Stochastic Two-temperature Nonequilibrium Ising model

Abstract

We investigate the nonequilibrium stationary state (NESS) of the two-dimensional Ising model under a stochastic dichotomous modulation of temperature, which alternates between $T_c \pm δ$ around the critical temperature $T_c$ at a rate $γ$. Both magnetization and energy exhibit non-monotonic dependence on $γ$, explained by a renewal approach in the slow-switching limit, while for small $δ$ dynamical response theory quantitatively captures the $γ$-dependence of the observables. In the fast-switching regime, the NESS appears Boltzmann-like with a $γ$-dependent effective temperature. However, a finite energy current flowing through the system from hot to cold reservoir confirms the intrinsic nonequilibrium nature of the dynamics.

Figures (11)

• Figure 1: Schematic diagram of two-dimensional Ising magnet driven by stochastic two-temperature bath. The right panel shows typical trajectories of $\sigma(t)$, $M(t)$ and $E(t)$.
• Figure 2: Plots of (a) average and (b) variance of the magnetization as functions of flip-rate $\gamma$ for different values of $\delta$. The horizontal solid lines indicate the corresponding equilibrium values at $T=T_{c}$ and the vertical dashed lines indicate the $\gamma^{\star}_{M}$. Here we have used $L=32$ and $\psi_{0}=0.01$.
• Figure 3: Plot of $\gamma^{\star}_{M}$ versus $\psi_{0}$ for a fixed small $\delta$. The symbols indicate data obtained from numerical simulation and the solid line denotes the best fit straight line. The shaded region indicates the error margin corresponding to the symbols. The inset illustrates how $\gamma^{\star}_{M}$ is extracted from the $\langle M\rangle$ vs $\gamma$ curves. Here we have used $L=32$ and $\delta=0.15$.
• Figure 4: Plots of (a) average and (b) variance of the energy as functions of flip-rate $\gamma$ for different values of $\delta$. The horizontal solid lines indicate the corresponding equilibrium values at $T=T_{c}$ and the vertical dashed lines indicate the $\gamma^{\star}_{E}$. Here we have used $L=32$ and $\psi_{0}=0.01$.
• Figure 5: Plots of (a) average magnetization $\langle M \rangle$ and (b) average energy $\langle E \rangle$ as functions of $\delta$ for different flip rates $\gamma$. The horizontal dashed lines mark the corresponding equilibrium values $M_{c}$ and $E_{c}$ at $T = T_{c}$. Red dashed parabolas are shown as guides to the eye, highlighting the quadratic behaviour at small $\delta$. Here we have used $L=32$ and $\psi_0=0.01$.