Lubashevsky Ihor, Friedrich Rudolf, Heuer Andreas
A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Str 38, 119991 Moscow, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Sep;80(3 Pt 1):031148. doi: 10.1103/PhysRevE.80.031148. Epub 2009 Sep 30.
The paper presents a multidimensional model for nonlinear Markovian random walks that generalizes the one we developed previously [I. Lubashevsky, R. Friedrich, and A. Heuer, Phys. Rev. E 79, 011110 (2009)] in order to describe the Lévy-type stochastic processes in terms of continuous trajectories of walker motion. This approach may open a way to treat Lévy flights or Lévy random walks in inhomogeneous media or systems with boundaries in the future. The proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear Langevin force whose intensity is proportional to the magnitude of the velocity for its large values. Based on the singular perturbation technique, the corresponding Fokker-Planck equation is analyzed and the relationship between the system parameters and the Lévy exponent is found. Following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval. In this way we overcome the problem of ascribing to Lévy random-walk non-Markov properties.
本文提出了一种用于非线性马尔可夫随机游走的多维模型，该模型推广了我们之前开发的模型[I. 卢巴舍夫斯基、R. 弗里德里希和A. 霍伊尔，《物理评论E》79, 011110 (2009)]，以便根据游走者运动的连续轨迹来描述 Lévy 型随机过程。这种方法可能为未来处理非均匀介质或有边界系统中的 Lévy 飞行或 Lévy 随机游走开辟一条道路。所提出的模型假设一个漫游粒子的速度受到线性摩擦力和非线性朗之万力的影响，对于较大速度值，非线性朗之万力的强度与速度大小成正比。基于奇异摄动技术，分析了相应的福克 - 普朗克方程，并找到了系统参数与 Lévy 指数之间的关系。实际上，遵循前一篇论文，我们还证明了漫游粒子的异常长位移是由粒子速度的极大波动引起的，其持续时间由系统参数决定，而不是由观测间隔的持续时间决定。通过这种方式，我们克服了将非马尔可夫性质归因于 Lévy 随机游走的问题。
