The EW equation, discussed in the previous chapter, was the first continuum equation used to study the growth of interfaces by particle deposition. The predictions of this linear theory change, however, when nonlinear terms are added to the growth equation. The first extension of the EW equation to include nonlinear terms was proposed by Kardar, Parisi and Zhang (KPZ). The KPZ equation, as it has come to be called, is capable of explaining not only the origin of the scaling form (2.8), but also the values of the exponents obtained for the BD model.

Although the KPZ equation cannot be solved in closed form due to its nonlinear character, a number of exact results have been obtained. Moreover, powerful approximation methods, such as dynamic renormalization group, can be used to obtain further insight into the scaling properties and exponents. In this chapter, we introduce the KPZ equation and present some of its key properties. The discussion will lead us to the exact values of the scaling exponents for one-dimensional interfaces. The renormalization group approach to the KPZ equation is then treated in the following chapter.

Construction of the KPZ equation

Although one cannot formally ‘derive’ the KPZ equation, one can develop a set of plausibility arguments using both (i) physical principles, which motivate the addition of nonlinear terms to the linear theory (5.6), and (ii) symmetry principles, as we did in the case of the EW equation.