Three basic issues concerning interface dynamics in nonequilibrium pattern formation

Abstract

In these lecture notes, we discuss at an elementary level three themes concerning interface dynamics that play a role in pattern forming systems: (i) We briefly review three examples of systems in which the normal growth velocity is proportional to the gradient of a bulk field which itself obeys a Laplace or diffusion type of equation (solidification, viscous fingers and streamers), and then discuss why the Mullins–Sekerka instability is common to all such gradient systems. (ii) Secondly, we discuss how underlying an effective interface or moving boundary description of systems with smooth fronts or transition zones, is the assumption that the relaxation time of the appropriate order parameter field(s) in the front region is much smaller than the time scale of the evolution of interfacial patterns. Using standard arguments we illustrate that this is generally so for fronts that separate two (meta)stable phases: in such cases, the relaxation is typically exponential, and the relaxation time in the usual models goes to zero in the limit in which the front width vanishes. (iii) We finally summarize recent results that show that so-called “pulled” or “linear marginal stability” fronts which propagate into unstable states have a very slow universal power-law relaxation. This slow relaxation makes the usual “moving boundary” or “effective interface” approximation for problems with thin fronts, like streamers, impossible.