Dynamics of feedback Ising model

TL;DR

We study a linear feedback Ising model (FIM) on a complete graph where the coupling depends on the instantaneous magnetization through ${\cal H}_{\mathrm{FIM}}=-h\sum_i s_i - \frac{1}{N}[1+\gamma m]\sum_{i<j} s_i s_j$, and analyze its Glauber-dynamics-driven non-equilibrium behavior. The mean-field analysis reveals temperature-induced bistability, a Maxwell curve, and cusp- and transcritical-type bifurcations under zero and nonzero external field, with a unique Maxwell temperature emerging for fixed field within a bistable window. Near critical sets, a Fokker-Planck reduction yields non-Gaussian equilibrium distributions and explicit barrier-based scaling laws for inter-well transition times, highlighting how feedback reshapes fluctuation statistics. The framework provides a minimal, tractable toolbox for modeling co-evolving macroscopic feedback in magnetic, ecological, climatic, and socio-economic systems, with potential extensions to networks and delayed feedback.

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.

Figures (13)

• Figure 1: The equilibrium magnetization $m_+, m_0, m_-$, top to bottom, versus $\beta=T^{-1}$ when $h=\frac{1}{5}$. The positive equilibrium $m_+(\beta)>0$ exists in the entire temperature range. Two negative equilibria are created in a saddle-node bifurcation at $\beta_c(h)$ implicitly determined by Eq. \ref{['eq:beta_h']}. At the bifurcation point, $m_c(h)=m_0(\beta_c)=m_-(\beta_c)=-\sqrt{1-\beta_c^{-1}}$. In the present case, $\beta_c(\frac{1}{5})\approx 1.72286$ and $m_c(\frac{1}{5})\approx 0.647742$. In the zero-temperature limit, $m_\pm(\infty) = \pm 1$ and $m_0(\infty)=-\frac{1}{5}$; generally, $m_0(\infty)=-h$ when $0<h<1$.
• Figure 2: The equilibrium magnetization $m$ versus the temperature $T$ in the FIM with $h=0$ and $\gamma=1$. The magnetization is implicitly given by Eq. \ref{['m-F:SS']} with $\gamma=1$. Solid curves: stable branches. Dashed curves: unstable branches. The stable values of the magnetization are $m=1$ and $m=-\frac{2}{3}$ when $T=0$.
• Figure 3: Phase diagrams of the FIM on the $(T,h)$-plane for (a) $\gamma=0$; (b) $\gamma=\frac{2}{3}$; (c) $\gamma=1$; and (d) $\gamma=\frac{4}{3}$. There are two stable equilibria $m_\pm$ and one unstable equilibrium $m_0$ satisfying $m_+>m_0>m_-$ in the wedge between $h_\pm(T)$ and one stable equilibrium outside this wedge. The Maxwell curve $h_*(T)$ where $m_\pm$ are equally probable is derived in Sec. \ref{['sec:Maxwell']}.
• Figure 4: Plot of $T_c$ (red) and $h_c$ (blue) as functions of $\gamma$. The dashed curves show asymptotic behavior predicted by Eqs. (\ref{['eq:Thc_g_0']}--\ref{['eq:Thc_g_inf']}).
• Figure 5: Bifurcation diagrams of the FIM on the $(h,m)$-plane for $\gamma=\frac{4}{3}$ and (a) $T=1.5$; (b) $T=2.25$; and (c) $T=3$.