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Lévy-walk-like Langevin dynamics affected by a time-dependent force

Yao Chen and Weihua Deng
Phys. Rev. E 103, 012136 – Published 28 January 2021
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Abstract
Authors
Article Text
  • INTRODUCTION
  • SUBORDINATOR
  • LÉVY WALK WITH A TIME-DEPENDENT FORCE
  • FIRST MOMENT
  • VELOCITY CORRELATION FUNCTION
  • EAMSD AND TAMSD
  • GENERALIZED KLEIN-KRAMERS EQUATION
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The Lévy walk is a popular and more ‘physical’ model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influence of external potentials at almost any time and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the Lévy walk in a time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including the nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, a consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for the space-dependent one.

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    • Received 30 November 2020
    • Accepted 13 January 2021

    DOI:https://doi.org/10.1103/PhysRevE.103.012136

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Nonequilibrium statistical mechanics
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Yao Chen and Weihua Deng*

    • School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China

    • *dengwh@lzu.edu.cn

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    Issue

    Vol. 103, Iss. 1 — January 2021

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    Images

    • Figure 1
      Figure 1

      Sample trajectories of the position process x(t) (solid blue line), velocity process v(t) (dashed black line), and inverse subordinator s(t) (dotted red line) of a Langevin system, Eq. (12), with F(t)=f0sin(ωt), α=0.8, f0=1, and ω=π.

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    • Figure 2
      Figure 2

      First moment 〈x(t)〉 for the stochastic process described by the Langevin equation, Eq. (12), with different α values. Other parameters are f0=1, ω=3, γ=0.5, and τ0=1. From top to bottom, the red squares, blue triangles, and green circles show the simulation results for α=0.5,0.8, and 1.5, respectively. The solid red line, dashed blue line, and dot-dashed green line represent the corresponding theoretical results presented in Eq. (21).

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    • Figure 3
      Figure 3

      Simulation results of the EAMSD and TAMSD for the stochastic process described by the Langevin equation, Eq. (12), with different α values and the parameters f0=1, ω=0.5, γ=2, and τ0=1. Red triangles and solid red lines represent, respectively, the simulation and theoretical results, Eq. (32), for the EAMSD; blue circles and solid blue lines show, respectively, the simulation and theoretical results, Eq. (34), for the TAMSD.

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