Master equation approach to synchronization in diffusion-coupled nonlinear oscillators

We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-diffusion systems through a master equation approach. In the limit of large system size, the probability density is analyzed by means of the eikonal approximation. This approximation yields a Hamilton-Jacobi equation for the stochastic potential, which may be reduced to coupled nonlinear diffusion equations for the phase of oscillation and (conjugate) "momentum." We give explicit expressions for the coefficients of these equations in terms of averages over the deterministic periodic orbit. For one-dimensional systems, we obtain an explicit solution for the stationary stochastic potential: the width in phase, which is defined as the root mean square fluctuation in phase, characterizes the roughness of phase locking, and diverges with the system size L according to a power law w approximately Lalpha, with alpha=1/2. To study higher-dimensional systems, we show that the eikonal approximations of the diffusion-coupled oscillator problem and the Kardar-Parisi-Zhang (KPZ) equation (in the limit of small noise intensity) are equivalent. The KPZ equation governs some forms of surface growth, and the height of a growing front corresponds to the phase (the 2pi periodicity in phase is ignored) in the diffusion-coupled oscillator problem. From the equivalence, we obtain the result that spatially synchronized states may exist only in systems with a spatial dimension greater than or equal to 3; for dimensions 1 and 2, a "rough" state exists in which the width (in phase) diverges algebraically with the system size, alpha>0.