非线性阻尼驱动的莱维飞行动力学性质

刘剑; 陈晓白; 徐登辉; 李熊; 陈晓松; 杨波

doi:10.7498/aps.65.160502

摘要

基于动力学连续时间无规行走方法，对受非线性阻尼驱动的莱维飞行在自由势场以及周期势场中的扩散行为进行了研究. 非线性阻尼取代斯托克斯阻尼，通过动力学连续时间无规行走方法体现在莱维随机行走粒子的每一步跳跃中. 结果显示，非线性阻尼的强阻尼耗散作用导致莱维飞行的超扩散行为衰减为正常扩散，粒子速度定态分布呈现双峰与单峰的相互转化. 周期势场的束缚作用会导致粒子扩散达到一个稳定态，而莱维粒子自身性质会使粒子存在极小概率跃出周期势阱的跳跃行为，表现为方均位移随时间的演化出现跃迁现象.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
北京工商大学理学院, 北京 100048;

2.
中国科学院理论物理研究所, 理论物理前沿重点实验室, 北京 100190

通信作者: 刘剑, liujian@mail.bnu.edu.cn

基金项目: 国家自然科学基金（批准号：11547231，61405003，11404013）资助的课题.

Authors and contacts

1.
School of Science, Beijing Technology and Business University, Beijing 100048, China;

2.
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author: Liu Jian, liujian@mail.bnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11547231, 61405003, 11404013).

参考文献

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[3]	Metzler R, Jeon J H, Cherstvy A G, Barkai E 2014 Phys. Chem. Chem. Phys. 16 24128
[4]	Bao J D, Zhuo Y Z 2003 Phys. Rev. Lett. 91 138104
[5]	Montroll E W, Weiss G H 1965 J. Math. Phys. 6 167
[6]	Scher H, Montroll E W 1975 Phys. Rev. B 12 2455
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[8]	Haus W, Kehr K W 1987 Phys. Rep. 150 263
[9]	Liu J, Yang B, Chen X S, Bao J D 2015 Eur. Phys. J. B 88 88
[10]	Zaburdaev V, Denisov S, Klafter J 2015 Rev. Mod. Phys. 87 483
[11]	Jager M, Weissing F J, Herman P M, Noler B A, Koppel J 2011 Science 332 1551
[12]	Harris T H, Banigan E J, Christian D A, Konradt C, Wojno E D T, Norose K, Wilson E H, John B, Weninger W, Luster A D, Liu A J, Hunter C A 2012 Nature 486 545
[13]	Barthelemy P, Bertolotti J, Wiersma D S 2008 Nature 453 495
[14]	Margolin G, Barkai E 2005 Phys. Rev. Lett. 94 080601
[15]	Barkai E, Garini Y, Metzler R 2012 Phys. Today 65 29
[16]	Jespersen S, Metzler R, Fogedby H C 1999 Phys. Rev. E 59 2736
[17]	Shlesinger M F, West B J, Klafter J 1987 Phys. Rev. Lett. 58 11
[18]	Klafter J, Blumen A, Shlesinger M F 1987 Phys. Rev. A 35 7
[19]	Mantegna R N, Stanley H E 1994 Phys Rev. Lett. 73 2946
[20]	Koponen I 1995 Phys. Rev. E 52 1197
[21]	Chechkin A V, Gonchar V, Klafter J, Metzler R 2005 Phys. Rev. E 72 010101
[22]	Lindner B 2010 New J. Phys. 12 063026
[23]	Bao J D, Liu J 2013 Phys. Rev. E 88 022153
[24]	Sagi Y, Brook M, Almog I, Davidson N 2012 Phys. Rev. Lett. 108 093002
[25]	Kessler D A, Barkai E 2012 Phys. Rev. Lett. 108 230602
[26]	Chechkin A V, Gonchar V, Klafter J, Metzler R, Tanatarov L 2002 Chem. Phys. 284 233
[27]	Chechkin A V, Klafter J, Gonchar V, Metzler R, Tanatarov L 2003 Phys. Rev. E 67 010102

施引文献

[1]	Bouchaud J-P, Georges A 1990 Phys. Rep. 195 127
[2]	Havlin S, Ben-Avrahm D 1987 Adv. Phys. 36 695
[3]	Metzler R, Jeon J H, Cherstvy A G, Barkai E 2014 Phys. Chem. Chem. Phys. 16 24128
[4]	Bao J D, Zhuo Y Z 2003 Phys. Rev. Lett. 91 138104
[5]	Montroll E W, Weiss G H 1965 J. Math. Phys. 6 167
[6]	Scher H, Montroll E W 1975 Phys. Rev. B 12 2455
[7]	Metzler R, Klafter J 2000 Phys. Rep. 339 1
[8]	Haus W, Kehr K W 1987 Phys. Rep. 150 263
[9]	Liu J, Yang B, Chen X S, Bao J D 2015 Eur. Phys. J. B 88 88
[10]	Zaburdaev V, Denisov S, Klafter J 2015 Rev. Mod. Phys. 87 483
[11]	Jager M, Weissing F J, Herman P M, Noler B A, Koppel J 2011 Science 332 1551
[12]	Harris T H, Banigan E J, Christian D A, Konradt C, Wojno E D T, Norose K, Wilson E H, John B, Weninger W, Luster A D, Liu A J, Hunter C A 2012 Nature 486 545
[13]	Barthelemy P, Bertolotti J, Wiersma D S 2008 Nature 453 495
[14]	Margolin G, Barkai E 2005 Phys. Rev. Lett. 94 080601
[15]	Barkai E, Garini Y, Metzler R 2012 Phys. Today 65 29
[16]	Jespersen S, Metzler R, Fogedby H C 1999 Phys. Rev. E 59 2736
[17]	Shlesinger M F, West B J, Klafter J 1987 Phys. Rev. Lett. 58 11
[18]	Klafter J, Blumen A, Shlesinger M F 1987 Phys. Rev. A 35 7
[19]	Mantegna R N, Stanley H E 1994 Phys Rev. Lett. 73 2946
[20]	Koponen I 1995 Phys. Rev. E 52 1197
[21]	Chechkin A V, Gonchar V, Klafter J, Metzler R 2005 Phys. Rev. E 72 010101
[22]	Lindner B 2010 New J. Phys. 12 063026
[23]	Bao J D, Liu J 2013 Phys. Rev. E 88 022153
[24]	Sagi Y, Brook M, Almog I, Davidson N 2012 Phys. Rev. Lett. 108 093002
[25]	Kessler D A, Barkai E 2012 Phys. Rev. Lett. 108 230602
[26]	Chechkin A V, Gonchar V, Klafter J, Metzler R, Tanatarov L 2002 Chem. Phys. 284 233
[27]	Chechkin A V, Klafter J, Gonchar V, Metzler R, Tanatarov L 2003 Phys. Rev. E 67 010102

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