Kinetic roughening of surfaces: Derivation, solution and application of linear growth equations

We present a comprehensive analysis of a linear growth model, which combines the characteristic features of the Edwards--Wilkinson and noisy Mullins equations. This model can be derived from microscopics and it describes the relaxation and growth of surfaces under conditions where the nonlinearities can be neglected. We calculate in detail the surface width and various correlation functions characterizing the model. In particular, we study the crossover scaling of these functions between the two limits described by the combined equation. Also, we study the effect of colored and conserved noise on the growth exponents, and the effect of different initial conditions. The contribution of a rough substrate to the surface width is shown to decay universally as $w_i(0) (\xi_s/\xi(t))^{d/2}$, where $\xi(t) \sim t^{1/z}$ is the time--dependent correlation length associated with the growth process, $w_i(0)$ is the initial roughness and $\xi_s$ the correlation length of the substrate roughness, and $d$ is the surface dimensionality. As a second application, we compute the large distance asymptotics of the height correlation function and show that it differs qualitatively from the functional forms commonly used in the intepretation of scattering experiments.