Kawagoe Kyle, Huber Greg, Pradas Marc, Wilkinson Michael, Pumir Alain, Ben-Naim Eli
Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA.
Department of Physics, University of Chicago, Chicago, Illinois 60637, USA.
Phys Rev E. 2017 Jul;96(1-1):012142. doi: 10.1103/PhysRevE.96.012142. Epub 2017 Jul 21.
We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one, and also one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail P(w)∼w^{-γ} for small weight w. We obtain the exponent γ analytically and find that it varies continuously with the two model parameters. The exponent γ can be positive or negative, so that in one range of parameters small-weight trails are abundant and in the complementary range they are rare.
我们研究了由包含聚集、碎片化和扩散的随机过程形成的轨迹的统计特性。在这个发生在一维空间的随机过程中，两条相邻的轨迹可能合并形成一条更大的轨迹，并且一条轨迹也可能分裂成两条。此外，轨迹进行扩散运动。该模型由两个参数定义，这两个参数量化了碎片化率和碎片大小。在长时间极限下，系统达到稳态，我们关注的是轨迹权重的极限分布。我们发现，对于小权重(w)，轨迹权重的密度具有幂律尾部(P(w) \sim w^{-\gamma})。我们通过解析得到了指数(\gamma)，并发现它随两个模型参数连续变化。指数(\gamma)可以是正的或负的，因此在一个参数范围内小权重轨迹很多，而在互补范围内它们很少见。
