Levy walks with variable waiting time: a ballistic case

Abstract

The L\'evy walk process for a lower interval of an excursion times distribution ($\alpha<1$) is discussed. The particle rests between the jumps and the waiting time is position-dependent. Two cases are considered: a rising and diminishing waiting time rate $\nu(x)$, which require different approximations of the master equation. The process comprises two phases of the motion: particles at rest and in flight. The density distributions for them are derived, as a solution of corresponding fractional equations. For strongly falling $\nu(x)$, the resting particles density assumes the $\alpha$-stable form (truncated at fronts), and the process resolves itself to the L\'evy flights. The diffusion is enhanced for this case but no longer ballistic, in contrast to the case for the rising $\nu(x)$. The analytical results are compared with Monte Carlo trajectory simulations. The results qualitatively agree with observed properties of human and animal movements.