The authors give a detailed study of truncated Lévy flights in super-diffusive transport in the presence of an external potential. The presented approach is based on the fractional Fokker-Planck equation and the replacement of the fractional diffusion operator by the tempered fractional diffusion operator. This approach allows for the study of the dependence of the steady-state probability density function and the current on the level of tempering \(\lambda\) and the order of the fractional derivative space \(\alpha\). The importance of Lévy processes is well-known in the literature. This also holds for truncated (tempered) Lévy distributions that allow for solutions of some problems connected with statistics in the case of power-law distributions (especially the infinite values of the second and higher moments). A review of the literature shows that the tempered distributions seem to be very popular and it is important for applications to explore the role of tempering. The authors use two complementary numerical techniques (the finite difference method and the Fourier-based spectral method) to study harmonic confining potentials and periodic potentials with broken spatial symmetry.
In the following sections of the paper, the authors present an introduction to tempered fractional diffusion (Section 2), perturbative and numerical steady-state solutions of the truncated fractional Fokker-Planck equation for a harmonic potential (Section 3). The next section is devoted to periodic potentials with broken spatial symmetry. In this section one can find also the results of numerical experiments.