Theory on Dynamics of Growing Surfaces with Self-Affinity

具有自仿射对称性的生长粗糙表面在自然界和科技领域中广泛存在。我们感兴趣的是这样一个事实，即表面的高度起伏满足一个关于时间和空间的标度律。这个项目的目的是获得作为基底维度d的函数的标度指数。我们期望，即使在远离平衡态的增长现象中也存在普适类，正如临界现象中所知的那样。Kardar，Parisi和Zhang（KPZ）通过引入随机微分方程带来了第一个理论突破。虽然该方程具有非常简单的形式，但从理论角度推导标度指数一直是一个非常困难的问题。我们指出KPZ方程中的白噪声假设对d>2是不成立的。在线性化的版本中，我们证明了不能接受的假设，以获得一个有限的宽度的表面，和表面应该是光滑的d>2。

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.