Memory-induced optimal switching in a dynamic Ising-Kuramoto model

Abstract

We study a globally coupled stochastic system in which each unit carries an Ising spin and a continuous phase. The spins undergo Glauber switching with relaxation rate $γ$, while the phases are driven by Ornstein-Uhlenbeck noise with correlation time $τ$. A spin-dependent phase interaction supports a composite coherent mode, $Z_m=N^{-1}\sum_jσ_j e^{iθ_j}$, that can order even when the ordinary phase coherence is small. Linear response about the nonmagnetic incoherent state gives the synchronization threshold $K_{\mathrm c}=[(μ+β_sγ)χ_γ]^{-1}$, where $χ_γ$ is a memory-weighted Laplace transform of the free phase correlation function. The competition between the increasing Glauber response and the decreasing memory susceptibility produces a finite switching rate at which $K_{\mathrm c}$ is minimized. We derive a general sufficient condition for such an interior optimum and show that finite-time Ornstein-Uhlenbeck correlations satisfy the required fast-switching asymptotics, whereas the white-noise limit need not. Finite-size simulations reproduce the nonmonotonic threshold, its shift toward slower switching as $τ$ increases, antiphase composite order in a nonmagnetic state, and mode mixing on a magnetized background.