Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter

We obtain the stationary probability distribution functions of the order parameter near onset for the one-dimensional real Ginzburg-Landau and Swift-Hohenberg equations with a fluctuating control parameter. A perturbative expansion in the intensity of the fluctuations leads to a hierarchy of Fokker-Planck equations for conditional probability distribution functions that relate components of the order parameter that evolve in different time scales. Successive integration leads to a Fokker-Planck equation for the slowest mode, which we solve analytically for the models studied. In all cases, the probability distribution function above onset is of the form (Formula presented) where (Formula presented) is the slow component of the order parameter and the values of δ and γ depend explicitly on the intensity of the fluctuations. Knowledge of (Formula presented) allows the calculation of an effective bifurcation threshold and of the moments of (Formula presented) above threshold.