2021/04/30 by Santanu Das, Anupam Kundu
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Brownian motion #Combinatorics #Correlation dimension #Diffusion #Diffusion and Search Dynamics #Dimension (graph theory) #Displacement (psychology) #Lévy flight #Mathematical analysis #Mathematics #Mean squared displacement #Physics #Population #Position (finance) #Quantum mechanics #Random walk #Random walker algorithm #Simple (philosophy) #Simple random sample #Statistical physics #Statistics #Stochastic processes and statistical mechanics #cond-mat.stat-mech #physics.class-ph #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1751-8121/ac2473
55 pages, 17 figures
arxiv created 2021/04/30 · openalex publication_date 2021/09/07 · arxiv updated 2021/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract Lévy walk (LW) process has been used as a simple model for describing anomalous diffusion in which the mean squared displacement of the walker grows non-linearly with time in contrast to the diffusive motion described by simple random walks or Brownian motion. In this paper we study a simple extension of the LW model in one dimension by introducing correlation among the velocities of the walker in different (flight) steps. Such correlation is absent in the LW model. The correlations are introduced by making the velocity at a step dependent on the velocity at the previous step in addition to the usual random noise (‘kick’) that the particle gets at random time intervals from the surrounding medium as in the LW model. Consequently the dynamics of the position becomes non-Markovian. We study the statistical properties of velocity and position of the walker at time t , both analytically and numerically. We show how different choices of the distribution of the random time intervals and the degree of correlation, controlled by a parameter r , affect the late time behaviour of these quantities.