Oscillatory behavior in a model of non-Markovian mean field interacting spins

TL;DR

This paper investigates a non-Markovian mean field spin system with memory effects, revealing potential oscillatory behavior and a Hopf bifurcation at a critical temperature through analytical and numerical methods.

Contribution

It introduces a non-Markovian extension of the Curie--Weiss model and analyzes the emergence of periodic behavior via linearization and spectral analysis.

Findings

Evidence of oscillatory behavior in the system.

Identification of a Hopf bifurcation at a critical temperature.

Numerical confirmation of theoretical predictions.

Abstract

We analyze a non-Markovian mean field interacting spin system, related to the Curie--Weiss model. We relax the Markovianity assumption by replacing the memoryless distribution of the waiting times of a classical spin-flip dynamics with a distribution with memory. The resulting stochastic evolution for a single particle is a spin-valued renewal process, an example of two-state semi-Markov process. We associate to the individual dynamics an equivalent Markovian description, which is the subject of our analysis. We study a corresponding interacting particle system, where a mean field interaction is introduced as a time scaling, depending on the overall magnetization of the system, on the waiting times between two successive particle's jumps. Via linearization arguments on the Fokker-Planck mean field limit equation, we give evidence of emerging periodic behavior. Specifically, numerical analysis on the discrete spectrum of the linearized operator, characterized by the zeros of an explicit holomorphic function, suggests the presence of a Hopf bifurcation for a critical value of the temperature, which is in accordance with the one obtained by simulating the N-particle system.