Langevin dynamics for Lévy walk with memory

Abstract

Memory effects, sometimes, can not be neglected. In the framework of continuous time random walk, memory effect is modeled by the correlated waiting times. In this paper, we derive the two-point probability distribution of the stochastic process with correlated increments as well as the one of its inverse process, and present the Langevin description of Lévy walk with memory, i.e., correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. The Langevin system exhibits sub-ballistic superdiffusion if the friction force is involved, while it displays super-ballistic diffusion or hyperdiffusion if there is no friction. It is discovered that the correlation of waiting times suppresses the diffusion behavior whether there is friction or not, and the stronger the correlation of waiting times becomes, the slower the diffusion is. In particular, the correlation function, correlation coefficient, ergodicity, and scaling property of the corresponding stochastic process are also investigated.

arXiv comment11 pages, 4 figures

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• Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivity
• Aging two-state process with Lévy walk and Brownian motion
• Langevin picture of anomalous diffusion processes in expanding medium
• Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
• Numerical algorithm for the model describing anomalous diffusion in expanding media