Cooperative Phenomena in Two-Dimensional Active Rotator Systems

Phase transitions of active rotator systems with short-range coupling are discussed. The constituents of the system which we call active rotators are represented by a phase model of a limit-cycle oscillator or an excitable element, i.e., dφ/dt=ω-bsinφ, (|ω/b|≷1). The rotators are subject to noises, and ferromagnetic type coupling is assumed between them. The effect of infinitesimal noises on a perfectly ordered motion is examined for various spatial dimensions by using a linear approximation. As a result, a macroscopic in-phase oscillation throughout the system turned out impossible for 1 and 2 dimensions, but possible for 3 dimensions. An interesting feature expected for a two-dimensional system is that there is a finite parameter region where characteristic length scale is absent. In order to see the latter feature in more detail, we performed a Langevin simulation for a two-dimensional system, and confirmed the existence of a Kosterlitz-Thouless type parameter region.