Automodel solutions for Lévy flight-based transport on a uniform background

Abstract

A wide class of non-stationary superdiffusive transport on a uniform background with a powerlaw decay, at large distances, of the step-length probability distribution function (PDF) is shown to possess an automodel solution. The solution for Green function is constructed using the scaling laws for the propagation front (relevant-to-superdiffusion average displacement) and asymptotic solutions far beyond and far in advance of the propagation front. These scaling laws are determined essentially by the long-free-path carriers (Lévy flights). The validity of the suggested automodel solution is proved by its comparison with numerical solutions in the one-dimensional (1D) case of the transport equation with a simple long-tailed PDF with various power-law exponents and in the 3D case of the Biberman-Holstein equation of the resonance radiation transfer for various (Doppler, Lorentz, Voight and Holtsmark) spectral line shapes.

Automodel Solutions of Biberman-Holstein Equation for Stark Broadening of Spectral Lines

“…It is seen that for the Lorentz-dominated line shapes, i.e., moderate and large values of the Voigt parameter a, the accuracy of automodel solution is high in almost the entire space of variables {ρ, t}, quite similar to the case [21,28] of the superdiffusive transport for a 1D model PDF with the power-law wings. However, for a small a, the accuracy dramatically falls down in the essential part of the space of variables {ρ, t}.…”

“…For the propagation front, ρ fr (t), we used in [21,28] the function which appears to be close to the time dependence of the mean displacement:…”

“…Here we append the results [21,27] of analyzing the automodel solution for the Holtsmark line shape with the results for the accuracy of automodel solution. Figure 6 shows the behavior of the automodel function Q G2 (29) and the relative deviation of the automodel solution from the exact one in the range from t min = 40 to t max = 10 3 , with the time step equal to 20, for 40 ≤ t ≤ 500, and 50, for 500 < t ≤ 1000.…”

“…For the time-dependent superdiffusive transport, recently, a wide class of non-steady-state superdiffusive transport on a uniform background with a power-law decay, at large distances, of the step-length probability distribution function (PDF) was shown [21] to possess an approximate automodel solution. The solution for the Green's function was constructed using the scaling laws for the propagation front (i.e., time dependence of the relevant-to-superdiffusion average displacement of the carrier) and asymptotic solutions far beyond and far ahead the propagation front.…”

“…It is important to note that the success of automodel solutions [21] for a wide class of non-stationary superdiffusive transport was achieved due to identification of the scaling laws in the case of radiative transfer in the Biberman-Holstein model in [11][12][13]17,21,[29][30][31][32][33].…”