Synchronization and fluctuations: Coupling a finite number of stochastic units

It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization. And yet it is worth noting that finite-number effects should be seriously taken into account since, in general, the limits N→∞ (where N is the number of units) and t→∞ (where t is time) do not commute. Mean-field theory implements the particular choice first N→∞ and then t→∞. Here we analyze an ensemble of three-state coupled stochastic units, which has been widely studied in the thermodynamic limit. We formally address the finite-N problem by deducing a Fokker-Planck equation that describes the system. We compute the steady-state solution of this Fokker-Planck equation (that is, finite N but t→∞). We use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions.