Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model
type: Item
instance of: scholarly article
title: Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model
zbMATH Open document ID: 0647.60106
DOI: 10.1007/BF01221108
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 81P20
zbMATH DE Number: 4056748
zbMATH Keywords: geometric spin system
zbMATH Keywords: long-range interaction
zbMATH Keywords: propagation of chaos
zbMATH Keywords: central limit theorem
zbMATH Keywords: infinite-dimensional Ornstein-Uhlenbeck process
zbMATH Keywords: critical fluctuation process
zbMATH Keywords: antiferromagnetic phase transition
author: Francis Comets
author: Karl-Theodor Eisele
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4056748