2019/11/13 by Paolo Dai Pra, Pra, Paolo Dai, Marco Formentin +3
Economics, Econometrics and Finance · Physics and Astronomy · #60K15 #60K35 #82C22 #82C26 #Complex Systems and Time Series Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Opinion Dynamics and Social Influence #Probability (math.PR) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1911.05373
openalex publication_date 2019/11/13 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We analyze a non-Markovian mean field interacting spin system, related to the\nCurie--Weiss model. We relax the Markovianity assumption by replacing the\nmemoryless distribution of the waiting times of a classical spin-flip dynamics\nwith a distribution with memory. The resulting stochastic evolution for a\nsingle particle is a spin-valued renewal process, an example of two-state\nsemi-Markov process. We associate to the individual dynamics an equivalent\nMarkovian description, which is the subject of our analysis. We study a\ncorresponding interacting particle system, where a mean field interaction is\nintroduced as a time scaling, depending on the overall magnetization of the\nsystem, on the waiting times between two successive particle's jumps. Via\nlinearization arguments on the Fokker-Planck mean field limit equation, we give\nevidence of emerging periodic behavior. Specifically, numerical analysis on the\ndiscrete spectrum of the linearized operator, characterized by the zeros of an\nexplicit holomorphic function, suggests the presence of a Hopf bifurcation for\na critical value of the temperature, which is in accordance with the one\nobtained by simulating the N-particle system.\n