Ising Model with Power Law Resetting

TL;DR

Ising model with power-law resetting investigates how heavy-tailed resetting influences nonequilibrium magnetisation under Glauber dynamics. The authors develop a renewal framework combining power-law inter-reset times with standard Ising relaxation in $1$D and $2$D to obtain magnetisation distributions $P_r(m,t)$ and their stationary limits. In $1$D, resetting induces a quasi-ferro state with persistent double peaks at $m=0$ and $m=m_0$, with no stationary state for $\alpha<1$ and a stationary double-peak for $\alpha>1$. In $2$D, for $T>T_C$ the system exhibits a persistent double-peak quasi-ferro state for all $\alpha$, while for $T<T_C$ there is a crossover at $\alpha^*=1-c$ between a single-peak ferro state and a double-peak ferro state, with critical dynamics at $T=T_C$ producing power-law regimes. The results highlight how non-Poissonian resetting qualitatively reshapes nonequilibrium phases in interacting many-body systems and provide a framework that could be extended to other models and higher dimensions.

Abstract

We investigate the nonequilibrium dynamics of the nearest-neighbour Ising model subjected to stochastic resetting, where the system is intermittently returned to an initial configuration with magnetisation $m_0$, with the inter-reset times drawn from the power law distribution $ατ_0^α/ τ^{α+1}$. The heavy-tailed resets generate magnetisation distributions that differ significantly from both equilibrium dynamics and the previously studied Ising model with exponentially distributed reset times. In two dimensions, for $T > T_C$, we find a quasi-ferro state for all $α$, marked by a double-peaked distribution that diverges at $m=0$ and $m=m_0$; no steady state exists for $α< 1$, while a stationary state emerges for $α> 1$. For $T < T_C$, power law resetting produces two distinct regimes separated by a crossover exponent $α^* = 1-c$: a single-peak ferromagnetic phase localised at $m_{eq}$ for $α< α^*$, and a dual-peak ferromagnetic phase with divergences at $m_{eq}$ and $m_0$ for $α> α^*$. Analytic results in one and two dimensions, supported by simulations, yield a rich phase diagram in the $(T,α)$ plane and reveal how heavy-tailed resetting generates nonequilibrium phases very different from those seen in the case of exponential resetting.

Figures (5)

• Figure 1: Phase diagram in the $(T,\alpha)$ plane for the 2D Ising model with power law stochastic resetting. The schematic shows the qualitative forms of the magnetisation distribution in the different regimes. For $T>T_c$, the system exhibits a quasi-ferromagnetic state for all $\alpha$, characterised by a double-peaked distribution with divergences at $m=0$ and $m=m_0$. For $T<T_c$, power law resetting gives rise to two distinct ferromagnetic phases separated by the crossover line $\alpha=\alpha^*$ (dashed orange), where $\alpha^*=1-c$ and $c$ is the Glauber dynamical exponent. For $\alpha<\alpha^*$, the distribution is single-peaked, diverging only at the equilibrium magnetisation $m_{eq}$. For $\alpha>\alpha^*$, a double-peaked ferromagnetic state emerges, with divergences at both $m_{eq}$ and $m_0$.
• Figure 2: Plot of the PDF of the magnetisation in the 1D Ising model with power law resetting for different values of $\alpha$: (a) Finite-time distribution at $t=12$ for $\alpha=0.5$ ($\alpha<1$), lattice size $L=2000$, $m_0=0.9$ and $\gamma=0.6$, averaged over $10^5$ realisations. (b) Steady-state distribution for $\alpha=1.5$ ($\alpha>1$), obtained with $L=1000$, $m_0=0.9,$ and $\gamma=0.6$, averaged over $10^6$ realisations. The red dashed lines correspond to the analytical predictions in Eq. \ref{['eq:8']} and Eq. \ref{['eq:10']}, and the blue curves correspond to the numerical simulations.
• Figure 3: Plot of the PDF of the average magnetisation in the 2D Ising model of lattice size $256\times 256$ with power law resetting at a temperature $T>T_C$ ($T=3.5$) (a) Finite-time distribution at $t=40$ for $\alpha=0.5$ ($\alpha<1$) and $m_0=0.99$, averaged over $10^6$ independent realisations. (b) Stationary distribution for $\alpha=1.5$ ($\alpha>1$) and $m_0=0.99$, averaged over $2\times10^8$ realisations.
• Figure 4: Probability density function (PDF) of the average magnetization in the two-dimensional Ising model with power law stochastic resetting for temperatures below the critical point ($T < T_c$). System size $L = 256\times256$. (a) Distribution at $t = 120$ for $\alpha = 0.1 < \alpha^*$, $T = 2.05$, and $m_0 = 0.99$, averaged over $3\times10^6$ realizations, with equilibrium magnetization $m_{\mathrm{eq}} = 0.8927$. (b) Distribution at $t = 120$ for $\alpha = 0.8 > \alpha^*$, $m_0 = 0.99$, and $m_{\mathrm{eq}} = 0.8927$, averaged over $10^5$ realizations. (c) Steady-state distribution for $\alpha = 1.5$, obtained with $m_0 = 0.7$ and averaged over $2\times10^7$ realizations.
• Figure 5: Probability density function (PDF) of the average magnetization in the two-dimensional Ising model with power law stochastic resetting at the critical temperature $T = T_c = 2.269$. System size $L = 256\times256$. (a) Distribution at $t = 100$ for $\alpha = 0.1$, $m_0 = 0.99$, averaged over $2.4\times10^6$ realizations. (b) Time-independent distribution for $\alpha = 1.5$, obtained with $m_0 = 0.99$ and averaged over $2\times10^7$ realizations.