Mesoscopic Theory for Coupled Stochastic Oscillators

The celebrated Ott-Antonsen Ansatz for coupled oscillators provides a useful framework for working with deterministic systems in the thermodynamic limit, but it fails to capture many features of stochastic systems. Several solutions have been recently proposed to accurately describe the behavior of the order parameters in coupled oscillator systems. However, a fluctuating description of such order parameters has been still elusive. In this Letter, I construct for the first time a general mesoscopic description of finite-size populations of oscillators subject to white noise. The theory allows one to derive Langevin equations for the Kuramoto-Daido order parameters, opening the door to study features of synchronization phase transitions and finite-size effects, which were inaccessible before. The analysis of the fluctuations in the stochastic Kuramoto model uncovered highly accurate, closed analytical expressions for the average Kuramoto order parameter which outperforms previous approaches.