Theory of adiabatic fluctuations: third-order noise

Abstract

We consider the response of a dynamical system driven by external adiabatic fluctuations. Based on the 'adiabatic following approximation' we have made a systematic separation of timescales to carry out an expansion in α|μ|^−L, where α is the strength of fluctuations and |μ| is the damping rate. We show that the probability distribution functions obey the differential equations of motion which contain third-order terms (beyond the usual Fokker-Planck terms) leading to non-Gaussian noise. The problem of adiabatic fluctuations in velocity space which is the counterpart of Brownian motion for fast fluctuations, has been solved exactly. The characteristic function and the associated probability distribution function are shown to be of stable form. The linear dissipation leads to a steady state which is stable and the variances and higher moments are shown to be finite.