A solvable model of noisy coupled oscillators with fully random interactions

Abstract

We introduce a solvable spherical model of coupled oscillators with fully random interactions and distributed natural frequencies. Using the dynamical mean-field theory, we derive self-consistent equations for the steady-state response and correlation functions. We show that any finite width of the natural-frequency distribution suppresses the finite-temperature spin-glass transition, because the resulting low-frequency singularity of the correlation function is incompatible with the spherical constraint. At zero temperature, however, a spin-glass phase persists for arbitrary frequency dispersion. This residual zero-temperature glassiness is likely a special feature of the spherical dynamics and would be destroyed by local nonlinearities. The model thus provides a solvable oscillator framework for studying how nonequilibrium perturbations suppress finite-temperature glassy freezing.

Figures (4)

• Figure 1: Phase diagram of the spherical model with ferromagnetic interactions. The filled and unfilled regions denote the synchronized and incoherent phases, respectively.
• Figure 2: Time correlation functions for (a) $\Delta=0$ and (b) $\Delta=0.1$ at several temperatures. Here $J=1$. For $\Delta=0$, the correlation approaches a nonzero plateau below the transition temperature $T_c=1$ (dashed line). For $\Delta>0$, it decays to zero for all temperatures.
• Figure 3: (a) $\Lambda=\lim_{\omega\to 0}C(\omega)$ for $J=1$ and several values of $\Delta$. For $\Delta=0$, $\Lambda$ diverges at the finite transition temperature $T_c=J$, whereas for $\Delta>0$ it diverges only as $T\to 0$. (b) Scaling plot of the same data. The red dotted line indicates $\Delta^3\Lambda\propto T^{-1}$.
• Figure 4: (a) Correlation functions for several values of $\Delta$ at $J=1$ and $T=0.5$. (b) Scaling plot of the same data. The late-time curves collapse when time is rescaled by $\Delta^3$.