Long-term fluctuations in globally coupled phase oscillators with general coupling: Finite size effects

Abstract

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D∼O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D∼O(1/N(a)) with a certain constant a>0 in the coherent regime and D∼O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.

Nonstandard scaling law of fluctuations in finite-size systems of globally coupled oscillators

“…In our discussion in Ref. [12], we suggested that the standard scaling law can be observed in a wide range of systems. We also found that, in the coherent regime of system (1), the asymptotic form of D for large N is represented as…”

“…On the other hand, many studies have also focused on the universal scaling law with respect to finite system size N . In fact, it is known that the variance of the order parameter generally decays with N in the order of 1/N independently of the details of the system elements [9][10][11][12]. Such a scaling law is regarded as standard because it applies to a wide range of models.…”

“…A synchronization transition is characterized by a change in the order parameter R(t) from almost zero to a finite value with an increase in the coupling strength K. In the thermodynamic limit N → ∞, the stationary state of R(t) in the incoherent (desynchronized) regime bifurcates at the critical coupling strength K = K c , above which the oscillators are coherent (synchronized). When N is finite, the effect of an increase in N on fluctuations can be different between the incoherent and coherent regimes [1,12]. To illustrate this, we introduce the diffusion coefficient of the time integral of R(t), characterizing long-term fluctuations of R(t) in systems of finite size [12].…”

“…When N is finite, the effect of an increase in N on fluctuations can be different between the incoherent and coherent regimes [1,12]. To illustrate this, we introduce the diffusion coefficient of the time integral of R(t), characterizing long-term fluctuations of R(t) in systems of finite size [12]. We consider the variance of the time integral t 0 R(s)ds, defined as follows:…”

“…which represents the deviation of t 0 R(s)ds from its mean value R t t on the sample average. The standard scaling law of D with respect to system size N is described as follows [12]:…”