Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom.
Medical Research Council, Laboratory of Molecular Biology, Neurobiology Division, Cambridge, United Kingdom.
Phys Rev E. 2023 Mar;107(3-1):034115. doi: 10.1103/PhysRevE.107.034115.
We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.
我们提出了一个具有随机跃迁概率的连续时间分数主方程，该跃迁概率跨越了随机游动者的种群，使得有效随机游动表现出总体自我增强。种群异质性产生了一种具有条件跃迁概率的随机游动，该概率随着之前走过的步数增加（自我增强）。通过这种方式，我们建立了具有异质总体的随机游动与具有强记忆的随机游动之间的联系，其中跃迁概率取决于步数的整个历史。我们通过从属关系找到分数主方程的总体平均解，该从属关系涉及在给定时间点计数步数的分数泊松过程和具有自我增强的基础离散随机游动。我们还找到了方差的精确解，即使分数指数趋于 1，方差也表现出超扩散。
