Goh Segun, Kwon H W, Choi M Y, Fortin J-Y
Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Dec;82(6 Pt 1):061115. doi: 10.1103/PhysRevE.82.061115. Epub 2010 Dec 8.
Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the evolution equation, to obtain such skew distributions as power-law, log-normal, and Weibull distributions, depending on the growth or division and production. Specifically, repeated production of elements of uniform size leads to power-law distributions, whereas production of elements with the size distributed according to the current distribution as well as no production of new elements results in log-normal distributions. Finally, division into two, or binary fission, bears Weibull distributions. Numerical simulations are also carried out, confirming the validity of the obtained solutions.
从一个主方程出发，我们推导出了演化系统中元素尺寸分布的演化方程，其中每个元素都可以生长、一分为二并产生新元素。然后我们探究演化方程的通解，以获得诸如幂律分布、对数正态分布和威布尔分布等偏态分布，这取决于生长、分裂和产生的情况。具体而言，重复产生大小均匀的元素会导致幂律分布，而产生尺寸根据当前分布的元素以及不产生新元素会导致对数正态分布。最后，一分为二，即二元裂变，会产生威布尔分布。我们还进行了数值模拟，证实了所得解的有效性。
