Theory on Dynamics of Growing Surfaces with Self-Affinity

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Growing rough surfaces with self-affine symmetry are widely seen in nature and technological fields. We are interesting in such a fact that the height fluctuation of surface satisfies a scaling law with respect to time and space. The purpose of this project is to obtain the scaling exponents as functions of substrate dimensionality d. We expect that there are universality classes even in growth phenomena far from equlibrium states, as known in critical phenomena. The first theoretical breakthrough has been brought by Kardar, Parisi and Zhang (KPZ) by introducing a stochastic differential equation. Although this equation has a very simple form, it has been a very difficult problem to derive the scaling exponents from a theoretical point of view. We pointed out that the white-noise assumption in the KPZ equation is not valid for d>2. In the linearized version, we proved that the assumption cannot be accepted to obtain a finite width of surfaces, and the surfaces should be smooth for d>2.