Systems near a critical point under multiplicative noise and the concept of effective potential

Abstract

This paper presents a general approach to and elucidates the main features of the effective potential, friction, and diffusion exerted by systems near a critical point due to nonlinear influence of noise. The model is that of a general many-dimensional system of coupled nonlinear oscillators of finite damping under frequently alternating influences, multiplicative or additive, and arbitrary form of the power spectrum, provided the time scales of the system's drift due to noise are large compared to the scales of unperturbed relaxation behavior. The conventional statistical approach and the widespread deterministic effective potential concept use the assumptions about a small parameter which are particular cases of the considered. We show close correspondence between the asymptotic methods of these approaches and base the analysis on this. The results include an analytical treatment of the system's long-time behavior as a function of the noise covering all the range of its table- and bell-shaped spectra, from the monochromatic limit to white noise. The trend is considered both in the coordinate momentum and in the coordinate system's space. Particular attention is paid to the stabilization behavior forced by multiplicative noise. An intermittency, in a broad area of the control parameter space, is shown to be an intrinsic feature of these phenomena.