The existence and dynamics of interfaces played a central role in the description of the domain-coarsening phenomena considered in the previous chapters. In the late stages of domain growth the random forces in the order parameter kinetic equations were suppressed and the interface dynamics was treated deterministically. In this chapter we provide a more detailed treatment of the effects of noise and diffusion on the structure of the interface. One may capture the essential physics of diffusively rough interfaces in a general model often called the Kardar–Parisi–Zhang (KPZ) equation (Kardar et al., 1986).

KPZ equation

Consider a front in a (d + 1)-dimensional system extended along x and moving, on average, in the x1 direction (see Fig. 16.1). The system is assumed to be infinitely extended along x1 and has linear dimension L along x. In contrast to the description in Chapter 7, we neglect the intrinsic thickness of the interface and investigate the effects of diffusion and noise on the dynamics of the interfacial profile. Referring to Fig. 16.1, let h(x, t) be the position of the interface as a function of x at time t, relative to an arbitrarily selected origin. Its mean position at time t is. We assume that the front propagates with velocity v in a direction normal to its interface; noise provides a destabilizing influence on the front while diffusion tends to remove any surface roughness.

A sketch of a portion of the interface profile h(x, t) as a function of x at time t is shown in Fig. 16.2.