Lévy flights with inertial term in various potentials are studied analytically and numerically in terms of the fractional Fokker-Planck equation. The probability density functions of the Lévy flight particle in the linear and harmonic potentials are exactly obtained and a transient contribution of inertial term is found in the case of linear potential, which can be neglected when evolution time is much longer than ${\gamma }^{-1}$ ($\gamma$ is the damping coefficient); for the harmonic potential, where the stationary state has still infinite variance of coordinate and its width is larger than that of the inertialess case for arbitrary Lévy index $\mu$ in the region of $0<\mu <2$, we still find that the distribution of velocity of the particle is also the Lévy-type one. Moreover, a crossover from bimodal to unimodal shape for the spatial distribution is shown in the anharmonic potential. Finally, we represent an analytical expression of the rate constant for the inertial Lévy flight particle escaping from a metastable potential by using the reactive flux method and show that the rate is a nonmonotonic function of the Lévy index.

• Received 9 August 2011

DOI:https://doi.org/10.1103/PhysRevE.84.051108