Abstract
Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.
Highlights
Self-similarity in the theory of transport phenomena means that the space-time evolution of the Green’s function, which by definition corresponds to the solution of a transport equation with an instant point source, is described for transport on a uniform background by a function of a single variable
We provide a survey of the progress in the theory of nonstationary radiative transfer in spectral lines in plasmas and gases, achieved due to the elaboration of a method of deriving an approximate analytic self-similar solution for a wide class of nonstationary superdiffusive transport on a uniform background in the framework of integral equation formalism [41,42,43,44,45,46,47,48,49,50]
We start with the case of an infinite velocity of carriers (c = ∞), which is associated with the case of Lévy flights, and continue with the case of a finite constant velocity of carriers (c = const = ∞), which is associated with the case of Lévy walks
Summary
Self-similarity in the theory of transport phenomena means that the space-time evolution of the Green’s function, which by definition corresponds to the solution of a transport equation with an instant point source, is described for transport on a uniform background by a function of a single variable. We provide a survey of the progress in the theory of nonstationary radiative transfer in spectral lines in plasmas and gases, achieved due to the elaboration of a method of deriving an approximate analytic self-similar solution for a wide class of nonstationary superdiffusive transport on a uniform background in the framework of integral equation formalism [41,42,43,44,45,46,47,48,49,50]. The method allows obtaining an approximate self-similar solution that is based on the following three building blocks, namely, scaling law for the propagation front, defined in a way relevant to superdiffusion, and asymptotics of the Green’s function far beyond and far in advance of the propagation front All these characteristics are determined by Lévy flights, in the case of infinite velocity of medium’s perturbation carriers, or Lévy walks, in the case of a finite fixed velocity of the carriers. The high accuracy of the suggested self-similar solution in a broad range of variables of the problem is proved by comparing with numerical solutions of transport equations
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