5. CGWW07.
    Cattiaux, P., Guillin, A., Wang, F.-Y., Wu, L.: Lyapunov conditions for logarithmic Sobolev and super Poincaré inequality, http://arxiv.org/abs/0712.0235[math.PR], 2007
  6. DFG06.
    Douc, R., Fort, G., Guillin, A.: Subgeometric rates of convergence of f-ergodic strong Markov processes, http://arxiv.org/abs/math/0605791v1[math.ST], 2006
  7. DMP+07.
    DeVille, R.E.L., Milewski, P.A., Pignol, R.J., Tabak, E.G., Vanden-Eijnden, E.: Nonequilibrium statistics of a reduced model for energy transfer in waves. Comm. Pure Appl. Math. 60(3), 439–461 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. DPZ96.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems, Vol. 229 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1996Google Scholar
  9. DV01.
    Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54(1), 1–42 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. EH00.
    Eckmann J.-P., Hairer M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. EH03.
    Eckmann J.-P., Hairer M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235(2), 233–253 (2003)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. EPR99a.
    Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Entropy production in nonlinear, thermally driven Hamiltonian systems. J. Statist. Phys. 95(1-2), 305–331 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. EPR99b.
    Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201(3), 657–697 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. FR05.
    Fort G., Roberts G.O.: Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15(2), 1565–1589 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. Hai05.
    Hairer, M.: A probabilistic argument for the controllability of conservative systems. http://arxiv.org/abs/math-ph/0506064v2, 2005
  16. HM08a.
    Hairer, M., Mattingly, J.: Slow energy dissipation in anharmonic oscillator chains. http://arxiv.org/abs/0712.3889v2[math-ph], 2009
  17. HM08b.
    Hairer, M., Mattingly J.: Yet another look at Harris’ ergodic theorem for Markov chains. http://arxiv.org/abs/0810.2777v1[math.PR], 2008
  18. HN04.
    Hérau F., Nier F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Rat. Mech. Anal. 171(2), 151–218 (2004)zbMATHCrossRefGoogle Scholar
  19. HN05.
    Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Vol. 1862 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2005Google Scholar
  20. Hör67.
    Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Hör85.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. III, Vol. 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1985Google Scholar
  22. MA94.
    MacKay R.S., Aubry S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7(6), 1623–1643 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. MT93.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communications and Control Engineering Series. London: Springer-Verlag London Ltd., 1993Google Scholar
  24. MTVE02.
    Milewski P.A., Tabak E.G., Vanden-Eijnden E.: Resonant wave interaction with random forcing and dissipation. Stud. Appl. Math. 108(1), 123–144 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  25. RT00.
    Rey-Bellet L., Thomas L.E.: Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Commun. Math. Phys. 215(1), 1–24 (2000)zbMATHCrossRefADSMathSciNetGoogle Scholar
  26. RT02.
    Rey-Bellet L., Thomas L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225(2), 305–329 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  27. RW01.
    Röckner M., Wang F.-Y.: Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Funct. Anal. 185(2), 564–603 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  28. Ver00.
    Veretennikov A.Y.: On polynomial mixing estimates for stochastic differential equations with a gradient drift. Teor. Veroyatnost. i Primenen. 45(1), 163–166 (2000)MathSciNetGoogle Scholar
  29. Ver06.
    Veretennikov, A.Y.: On lower bounds for mixing coefficients of Markov diffusions. In: From Stochastic Calculus to Mathematical Finance. Berlin: Springer, 2006, pp. 623–633Google Scholar
  30. Vil07.
    Villani C.: Hypocoercive diffusion operators. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10(2), 257–275 (2007)zbMATHMathSciNetGoogle Scholar
  31. Vil08.
    Villani, C.: Hypocoercivity, 2008 To appear in Memoirs Amer. Math. Soc.Google Scholar
  32. VK04.
    Veretennikov A.Y., Klokov S.A.: On the subexponential rate of mixing for Markov processes. Teor. Veroyatn. Primen. 49(1), 21–35 (2004)MathSciNetGoogle Scholar
  33. Won66.
    Wonham W.M.: Liapunov criteria for weak stochastic stability. J. Diff. Eqs. 2, 195–207 (1966)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematics InstituteThe University of WarwickCoventryUnited Kingdom
  2. 2.Courant InstituteNew York UniversityNew YorkUSA

Personalised recommendations