Dynamics of a randomly kicked particle

TL;DR

This paper extends the Levy walk model by introducing velocity correlations, resulting in a non-Markovian dynamics that affects the statistical properties of the particle's position and velocity over time.

Contribution

It introduces a correlated velocity component into the Levy walk model, creating a non-Markovian process and analyzing its effects on diffusion behavior.

Findings

Correlation parameter r influences late-time dynamics

Velocity and position distributions are affected by correlation

Analytical and numerical methods reveal non-Markovian effects

Abstract

Levy 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.