Amplitude equation for N -SPDEs with cubic nonlinearity in mean field limit *

Abstract We derive the amplitude equation for a system of N stochastically forced partial differential equations with cubic nonlinearity near a bifurcation point. The system features linear mean-field coupling, which introduces collective interactions among the components. By decomposing the dynamics into a slow mean-field component and N fast fluctuation modes, we employ an averaging method to derive the amplitude equation, valid both for large N and in the mean-field limit ( N → ∞ ). Our analysis reveals how the interplay between mean-field interactions and additive noise influences the system’s macroscopic behavior near criticality. These results provide a framework for understanding pattern formation and phase transitions in stochastically driven, collectively interacting systems.