arXiv 1807.077952018-07-20cond-mat.stat-mech

膨胀介质中的连续时间随机游动与Fokker-Planck方程

Continuous time random walks and Fokker-Planck equation in expanding media

F. Le Vot, S. B. Yuste

备注：: 14 pages, 5 figures
发表信息：: Phys. Rev. E 98, 042117 (2018)
DOI：: 10.1103/PhysRevE.98.042117

摘要：

我们考虑一个连续的随机游走模型，用于描述粒子在均匀膨胀（或收缩）介质中扩散时受到外力的正常和异常扩散。给出了在给定位置和时间找到粒子的概率分布函数与单步跳跃长度和等待时间的一般方程。当粒子的跳跃长度具有有限方差时，该方程的形式为广义Fokker-Planck方程。在等待时间为重尾的情况下，该广义方程变成分数阶Fokker-Planck方程。这些方程使我们能够研究膨胀、扩散和外力之间的关系。我们建立的条件下，占主导地位的贡献运输源于扩散运输，而不是从漂移由于介质膨胀。我们发现，在一个恒定的外力下，在膨胀介质中描述我们的连续随机游走模型的反常扩散过程是不是伽利略不变，违反广义爱因斯坦关系，并导致传播是定性不同的静态介质中发现的。我们的结果得到了数值模拟的支持。

英文摘要：

We consider a continuous random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates the probability distribution function (pdf) of finding a particle at a given position and time to the single-step jump length and waiting time pdfs is provided. The equation takes the form of a generalized Fokker-Planck equation when the jump length pdf of the particle has a finite variance. This generalized equation becomes a fractional Fokker-Planck equation in the case of a heavy-tailed waiting time pdf. These equations allow us to study the relationship between expansion, diffusion and external force. We establish the conditions under which the dominant contribution to transport stems from the diffusive transport rather than from the drift due to the medium expansion. We find that anomalous diffusion processes under a constant external force in an expanding medium described by means of our continuous random walk model are not Galilei invariant, violate the generalized Einstein relation, and lead to propagators that are qualitatively different from the ones found in a static medium. Our results are supported by numerical simulations.

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