Dynamics and thermodynamics of a model with long-range interactions


The dynamics and thermodynamics of particles/spins interacting via long-range forces display several unusual features compared with systems with short-range interactions. The Hamiltonian mean field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular, the model shows a second-order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact, the numerical simulations show the presence of quasi-stationary states (QSS’s), i.e. metastable states that become stable if the thermodynamic limit is taken before the infinite time limit. The QSS’s differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heats, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly decaying correlations, and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSS’s have recently been interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which does not preclude to the previous one.

This is a preview of subscription content, log in to check access.


  1. 1

    Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31, 253-258 (1925)

    Google Scholar 

  2. 2

    Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds): Dynamics and Thermodynamics of Systems with Long Range Interactions (Lecture Notes in Physics, Vol. 602), Springer Berlin Heidelberg New York (2002)

  3. 3

    Kac, M., Uhlenbeck, G., Hemmer, P.C.: On the Van der Waals theory of vapor-liquid equilibrium. J. Math. Phys. 4, 216-228 (1963)

    MATH  Google Scholar 

  4. 4

    Anteneodo, C., Tsallis, C.: Breakdown of exponential sensitivity to initial conditions: role of the range of interactions. Phys. Rev. Lett. 80, 5313-5316 (1998); Tamarit, F., Anteneodo, C.: Rotators with long-range interactions: connection with the mean-field approximation. Phys. Rev. Lett. 84, 208-211 (2000)

    Article  Google Scholar 

  5. 5

    Campa, A., Giansanti, A., Moroni, D.: Canonical solution of classical magnetic models with long-range couplings. J. Phys A: Math. Gen. 36, 6897-6921 (2003) and references therein

    Article  Google Scholar 

  6. 6

    Dauxois, T., Latora, V., Rapisarda, A., Ruffo, S., Torcini, A.: In: Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds) Dynamics and Thermodynamics of Systems with Long Range Interactions (Lecture Notes in Physics, Vol. 602), Springer Berlin Heidelberg New York (2002), p.458 and references therein

  7. 7

    Latora, V., Rapisarda, A., Ruffo, S.: Superdiffusion and out-of-equilibrium chaotic dynamics with many degrees of freedom. Phys. Rev. Lett. 83, 2104-2107 (1999)

    Article  Google Scholar 

  8. 8

    Latora, V., Rapisarda, A.: Microscopic dynamics of a phase transition: equilibrium vs out-of-equilibrium regime. Nucl. Phys. A 681, 406-416 (2001)

    Article  MATH  Google Scholar 

  9. 9

    Latora, V., Rapisarda, A., Tsallis, C.: Non-Gaussian equilibrium in a long-range Hamiltonian system. Phys. Rev. E 64, 056134-056138 (2001); Latora, V., Rapisarda, A., Tsallis, C.: Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system. Physica A 305, 129-136 (2002)

    Article  Google Scholar 

  10. 10

    Montemurro, M.A., Tamarit, F.A., Anteneodo, C.: Aging in an infinite-range Hamiltonian system of coupled rotators. Phys. Rev. E 67, 031106-031109 (2003)

    Article  Google Scholar 

  11. 11

    Pluchino, A., Latora, V., Rapisarda, A.: Metastable states, anomalous distributions and correlations in the HMF model. Physica D (2004) (in press) [cond-mat/0303081]

  12. 12

    Moyano, L.G., Baldovin, F., Tsallis, C.: Zeroth principle of thermodynamics in aging quasistationary states. [cond-mat/0305091] (submitted)

  13. 13

    Yamaguchi, Y.Y.: Relaxation and diffusion in a globally coupled Hamiltonian system. Phys. Rev. E 68, 066210-066219 (2003)

    Article  Google Scholar 

  14. 14

    Cabral, B.J.C., Tsallis, C.: Metastability and weak mixing in classical long-range many-rotator systems. Phys. Rev. E 66, 065101-065104 (R) (2002)

    Article  Google Scholar 

  15. 15

    Tsallis, C., Rapisarda, A., Latora, V., Baldovin, F.: Nonextensivity: from low-dimensional maps to Hamiltonian systems. In: Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds) Dynamics and Thermodynamics of Systems with Long Range Interactions (Lecture Notes in Physics, Vol. 602), Springer Berlin Heidelberg New York (2002), p.140-162

  16. 16

    Borges, E., Tsallis, C.: Negative specific heat in a Lennard-Jones-like gas with long range interactions. Physica A 305, 148-151 (2002)

    Article  MATH  Google Scholar 

  17. 17

    Nobre, F.D., Tsallis, C.,: Classical infinite-range-interaction Heisenberg ferromagnetic model: metastability and sensitivity to initial conditions. [cond-mat/0301492]

  18. 18

    Tsallis, C.: Possible generalization of Botlzmann-Gibbs statistics. J. Stat. Phys. 52, 479-487 (1988); for recent reviews, see: Tsallis, C.: Nonextensive statistical mechanics and thermodynamics. In: Abe, S., Okamoto, Y. (eds) Lecture Notes in Physics, Springer Berlin Heidelberg New York (2001); Kaniadakis, G., Lissia, M., Rapisarda, A. (eds): Proceedings of NEXT2001, special issue of Physica A 305 (2002); Tsallis, C., M. Gell-Mann, M., (eds) Nonextensive Entropy - Interdisciplinary Applications, Oxford University Press (2004); see also http://tsallis.cat.cbpf.br/biblio.htm for a regularly updated and complete bibliography

    MathSciNet  MATH  Google Scholar 

  19. 19

    For the recent debate about nonextensive thermodynamics, see: Cho, A.: A fresh take on disorder, or disorderly science? Science 297, 1268-1269 (2002); Abe, S., Rajagopal, A.K., Plastino, A., Latora, V., Rapisarda, A., Robledo, A.: Revisiting disorder and Tsallis statistics. Science 300, 249-251 (2003)

    Google Scholar 

  20. 20

    Bouchaud, J.P., Cugliandolo, L.F., Kurchan, J., Mezárd, M.M.: Out-of-equilibrium Dynamics in Spin-Glasses and other Glassy Systems. In: Young, A.P. (ed) Spin glasses and random fields. World Scientific Singapore (1998)

  21. 21

    Pluchino, A., Latora, V., Rapisarda, A.: Glassy phase in the Hamiltonian mean field model. [cond-mat/0306374] (submitted)

  22. 22

    Boghosian, B.M.: Thermodynamic description of the relaxation of two-dimensional turbulence using Tsallis statistics. Phys. Rev. E 53, 4754-4763 (1996); Coraddu, M., Lissia, M., G. Mezzorani, G., Quarati, P.: Deuterium burning in Jupiter interior. Physica A 305, 282-286 (2002)

    Article  Google Scholar 

  23. 23

    Beck, C.: Dynamical foundations of nonextensive statistical mechanics. Phys. Rev. Lett. 87, 180201-180204 (2001) and references therein

    Google Scholar 

  24. 24

    Beck, C., Cohen, E.G.D.: Superstatistics. Physica A 322, 267-275 (2003)

    Article  MATH  Google Scholar 

  25. 25

    Lyra, M.L., Tsallis, C.: Nonextensivity and multifractality in low-dimensional dissipative systems. Phys. Rev. Lett. 80, 53-56 (1998); Tirnakli, U., Tsallis, C., Lyra, M.L.: Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity. Eur. Phys. J. B 10, 309-315 (1999); de Moura, F.A.B.F., Tirnakli, U., Lyra, M.L.: Convergence to the critical attractor of dissipative maps: log-periodic oscillations, fractality, and nonextensivity. Phys. Rev. E 62, 6361-6365 (2000)

    Article  Google Scholar 

  26. 26

    Latora, V., Baranger, M., Rapisarda, A., Tsallis, C.: The rate of entropy increase at the edge of chaos. Phys. Lett. A 273, 97-103 (2000); Borges, E.P., Tsallis, C., Ananos, G.F.J., de Oliveira, P.M.C.: Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos. Phys. Rev. Lett. 89, 254103-254106 (2002)

    Article  MathSciNet  Google Scholar 

  27. 27

    Baldovin, F., Robledo, A.: Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. Phys. Rev. E 66, 045104(R)-045107(R) (2002); Baldovin, F., Robledo, A.: Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions. Europhys. Lett. 60, 518-524 (2002)

    Article  Google Scholar 

  28. 28

    Baldovin, F., Tsallis, C., Schulze, B.: Nonstandard entropy production in the standard map. Physica A 320, 184-192 (2003)

    Article  MATH  Google Scholar 

  29. 29

    Bediaga, I., Curado, E.M.F., Miranda, J.: A nonextensive thermodynamical equilibrium approach in e + e hadrons. Physica A 286, 156-163 (2000); Wilk, G., Wlodarczyk, Z.: Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and Lévy distributions. Phys. Rev. Lett. 84, 2770-2773 (2000)

    Article  Google Scholar 

  30. 30

    Tsallis, C., Anjos, J.C., Borges, E.P.: Fluxes of cosmic rays: a delicately balanced anomalous thermal equilibrium. Phys. Lett. A 310, 372-379 (2003)

    Article  Google Scholar 

  31. 31

    Tsallis, C., Bukman, D.J.: Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis. Phys. Rev. E 54, R2197-R2200 (1996)

    Google Scholar 

  32. 32

    Daniels, K.E., Beck, C., Bodenschatz, E.: Defect turbulence and generalized statistical mechanics. [cond-mat/0302623]

  33. 33

    Kurchan, J.: Elementary constraints on autocorrelation function scalings. Phys. Rev. E 66, 017101-017104 (2002)

    Article  Google Scholar 

  34. 34

    Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792-1796 (1975); Sherrington, D., Kirkpatrick, S.: Infinite-ranged models of spin-glasses. Phys. Rev. B 17, 4384-4403 (1978)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to A. Rapisarda.

Additional information

Received: 8 July 2003, Accepted: 27 October 2003, Published online: 11 February 2004


05.70.Fh, 89.75.Fb, 64.60.Fr, 75.10.Nr

Correspondence to: A. Rapisarda

Communicated by M. Sugiyama

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pluchino, A., Latora, V. & Rapisarda, A. Dynamics and thermodynamics of a model with long-range interactions. Continuum Mech. Thermodyn. 16, 245–255 (2004). https://doi.org/10.1007/s00161-003-0170-0

Download citation


  • phase transitions
  • Hamiltonian dynamics
  • long-range interaction
  • out-of-equilibrium statistical mechanics