Communications in Mathematical Physics

, Volume 118, Issue 4, pp 531–567 | Cite as

Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model

  • F. Comets
  • Th. Eisele


We study the dynamics of geometric spin system on the torus with long-range interaction. As the number of particles goes to infinity, the process converges to a deterministic, dynamical magnetization field that satisfies an Euler equation (law of large numbers). Its stable steady states are related to the limits of the equilibrium measures (Gibbs states) of the finite particle system. A related equation holds for the magnetization densities, for which the property of propagation of chaos also is established. We prove a dynamical central limit theorem with an infinite-dimensional Ornstein-Uhlenbeck process as a limiting fluctuation process. At the critical temperature of a ferromagnetic phase transition, both a tighter quantity scaling and a time scaling is required to obtain convergence to a one-dimensional critical fluctuation process with constant magnetization fields, which has a non-Gaussian invariant distribution. Similarly, at the phase transition to an antiferromagnetic state with frequencyp0, the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequencyp0 and has a non-Gaussian invariant distribution on these fields. Finally, we compute the critical fluctuation process in the infinite particle limit at a triple point, where a ferromagnetic and an antiferromagnetic phase transition coincide.


Critical Temperature Constant Magnetization Field Stable Steady State Gibbs State Dynamical Magnetization 
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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • F. Comets
    • 1
  • Th. Eisele
    • 2
  1. 1.Laboratoire de Stratistique Appliquée UA CNRS 743Université de Paris-SudOrsay CédexFrance
  2. 2.Institut für Angewandte MathematikUniversität HeidelbergHeidelberg 1Germany

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