Synchronization is observed in a variety of physical systems, from electrical networks to flocking birds. These complex systems are typically described at the microscopic level by a large number of interacting basic units. A common approach to studying such systems is to take the infinite- limit and analyze the one-particle distribution function. However, when is large but finite, these calculations are no longer strictly valid, and it becomes crucial to properly account for finite- fluctuations. The study of these fluctuations has been largely numerical. From a theoretical perspective, while finite- fluctuations due to quenched disorder have been studied for few specific cases, the effects of finite- fluctuations arising from annealed disorder remain largely unexplored.
In this work, we analyze a system with both quenched and annealed disorder and systematically reduce the large-deviation principle (LDP) for the one-particle distribution function to an LDP for the macroscopic order parameter. For a variety of models, both equilibrium and non-equilibrium, we observe that the time evolution of the order parameter for a finite-N system follows a Brownian motion in an effective two-dimensional potential, with the Brownian noise vanishing in the infinite- limit. We compare our theoretical results with direct numerical simulations and find excellent agreement.