Non-Equilibrium Steady States for Chains of Four Rotors

Abstract

We study a chain of four interacting rotors (rotators) connected at both ends to stochastic heat baths at different temperatures. We show that for non-degenerate interaction potentials the system relaxes, at a stretched exponential rate, to a non-equilibrium steady state (NESS). Rotors with high energy tend to decouple from their neighbors due to fast oscillation of the forces. Because of this, the energy of the central two rotors, which interact with the heat baths only through the external rotors, can take a very long time to dissipate. By appropriately averaging the oscillatory forces, we estimate the dissipation rate and construct a Lyapunov function. Compared to the chain of length three (considered previously by C. Poquet and the current authors), the new difficulty with four rotors is the appearance of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.

References

  1. 1

    Aoki K., Lukkarinen J., Spohn H.: Energy transport in weakly anharmonic chains. J. Stat. Phys. 124, 1105–1129 (2006)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  2. 2

    Bernardin, C., Huveneers, F., Lebowitz, J.L., Liverani, C., Olla, S.: Green-kubo formula for weakly coupled systems with noise. Commun. Math. Phys. 334, 1377–1412

  3. 3

    Bonetto F., Lebowitz J.L., Rey-Bellet L.: Fourier’s law: a challenge to theorists. In: Mathematical physics 2000 (London: Imp. Coll. Press, 2000), pp. 128–150

  4. 4

    Bricmont J., Kupiainen A.: Towards a derivation of Fouriers law for coupled anharmonic oscillators. Commun. Math. Phys. 274, 555–626 (2007)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  5. 5

    Carmona P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stoch. Process. Appl. 117, 1076–1092 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6

    Cuneo N., Eckmann J.-P., Poquet C.: Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors. Nonlinearity 28, 2397–2421 (2015)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  7. 7

    De Roeck W., Huveneers F.: Asymptotic localization of energy in nondisordered oscillator chains. Commun. Pure Appl. Math. 68, 1532–1568 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8

    Douc R., Fort G., Guillin A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Stoch. Process. Appl. 119, 897–923 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9

    Eckmann J.-P., Hairer M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164 (2000)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  10. 10

    Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Entropy production in nonlinear, thermally driven hamiltonian systems. J. Stat. Phys. 95, 305–331 (1999)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  11. 11

    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, 657–697 (1999)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  12. 12

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

    ADS  Article  MATH  MathSciNet  Google Scholar 

  13. 13

    Eleftheriou M., Lepri S., Livi R., Piazza F.: Stretched-exponential relaxation in arrays of coupled rotators. Phys. D 204, 230–239 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14

    Gallavotti, G., Iacobucci, A., Olla, S.: Nonequilibrium stationary state for a damped rotator (2013) arXiv:1310.5379

  15. 15

    Gendelman O.V., Savin A.V.: Normal heat conductivity of the one-dimensional lattice with periodic potential of nearest-neighbor interaction. Phys. Rev. Lett. 84, 2381–2384 (2000)

    ADS  Article  Google Scholar 

  16. 16

    Giardinà C., Livi R., Politi A., Vassalli M.: Finite thermal conductivity in 1d lattices. Phys. Rev. Lett. 84, 2144 (2000)

    ADS  Article  Google Scholar 

  17. 17

    Hairer M.: How hot can a heat bath get?. Commun. Math. Phys. 292, 131–177 (2009)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  18. 18

    Hairer M., Mattingly J.C.: Slow energy dissipation in anharmonic oscillator chains. Commun. Pure Appl. Math. 62, 999–1032 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19

    Iacobucci A., Legoll F., Olla S., Stoltz G.: Negative thermal conductivity of chains of rotors with mechanical forcing. Phys. Rev. E 84, 061108 (2011)

    ADS  Article  Google Scholar 

  20. 20

    Lefevere R., Schenkel A.: Normal heat conductivity in a strongly pinned chain of anharmonic oscillators. J. Stat. Mech. Theory Exp. 2006, L02001 (2006)

    Article  Google Scholar 

  21. 21

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

    ADS  Article  MathSciNet  Google Scholar 

  22. 22

    Meyn, S., Tweedie, R.L.: Markov chains and stochastic stability (Cambridge University Press, Cambridge, 2009), second edition. With a prologue by Peter W. Glynn.

  23. 23

    Meyn S.P., Tweedie R.L.: Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25, 518–548 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24

    Pardoux E., Veretennikov A.Y.: On the Poisson equation and diffusion approximation. I. Ann. Probab. 29, 1061–1085 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25

    Pardoux E., Veretennikov A.Y.: On Poisson equation and diffusion approximation. II. Ann. Probab. 31, 1166–1192 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26

    Pardoux E., Veretennikov A.Y.: On the Poisson equation and diffusion approximation. III. Ann. Probab. 33, 1111–1133 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27

    Rey-Bellet L., Thomas L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225, 305–329 (2002)

    ADS  Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to J.-P. Eckmann.

Additional information

Communicated by H. Spohn

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cuneo, N., Eckmann, JP. Non-Equilibrium Steady States for Chains of Four Rotors. Commun. Math. Phys. 345, 185–221 (2016). https://doi.org/10.1007/s00220-015-2550-2

Download citation

Keywords

  • Invariant Measure
  • Lyapunov Function
  • External Rotor
  • Heat Bath
  • Central Rotor