Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases

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.

New Approach to Cross-Correlation Reflectometry Diagnostics of Nonlocality of Plasma Turbulence

“…In conclusion, we note that the present work is the result of applying a general theoretical approach [28][29][30][31][32][33] to the practical problem of controlled thermonuclear fusion and demonstrates the relevance of the development of analytical approaches in the theory of nonlocal transfer (a modern review of the status and history of analytical approaches related to the theory of nonlocal transfer of radiation can be found in [53]).…”

“…Accounting for the finite speed of light is typical for problems of radiation transfer in astrophysical objects [10][11][12][13], but the problems of nonstationary nonlocal transfer were considered only for the infinite speed of light, and the finiteness of the speed of light was taken into account in the problems of transfer of monochromatic radiation, which is not a superdiffusion. Problems of nonstationary nonlocal transport for a finite fixed velocity of carriers were recently solved, in [29][30][31][32][33], including the tracing in [31] of the transition between transport modes dominated by flights or Lévy walks, i.e., between modes, when, respectively, it is appropriate or inappropriate to neglect the effects of delay due to the finiteness of the velocity of the carriers.…”

“…Analytical methods are well developed for problems of stationary nonlocal transfer, such as, for example, the theory of resonant radiation transfer with complete frequency redistribution in the act of absorption and emission of a photon by an atom or ion in a gas or plasma [15][16][17][18][19][20][21][22][23][24][25][26][27]. For more complex problems, namely nonstationary nonlocal transport, a method for calculating the characteristics of nonlocal transport of interest (primarily an approximate description of the Green's function characterizing the dynamics of the propagation front of the perturbation of a medium from an instantaneous point source) was developed in [28] in the limit of infinite speed of light and generalized to the case taking into account the delay (i.e., the finite speed of light) in [29][30][31][32]; see also review [33]. This method for describing nonstationary superdiffusion processes is applied in this work to describe the kinetics of moving density fluctuations in a turbulent medium.…”

Lévy Walks as a Universal Mechanism of Turbulence Nonlocality

“…The current state of the concept of Lévy flights and walks can be found in an extensive review [11], covering various fields of science. The latest results for Green's function of the problem of nonstationary transfer of resonant radiation in the Biberman-Holstein model are presented in [38], including a generalization of the interpolated self-similarity method [39] to the case of a finite speed of light or other medium excitation carriers in [40][41][42]. Further progress in this direction led in [43] to the derivation of a unified approximate analytical description of the front of the nonstationary Green's function for transfers in the Lévy flights and Lévy walks modes.…”