The European Physical Journal B

, Volume 64, Issue 3–4, pp 355–363 | Cite as

Equilibrium and nonequilibrium properties of systems with long-range interactions

  • S. Ruffo


We briefly review some equilibrium and nonequilibrium properties of systems with long-range interactions. Such systems, which are characterized by a potential that weakly decays at large distances, have striking properties at equilibrium, like negative specific heat in the microcanonical ensemble, temperature jumps at first order phase transitions, broken ergodicity. Here, we mainly restrict our analysis to mean-field models, where particles globally interact with the same strength. We show that relaxation to equilibrium proceeds through quasi-stationary states whose duration increases with system size. We propose a theoretical explanation, based on Lynden-Bell’s entropy, of this intriguing relaxation process. This allows to address problems related to nonequilibrium using an extension of standard equilibrium statistical mechanics. We discuss in some detail the example of the dynamics of the free electron laser, where the existence and features of quasi-stationary states is likely to be tested experimentally in the future. We conclude with some perspectives to study open problems and to find applications of these ideas to dipolar media.


05.20.-y Classical statistical mechanics 05.70.Fh Phase transitions: general studies 05.45.-a Nonlinear dynamics and chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Dynamics and thermodynamics of systems with long-range interactions (Springer, Berlin, 2001)Google Scholar
  2. 2.
    L. Onsager, Il Nuovo Cimento (Suppl.) 6, 279 (1949)CrossRefMathSciNetGoogle Scholar
  3. 3.
    V.A. Antonov, Leningrad Univ. 7, 135 (1962) [translation in IAU Symposium 113, 525 (1995)]; D. Lynden-Bell, R. Wood, Mon. Not. R. Astr. Soc. 138, 495 (1968); P. Hertel, W. Thirring, Ann. Phys. 63, 520 (1971)ADSGoogle Scholar
  4. 4.
    A. Campa, S. Ruffo, H. Touchette, Physica A 385, 233 (2007)CrossRefADSGoogle Scholar
  5. 5.
    J. Barré, D. Mukamel, S. Ruffo, Phys. Rev. Lett. 87, 030601 (2001)CrossRefADSGoogle Scholar
  6. 6.
    D. Mukamel, S. Ruffo, N. Schreiber, Phys. Rev. Lett. 95, 240604 (2005)CrossRefADSGoogle Scholar
  7. 7.
    M.K.H. Kiessling, J.L. Lebowitz, Lett. Math. Phys. 42, 43 (1997), and references therein; R.S. Ellis, K. Haven, B. Turkington, J. Stat. Phys. 101, 999 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    I. Ispolatov, E.G.D. Cohen, Phys. A 295, 475 (2001)MATHCrossRefGoogle Scholar
  9. 9.
    J. de Luca, A.J. Lichtenberg, S. Ruffo, Phys. Rev. E 60, 3781 (1999)CrossRefADSGoogle Scholar
  10. 10.
    Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Physica A 337, 36 (2004)CrossRefADSGoogle Scholar
  11. 11.
    P.H. Chavanis, Ph.D. thesis, ENS Lyon (1996); P.H. Chavanis, J. Sommeria, R. Robert, Astroph. J. 471, 385 (1996)Google Scholar
  12. 12.
    F. Hohl, J.W. Campbell, Collective motion of a one-dimensional self-gravitating system, NASA Technical Note TN D-5540, November (1969)Google Scholar
  13. 13.
    T. Yamashiro, N. Gouda, M. Sakagami, Prog. Theor. Phys. 88, 269 (1992)CrossRefADSGoogle Scholar
  14. 14.
    M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995)CrossRefADSGoogle Scholar
  15. 15.
    J. Barré, T. Dauxois, G. De Ninno, D. Fanelli, S. Ruffo, Phys. Rev. E 69, 045501 (2004)CrossRefADSGoogle Scholar
  16. 16.
    P.H. Chavanis, Eur. Phys. J. B 53, 487 (2006); A. Antoniazzi, D. Fanelli, J. Barré, P.-H. Chavanis, T. Dauxois, S. Ruffo, Phys. Rev. E 75, 011112 (2007)CrossRefADSGoogle Scholar
  17. 17.
    P. de Buyl, D. Mukamel, S. Ruffo, in Unsolved Problems of Noise and Fluctuations, AIP Conference Proceedings 800, 533 (2005)CrossRefADSGoogle Scholar
  18. 18.
    P.H. Chavanis, J. Vatteville, F. Bouchet, Eur. Phys. J. B 46, 61 (2005)CrossRefADSGoogle Scholar
  19. 19.
    W. Braun, K. Hepp, Comm. Math. Phys. 56, 101 (1977)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    A. Antoniazzi, F. Califano, D. Fanelli, S. Ruffo, Phys. Rev. Lett. 98, 150602 (2007)CrossRefADSGoogle Scholar
  21. 21.
    F. Baldovin, E. Orlandini, Phys. Rev. Lett. 96, 240602 (2006); F. Baldovin, E. Orlandini, Phys. Rev. Lett. 97, 100601 (2006)CrossRefADSGoogle Scholar
  22. 22.
    A. Campa, D. Mukamel, A. Giansanti, S. Ruffo, Phys. A 365, 120 (2006)CrossRefGoogle Scholar
  23. 23.
    F. Bouchet, T. Dauxois, Phys. Rev. E 72, 045103(R) (2005)CrossRefADSGoogle Scholar
  24. 24.
    P.H. Chavanis, Phys. A 361, 55 (2006); P.H. Chavanis, Phys. A 361, 81 (2006); doi:10.1016/j.physa.2007.10.026, doi:10.1016/j.physa.2007.10.034CrossRefMathSciNetGoogle Scholar
  25. 25.
    V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64, 056134 (2001); A. Pluchino, A Rapisarda, C. Tsallis, Europhys. Lett. 80, 26002 (2007)CrossRefADSGoogle Scholar
  26. 26.
    A. Antoniazzi, D. Fanelli, S. Ruffo, Y.Y. Yamaguchi, Phys. Rev. Lett. 99, 040601 (2007)CrossRefADSGoogle Scholar
  27. 27.
    F. Bouchet, T. Dauxois, D. Mukamel, S. Ruffo, Phase space gaps and ergodicity breaking in systems with long range interactions, e-print arXiv:0711.0268Google Scholar
  28. 28.
    F. Borgonovi, G.L. Celardo, M. Maianti, E. Pedersoli, J. Stat. Phys. 116, 1435 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    W.B. Colson, Phys. Lett. A 59, 187 (1976); R. Bonifacio et al., Opt. Comm. 50, 373 (1984)CrossRefADSGoogle Scholar
  30. 30.
    J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, J. Stat. Phys. 119, 677 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Y. Elskens, D.F. Escande, Microscopic Dynamics of Plasmas and Chaos (IoP Publishing, Bristol, 2003)MATHGoogle Scholar
  32. 32.
    A. Antoniazzi, G. De Ninno, D. Fanelli, A. Guarino, S. Ruffo, J. Phys.: Conf. Ser. 7, 143 (2005)CrossRefADSGoogle Scholar
  33. 33.
    A. Antoniazzi, R.S. Johal, D. Fanelli, S. Ruffo, Comm. Nonlin. Sci. Num. Simul. 13, 2 (2008)MATHCrossRefADSGoogle Scholar
  34. 34.
    J. Barré, F. Bouchet, J. Stat. Phys. 118, 1073 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    C. von Cube et al., Phys. Rev. Lett. 93, 083601 (2004); C. von Cube et al., Fortschr. Phys. 54, 726 (2006)CrossRefADSGoogle Scholar
  36. 36.
    L.Q. English, M. Sato, A.J. Sievers, Phys. Rev. B 67, 024403 (2003)CrossRefADSGoogle Scholar
  37. 37.
    A. Campa, R. Khomeriki, D. Mukamel, S. Ruffo, Phys. Rev. B 76, 064415 (2007)CrossRefADSGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Dipartimento di Energetica “S. Stecco” and CSDCUniversità di FirenzeFirenzeItaly
  2. 2.INFN, Sezione di FirenzeFirenzeItaly

Personalised recommendations