Stages of relaxation of patterns and the role of stochasticity in the final stage

The disorder function formalism (Gunaratne et al 1998 Phys. Rev. E 57 5146) is used to show that pattern relaxation in an experiment on a vibrating layer of brass beads occurs in three distinct stages. During stage I, all length scales associated with moments of the disorder grow at a single universal rate, given by L(t) ~ t0.5. In stage II, pattern evolution is non-universal and includes a range of growth indices. Relaxation in the final stage is characterized by a single, non-universal index. We use analysis of patterns from the Swift–Hohenberg equation to argue that mechanisms that underlie the observed pattern evolution are linear spatio-temporal dynamics (stage I), nonlinear saturation (stage II), and stochasticity (stage III).