Title: Dynamics of feedback Ising model Show Chinese title

Title: 反馈伊辛模型的动力学

Abstract: We study the dynamics of a mean-field Ising model whose coupling depends on the magnetization via a linear feedback function. A key feature of this linear feedback Ising model (FIM) is the possibility of temperature-induced bistability, where a temperature increase can favor bistability between two phases. We show that the linear FIM provides a minimal model for a transcritical bifurcation as the temperature varies. Moreover, there can be two or three critical temperatures when the external magnetic field is non-negative. In the bistable region, we identify a Maxwell temperature where the two phases are equally probable, and we find that increasing the temperature favors the lower phase. We show that the probability distribution becomes non-Gaussian on certain time intervals when the magnetization converges algebraically at either zero temperature or critical temperatures. Near critical points in the parameter space, we derive a Fokker-Planck equation, construct the families of equilibrium distributions, and formulate scaling laws for transition rates between two stable equilibria. The linear FIM provides considerable freedom to control steady-state bifurcations and their associated equilibrium distributions, which can be desirable for modeling feedback systems across disciplines. Show Chinese abstract

Abstract: 我们研究了一种平均场伊辛模型的动力学，其耦合通过一个线性反馈函数依赖于磁化强度。 这种线性反馈伊辛模型（FIM）的一个关键特征是温度引起的双稳态，其中温度升高可以促进两个相之间的双稳态。 我们表明，当温度变化时，线性FIM提供了一个转临界分岔的最小模型。 此外，当外部磁场为非负时，可能存在两个或三个临界温度。 在双稳态区域，我们确定了一个麦克斯韦温度，在该温度下两个相的概率相等，并且我们发现增加温度有利于较低的相。 我们表明，当磁化强度在零温度或临界温度处代数收敛时，在某些时间间隔内概率分布变为非高斯分布。 在参数空间中的临界点附近，我们推导出福克-普朗克方程，构建平衡分布族，并制定两个稳定平衡之间的跃迁速率的标度定律。 线性FIM提供了控制稳态分岔及其相关平衡分布的极大自由度，这在跨学科建模反馈系统时可能是有益的。