Pattern Formation Under Multiplicative Noise

Abstract

Many spatially extended systems can undergo nonequilibrium transitions that lead to inhomogeneous states exhibiting some kind of stable pattern [CH93] or dissipative structure [Man90] that subsists as long as energy is externally introduced into the system. Examples of such patternforming processes out of equilibrium are convective rolls originating in a Rayleigh—Bénard cell heated from below [Cha81] and transverse structures in a wide-aperture laser beam [NM92]. In this chapter we analyze the role of spatiotemporal fluctuations on the dynamics of pattern formation. In analogy to what was observed in Chapter 3, we first show in Section 5.1 that multiplicative noise can induce order by advancing the appearance of a deterministic pattern-forming bifurcation in the Swift—Hohenberg model. In a second kind of model, presented in Section 5.2, noise can be seen to induce pure stochastic patterns, which do not exist in the absence of fluctuations.