人口迁移驱动的广义总量增长模型--渐近分布、幂律和表观分形
Generalized Model of Migration-Driven Aggregate Growth - Asymptotic Distributions, Power Laws and Apparent Fractality
摘要：
将以幂律交换速率（$\sim{n}^α$）计算的$n$团簇间单体交换驱动聚集的速率方程简化为Fokker-Planck方程。在无偏集结条件下，得到了它的精确解，并与其它模型的渐近结论一致。从精确解到威布尔分布/正态分布/指数分布，再到幂律分布，都存在渐近过渡。发现中间渐近尺寸分布是指数$α$的函数，并且从正态分布（$α=0$）通过Weibull分布（$0<α<1$）变化到指数分布（$α=1$），这给出了用于连接这些基本统计分布的新系统。在初始速率方程的基础上，对无偏聚集模型进行了模拟，而没有简化为Fokker-Planck方程。确认了精确解，通过分析累积分布函数和平均簇大小来确定威布尔分布（对于$0<α<1$）的形状和尺度参数，这是非常重要的，因为它们可以在实验中测量并允许识别聚集动力学的细节（如$α$）。在实际意义上，对聚集团分布的\emph{evolving series}进行标度分析，可以得到比\emph{solitary}分布更可靠的参数估计。据推测，实验观察到的一些表观功率和分形规律可能是这种简单的迁移驱动的聚集动力学的表现。
英文摘要：
The rate equation for exchange-driven aggregation of monomers between clusters of size $n$ by power-law exchange rate ($\sim{n}^α$), where detaching and attaching processes were considered separately, is reduced to Fokker-Planck equation. Its exact solution was found for unbiased aggregation and agreed with asymptotic conclusions of other models. Asymptotic transitions were found from exact solution to Weibull/normal/exponential distribution, and then to power law distribution. Intermediate asymptotic size distributions were found to be functions of exponent $α$ and vary from normal ($α=0$) through Weibull ($0<α<1$) to exponential ($α=1$) ones, that gives the new system for linking these basic statistical distributions. Simulations were performed for the unbiased aggregation model on the basis of the initial rate equation without simplifications used for reduction to Fokker-Planck equation. The exact solution was confirmed, shape and scale parameters of Weibull distribution (for $0<α<1$) were determined by analysis of cumulative distribution functions and mean cluster sizes, which are of great interest, because they can be measured in experiments and allow to identify details of aggregation kinetics (like $α$). In practical sense, scaling analysis of \emph{evolving series} of aggregating cluster distributions can give much more reliable estimations of their parameters than analysis of \emph{solitary} distributions. It is assumed that some apparent power and fractal laws observed experimentally may be manifestations of such simple migration-driven aggregation kinetics even.
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