Cattivelli Luca, Agliari Elena, Sartori Fabio, Cassi Davide
Scuola Normale Superiore, Pisa, Italy.
Dipartimento di Matematica, Sapienza Università di Roma, Rome, Italy.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):042156. doi: 10.1103/PhysRevE.92.042156. Epub 2015 Oct 28.
We consider a particle performing a stochastic motion on a one-dimensional lattice with jump lengths distributed according to a power law with exponent μ+1. Assuming that the walker moves in the presence of a distribution a(x) of targets (traps) depending on the spatial coordinate x, we study the probability that the walker will eventually find any target (will eventually be trapped). We focus on the case of power-law distributions a(x)∼x(-α) and we find that, as long as μ<α, there is a finite probability that the walker will never be trapped, no matter how long the process is. This result is shown via analytical arguments and numerical simulations which also evidence the emergence of slow searching (trapping) times in finite-size system. The extension of this finding to higher-dimensional structures is also discussed.
我们考虑一个在一维晶格上进行随机运动的粒子，其跳跃长度根据指数为μ + 1的幂律分布。假设步行者在依赖于空间坐标x的目标（陷阱）分布a(x)的情况下移动，我们研究步行者最终找到任何目标（最终被困住）的概率。我们关注幂律分布a(x)∼x(-α)的情况，并且发现，只要μ < α，无论过程持续多长时间，步行者都有永远不被困住的有限概率。这个结果通过解析论证和数值模拟得到证明，数值模拟还证明了在有限尺寸系统中出现缓慢的搜索（捕获）时间。还讨论了这一发现对更高维结构的扩展。
