Dynamical mechanism of Lévy flight driven by the nonlinear friction
Authors: Liu Jian  , Chen Xiao-Bai  , Xu Deng-Hui  , Li Xiong  , Chen Xiao-Song  , Yang Bo  ,   School of Science, Beijing Technology and Business University, Beijing 100048, China;,   State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
Published: December 2020

As a basic problem, anomalous diffusions in various fields of physics and related science have been studied for several decades. One of the topic problems of anomalous diffusion is Lévy flight, which is employed as the statistical model to solve the problems in various fields. Therefore, studying the dynamical mechanism of Lévy flight, especially in the existence of external potential, is of importance for relative theoretical and experimental research. In this paper, within the framework of dynamical continuous time random walk method, the Lévy flight diffusive behaviors and dynamical mechanisms driven by nonlinear friction are studied in the force-free potential and periodic potential. The nonlinear friction instead of Stokes friction is considered in each step of Lévy random walker through the dynamical continuous time random walk method. In the force-free potential, the nonlinear friction term can be considered to be inharmonic potential in the velocity space which can restrain the velocity of random walker, so the anomalous superdiffusion of Lévy flight turns into a behavior in the normal case because of the strong dissipative effect of the nonlinear friction. Due to the introduction of the nonlinear friction, the velocity steady probability density distribution behaves as transitions between bimodal shape and unimodal shape, which is detrmined by the Lévy index μ and the friction indexes γ0 and γ2. The bimodality is most pronounced at μ =1, with μ increasing the bimodality becomes weaker, and vanishes at μ =2 which is the Gaussian case. Besides, there is a critical value γ0c=0.793701, which also determines the bimodal behaviors. For γ0=0 the bimodality is most pronounced, as γ0 increases it smooths out and turns into a unimodal one for γ0 > γ0c. In the existence of periodic potential, the Lévy random walker can be captured by the periodical potential due to the introduction of nonlinear friction, which behaves as the mean square displacement x2(t)> of the random walker and can reach a steady state quite quickly after a short lag time. However, the restraint is not equivalent to truncation procedures. Since the velocity of random walker obeys Lévy distribution, there is still extremely large jump length for random walker with extremely small probability. When the extremely large jump length is long enough and the barrier height U0 is not comparably high, the random walker can cross the barrier height of the periodic potential and jump out of the periodic potential, which behaves as the mean square displacement x2(t)> and a leap from a steady state to another one appears. However, the restraint on the random walker from the nonlinear friction always exists, so the random walker is captured again by the periodic potential, which means that the mean square displacement comes into a steady state again.