Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model.(English)Zbl 0647.60106
We study the dynamics of a 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 limit 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 frequency \(p_ 0\), the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequency \(p_ 0\) 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.
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 frequency \(p_ 0\), the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequency \(p_ 0\) 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.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 81P20 | Stochastic mechanics (including stochastic electrodynamics) |
Keywords:
geometric spin system;long-range interaction;propagation of chaos;central limit theorem;infinite-dimensional Ornstein-Uhlenbeck process;critical fluctuation process;antiferromagnetic phase transitionCite
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