Stationary power-law distribution in a generalized aggregation system with injection

在这项研究中，我们研究了带电粒子在晶格上随机扩散和聚集的广义聚集过程。对于一维平均场模型，我们证明了当随机带电粒子被连续注入时，存在一个统计不变的稳态。稳态电荷分布服从幂函数分布，其指数与注入类型和空间尺寸有关。系统对扰动(即松弛)的响应表现为幂指数衰减或压缩指数衰减。作为该模型的应用，我们引入了一个由水蚀形成的地貌模型。水流汇聚形成河流网络，结果是地表被侵蚀。结果表明，地貌在侵蚀作用下会自动生成垂直几何图形。

In this research we studied a generalized aggregation process in which charged particles diffuse and coalesce randomly on a lattice. For 1-dimensional and mean-field models, we show that there exists a statistically-invariant steady state when randomly charged particles are constantly injected. The steady state charge distribution obeys a power law with exponent depending both on the type of the injection and the spatial dimension. The response of the system to a perturbation(i. e., relaxation)is characterized by either a power law decay or a compressed exponential decay.As an application of this model we introduced a model of landform created by water erosion. Water flows aggregate making river networks and the earth's surface is eroded as a result. We showed that the landform automatically makes Practal geometry by the effect of erosion.