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 a 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-depending on the magnetization of the system-is introduced as a time scaling 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. The presence of a Hopf bifurcation in the limit equation matches the emergence of a periodic behavior obtained by simulating the N -particle system.

中文翻译：

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非马尔可夫平均场相互作用自旋模型中的振荡行为

我们分析了与居里-魏斯模型相关的非马尔可夫平均场相互作用自旋系统。我们通过用有记忆的分布替换经典自旋翻转动力学的等待时间的无记忆分布来放宽马尔可夫假设。由此产生的单个粒子的随机演化是一个自旋值更新过程，一个二态半马尔可夫过程的例子。我们将个体动力学与等效的马尔可夫描述联系起来，这是我们分析的主题。我们研究了相应的相互作用粒子系统，其中平均场相互作用（取决于系统的磁化强度）被引入作为两个连续粒子跳跃之间等待时间的时间缩放。通过 Fokker-Planck 平均场极限方程的线性化参数，我们提供了新出现的周期性行为的证据。具体来说，对线性化算子的离散谱进行数值分析，其特征是显式全纯函数的零点，表明存在温度临界值的 Hopf 分岔。极限方程中 Hopf 分岔的存在与通过模拟 N 粒子系统获得的周期性行为的出现相匹配。