Langevin equations and surface growth

The three approaches usually used to study surface growth are (i) master equation, (ii) stochastic Langevin equation, and (iii) microscopic models. All of them give the same scaling exponents. Recently, Vvedensky et al. [Phys. Rev. E 48, 852 (1993)] derived a stochastic Langevin equation from a master equation of the birth and death type, for the epitaxial growth, demonstrating the equivalence of both approaches. In this paper a stochastic Langevin equation is derived from a discrete model. The results are the same as those obtained by Vvedensky el al. demonstrating that the three approaches are equivalent. As a nontrivial example, our procedure is used to derive the Kardar-Parisi-Zhang (KPZ) equation from the ballistic deposition process. This model with vacancies and overhangs is very difficult to handle, due to the algebraic complications that arise when the master equation approach is used.