Synchronization of thermodynamically consistent stochastic phase oscillators

TL;DR

This work analyzes two coupled stochastic phase oscillators implemented as a Markov jump process over $N$ discrete states, which reduces to a one-dimensional phase-difference dynamic. In the thermodynamic limit, the system undergoes a continuous nonequilibrium synchronization transition analogous to a Kuramoto system, but with rich nonequilibrium thermodynamics and fluctuation phenomena. A key finding is the absence of a universal extremum-dissipation principle: synchronization can both increase or decrease dissipation depending on parameters, and linear response emerges only in the stochastic, finite-size regime with a scaling $\vartheta \sim N^{-1/3}$ near criticality. The study reveals universal finite-size scalings for frequency detuning, giant-phase-diffusion-like fluctuations ($\llangle \varphi \rrangle \propto N^{2/3}$), and a distinctive transition in information metrics: mutual information shifts from logarithmic in $N$ to $N$-independent, while information flow becomes intensive in the synchronized state and vanishes in the unsynchronized state. These results shed light on the thermodynamics and information processing of coupled oscillators and are likely relevant to limit-cycle systems after phase reduction.

Abstract

We consider a toy model of two kinetically coupled stochastic oscillators whose dynamics is described as a Markov jump process among $N$ discrete phase states. For large $N$, it maps onto the deterministic two-oscillator Kuramoto model of synchronization. Despite its simplicity, we postulate its relevance for understanding more complex and realistic oscillator systems. In the thermodynamic limit, the model exhibits a continuous nonequilibrium phase transition between the unsynchronized and synchronized states. We show that this transition is not governed by any extremum dissipation principle -- depending on system parameters, synchronization may either reduce or enhance the dissipation. Close to the phase transition, we observe a divergent behavior of fluctuations and responses with $N$ and characterize their universal scaling behavior. In particular, the covariances of the oscillator phases and the local entropy productions are shown to diverge towards $-\infty$, a phenomenon that has not been reported before. Finally, we study the behavior of information-theoretic quantities, demonstrating that mutual information and information flow between oscillators display different scaling with $N$ in synchronized and unsynchronized states, and thus can act as order parameters of synchronization.

Figures (15)

• Figure 1: Illustration of the model. Top: two coupled discrete-phase oscillators $X$ and $Y$, each consisting of $N$ discrete states, are in states $x$ and $y$. These states correspond to discrete phases $\theta_X=2 \pi x/N$ and $\theta_Y=2\pi y/N$. Bottom: the interaction between the oscillators depends only on the phase difference $\varphi \equiv \theta_Y-\theta_X = 2\pi(y-x)/N$.
• Figure 2: The observed frequencies $\bar{\Omega}_\alpha$ for (a) the symmetric coupling $a_X=a_Y=0.5$ and (b) the asymmetric coupling $a_X=0.9$, $a_Y=0$. The finite-size results are given by the master equation, while the results for $N \rightarrow \infty$ are given by the deterministic approach. The vertical gray lines denote the phase transition from unsynchronized (left) to synchronized (right) state. The insets show the behavior of $\bar{\Omega}_\alpha$ close to the phase transition point, in the region enclosed by a small rectangle. Other parameters: $f_Y=2$, $\Gamma_X=\Gamma_Y=1$, $\beta=1$.
• Figure 3: Demonstration of the validity of the scaling formula \ref{['eq:scalingdetuning']} for $\xi=f_X$. Rescaled frequency detuning $\vartheta N^{1/3}/(2\pi D_0^{1/3} \mu^{2/3})$ gradually converges with $N$ to a universal scaling function $U(\gamma)$. We consider symmetric coupling $a_X=a_Y=0.5$ and other parameters as in Fig. \ref{['fig:freq']}: $f_Y=2$, $\Gamma_X=\Gamma_Y=1$, $\beta=1$.
• Figure 4: Demonstration of the $\propto N^{1/3}$ scaling of the maximum response of the frequency detuning, $\max_{f_X} |d_{f_X} \vartheta|$. It is plotted in the log-log scale for (a) the symmetric coupling $a_X=a_Y=0.5$ and (b) the asymmetric coupling $a_X=0.9$, $a_Y=0$. The points represent the master equation results. The black solid line in (a) represents Eq. \ref{['eq:maxresponse']}, while in (b) it represents $\propto N^{1/3}$ scaling fitted to cross the point for $N=5120$. Other parameters as in Fig. \ref{['fig:freq']}: $f_Y=2$, $\Gamma_X=\Gamma_Y=1$, $\beta=1$.
• Figure 5: Intensive global entropy production rate $\dot{\sigma}$, calculated in the deterministic limit, as a function of $f_X$ for (a) the symmetric coupling $a_X=a_Y=0.5$ and (b) the asymmetric coupling $a_X=0.9$, $a_Y=0$. The vertical gray lines denote the phase transition from unsynchronized (left) to synchronized (right) state. The dashed line represents the extrapolation of trend from the synchronized state. Other parameters are as in Fig. \ref{['fig:freq']} ($f_Y=2$, $\Gamma_X=\Gamma_Y=1$, $\beta=1$), but the results are plotted for a smaller range of $f_X$ for better visibility.