Theory on Dynamics of Growing Surfaces with Self-Affinity

具有自仿射对称性的粗糙表面生长在自然界和技术领域广泛存在。我们对这样一个事实很感兴趣，即表面的高度波动满足一个关于时间和空间的标度定律。本项目的目的是获得作为基底维数d的函数的标度指数。我们期望即使在远离平衡状态的生长现象中也存在普适性类，如在临界现象中所知。第一个理论突破是由Kardar， Parisi和Zhang （KPZ）通过引入随机微分方程带来的。虽然这个方程的形式很简单，但从理论的角度推导出标度指数是一个很困难的问题。我们指出，KPZ方程中的白噪声假设对于d>2是无效的。在线性化的版本中，我们证明了不能接受得到有限宽度曲面的假设，并且曲面在d>2内应该是光滑的。

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.