Dynamics and thermodynamics of a model with long-range interactions

Abstract.

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.

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Correspondence to A. Rapisarda.

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

PACS:

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

Correspondence to: A. Rapisarda

Communicated by M. Sugiyama

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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

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Keywords:

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