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

^{1}

^{2}

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^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{fr}(t) ~ (Dt)

^{1/2}.

## 2. Materials and Methods

#### 2.1. Approximate Self-Similar Solution of the Biberman–Holstein (BH) Equation (BH Step-Length PDF, $c=\infty $, 3D Case)

#### 2.1.1. Biberman–Holstein Equation and Its General Solution

**r**, t), and spectral intensity of resonance radiation. This system is reduced to a single equation for f(

**r**, t), which turns out to be an integral equation that cannot be reduced to a differential equation of diffusion type:

_{v}and the absorption coefficient k

_{v}(for the theory of spectral line shapes, see [51,52,53,54,55,56,57]). In homogeneous media, G depends on the distance between the points of emission and absorption of the photon:

_{0}r and P = p/k

_{0}, where k

_{0}is the absorption coefficient for photons at the frequency, corresponding to the line shape center. This gives for ρ ≠ 0

#### 2.1.2. Propagation Front and Asymptotics of Biberman–Holstein Equation Green’s Function

**r**, t) = δ(

**r**)δ(t). The respective scalings for various line-broadening mechanisms strongly deviate from the diffusion law (see [11,12,13,14,15,16]). For Doppler and Lorentz line shapes, the results [12] may be written in the unified form [14,15] (in dimensionless units):

_{fr}(t):

_{fr}(ρ). Note that Equation (7) is substantiated for large values of dimensionless time, whereas for t ~ 1, it is simply interpolated to an obvious condition ρ

_{fr}(0) = 0.

_{fr}(t) $\gg $ 1 (or, equivalently, for a short time, 1 $\ll $ t $\ll $ t

_{fr}(ρ)), the asymptotics of the Green’s function [12] for Doppler and Lorentz line shapes may be written in the following generalized form:

_{fr}(t), or equivalently t $\gg $ t

_{fr}(ρ) $\gg $ 1, atoms quickly exchange photons in the core of the spectral line shape. This allows us to estimate the Green’s function assuming the local uniformity of the excitation. The respective quasi-plateau solution in the 3D case takes the form:

_{fr}(t) is defined by Equation (7). Numerical calculations of the exact Green’s function [12] show that Equation (11) gives a good scaling for the time dependence of the asymptotic behavior of the Green’s function for various spectral line shapes. However, the absolute values of the plateau in Equation (11) may differ from the asymptotics of the exact Green’s function by a constant. For the Doppler line shape, this constant is of the order of unity, and for the Lorentz line shape, it reaches ∼200. The large value of the constant in the latter case may be explained by the longer tail of the step-length PDF that, in turn, stems from the wider wings of the Lorentz spectral line shape.

#### 2.1.3. Approximate Self-Similar Solution

^{−1}is the function reciprocal to the G function,

_{G}

_{1}and Q

_{G}

_{2}functions on, respectively, space coordinate and time. The results of the validation of the self-similar solution, including the reconstruction of the function g from the comparison of function (12) with computations of the Green’s function for the Doppler, Lorentz, Voigt, and Holtsmark line shapes, are given in Section 3.3.

#### 2.2. A Method of Deriving an Self-similar Green’s Function for Lévy Flights (Simple Step-Length PDF, C = ∞, 1D Case)

_{fr}(t) $\gg $ 1, where the propagation front is defined by Equation (7) (or, equivalently, for a short time, 1 $\ll $ t $\ll $ t

_{fr}(ρ)), the density is determined by the direct population by the carriers emitted by the source. This gives a simple relation (cf. (10))

_{fr}(t), or equivalently t $\gg $ t

_{fr}(ρ)) $\gg $ 1, may be found taking into account the above-mentioned frequent exchange with short-free-path carriers. The latter produces local uniformity of the density. Assuming a plateau-like spatial distribution around the origin, one has (cf. (11)):

_{fr}(t) = ρ(t, s = 1).

#### 2.3. A Method of Deriving a Self-Similar Green’s Function for Lévy Walks (Simple Step-Length PDF, $c=\mathit{const}\ne \infty $, 1D, 2D and 3D Cases)

#### 2.3.1. General Solution of the Time-Dependent Superdiffusive Transport Equation

#### 2.3.2. Asymptotics of the Green’s Function Far Ahead and Far behind the Perturbation Front

#### 2.3.3. Integral Characteristics of Green’s Function

#### 2.3.4. Approximate Self-Similar Solution and Test of Its Accuracy

#### 2.4. Distributed Computing Parameters and Implementation for Verification for Accuracy of the Approximate Self-Similar Solution

#### 2.4.1. A Test of the Proposed Self-Similar Solution for a Simple Model PDF, 1D Case

_{exact}(ρ,t) were calculated. The values of t were evenly spaced on a log scale (100 points per power) in the range from t

_{min}= 30 to t

_{max}= 10

^{8}, creating the numerical mesh of a total of 653 points. The values of s were also evenly spaced on a log scale (50 points per power) in the range from s

_{min}= 0.01 to s

_{max}= 1000, creating the numerical mesh of a total of 501 points. The respective values of ρ are defined by these two numerical meshes using (34). The values of γ were evenly spaced in the range from γ

_{min}= 0.5 to γ

_{max}= 1.5, creating the numerical mesh of a total of 101 points. We used distributed computing to calculate f

_{exact}(ρ,t) on these meshes.

#### 2.4.2. A Test of the Self-Similar Solution of Biberman–Holstein Equation

_{exact}(ρ,t). The values of t were evenly spaced on a log scale in the range from t

_{min}= 30 to t

_{max}= 10

^{8}. The values of s were also evenly spaced on a log scale. For the Voigt line shape, depending on the value of a, the lower limit for s was set in the range from 0.00003 (small a) to 0.005 (large a), whereas the upper limit was set to 1000 for all values of a. The respective values of ρ are defined by the t and s numerical meshes, using (18).

## 3. Results

#### 3.1. Illustration of Nonlocality by Monte Carlo Calculations of Trajectories

#### 3.2. A Test of the Proposed Self-Similar Solution (Simple Step-Length PDF, $c=\infty $, 1D Case)

_{10%}, as a function of time only, i.e., for the entire range of space coordinates, is shown in Figure 3.

_{w}(s,t) (33) are shown for different values of t in the range from t

_{10%}(γ) to t

_{max}= 10

^{8}. The normalized functions Q

_{w}(s,t)/{Q

_{w}}

_{av}(s), where subscript av denotes averaging over time from t

_{min}= 30 to t

_{max}= 10

^{8}, and the relative errors of the self-similar solution f

_{auto}(ρ,t)/f

_{exact}(ρ,t) are shown for the same range of time.

_{min}, 0} for γ < 1.3 to the region s~0.2 for γ > 1.3 (Figure 4b, Figure 5b and Figure 6b). This corresponds to a kink in the curve in Figure 3, marked with a vertical line. The self-similar function g(s), as a function of a single variable, is formed with high accuracy in the very wide range of {t, ρ} space (Figure 4a, Figure 5a and Figure 6a). The highest superdiffusivity, being produced by the longest tail of the PDF, needs, as expected, the longest computation time of the exact solution, whereas the applicability of the self-similar solutions is limited only by the expected violation at not large values of time. This illustrates the importance of self-similar solutions for the most time-consuming problems.

#### 3.3. Verification for Accuracy of the Approximate Self-Similar Solution for Various Spectral Line Shapes (Biberman–Holstein Step-Length PDF, $c=\infty $, 3D Case)

#### 3.3.1. Lorentz Spectral Line Shape

_{ν}and respective absorption coefficient k

_{ν}are taken in the form [12]:

_{G}

_{1}and Q

_{G}

_{2}functions on, respectively, space coordinate and time. The results of the validation of the self-similar solution and the reconstruction of function g from comparison with exact solution (5), using (51) for ε

_{ν}and k

_{ν}, with the help of Equations (15) and (16) are shown in Figure 7a,b.

#### 3.3.2. Doppler Spectral Line Shape

_{ν}and respective absorption coefficient k

_{ν}are taken in the form [12]:

_{ν}and k

_{ν}, with the help of Equations (15) and (17) are shown in Figure 7c,d.

#### 3.3.3. Voigt Spectral Line Shape

_{ν}is taken in the form [51]:

_{ν}(a) has the form

_{0}is the density of absorbing atoms, λ is the wavelength of a photon, and ${g}_{i}$ is the statistical weight of the i-th level.

_{G2}(s,t) (17) are shown for different values of t in the range from t

_{min}= 30 to t

_{max}= 10

^{6}. The normalized values of these functions are defined as Q

_{G2}(s,t)/{Q

_{G2}}

_{av}(s), where subscript av denotes averaging over time from t

_{min}= 30 to t

_{max}= 10

^{8}. These normalized functions and the relative errors of the self-similar solution f

_{auto}(ρ,t)/f

_{exact}(ρ,t) are shown for the same range of time.

#### 3.3.4. Holtsmark Spectral Line Shape

_{ν}and k

_{ν}, which enter the function J(p), for linear Stark effect may be expressed in the form (cf. [54]):

_{ν}and k

_{ν}, ${\nu}^{\prime}=\left(\nu -{\nu}_{0}\right)/\mathsf{\Delta}{\nu}_{H}$, takes the form:

_{G2}(17) and the relative deviation of the self-similar 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.

#### 3.4. Verification for Accuracy of the Approximate Self-Similar Solution for Lévy Walks (Simple Step-Length PDF, $c=\mathit{const}\ne \infty $, 1D, 2D and 3D Cases)

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Shlesinger, M.; Zaslavsky, G.M.; Frisch, U. (Eds.) Lévy Flights and Related Topics in Physics; Springer: Berlin, Germany, 1995; ISBN 978-3-662-14048-2. [Google Scholar]
- Dubkov, A.A.; Spagnolo, B.; Uchaikin, V.V. Lévy flight superdiffusion: An introduction. Int. J. Bifurc. Chaos
**2008**, 18, 2649–2672. [Google Scholar] [CrossRef] [Green Version] - Klafter, J.; Sokolov, I.M. Anomalous diffusion spreads its wings. Phys. World
**2005**, 18, 29–32. [Google Scholar] [CrossRef] - Eliazar, I.I.; Shlesinger, M.F. Fractional motions. Phys. Rep.
**2013**, 527, 101–129. [Google Scholar] [CrossRef] - Shlesinger, M.F.; Klafter, J.; Wong, Y.M. Random walks with infinite spatial and temporal moments. J. Stat. Phys.
**1982**, 27, 499–512. [Google Scholar] [CrossRef] - Zaburdaev, V.Y.; Chukbar, K.V. Enhanced superdiffusion and finite velocity of Lévy flights. J. Exp. Theor. Phys.
**2002**, 94, 252–259. [Google Scholar] [CrossRef] - Zaburdaev, V.; Denisov, S.; Klafter, J. Lévy walks. Rev. Mod. Phys.
**2015**, 87, 483. [Google Scholar] [CrossRef] [Green Version] - Mandelbrot, B.B. The Fractal Geometry of Nature; W. H. Freeman: New York, NY, USA, 1982; ISBN 0-7167-1186-1189. [Google Scholar]
- Biberman, L.M. On the diffusion theory of resonance radiation. Zh. Eksp. Teor. Fiz.
**1947**, 17, 416. [Google Scholar] - Holstein, T. Imprisonment of Resonance Radiation in Gases. Phys. Rev.
**1947**, 72, 1212–1233. [Google Scholar] [CrossRef] - Biberman, L.M.; Vorob’ev, V.S.; Yakubov, I.T. Kinetics of Nonequilibrium Low Temperature Plasmas; Consultants Bureau: New York, NY, USA, 1987; ISBN 978-1-4684-1667-1. [Google Scholar]
- Veklenko, B.A. Green’s Function for the Resonance Radiation Diffusion Equation. Sov. Phys. JETP
**1959**, 36, 138–142. [Google Scholar] - Biberman, L.M. Approximate method of describing the diffusion of resonance radiation. Dokl. Akad. Nauk. SSSR Ser. Phys.
**1948**, 59, 659. (In Russian) [Google Scholar] - Kogan, V.I. A Survey of Phenomena in Ionized Gases (Invited Papers). In Proceedings of the ICPIG’67, Vienna, Austria, 27 August–2 September 1968; p. 583. (In Russian). [Google Scholar]
- Kogan, V.I. Encyclopedia of Low Temperature Plasma. Introduction Volume; Fortov, V.E., Ed.; Nauka/Interperiodika: Moscow, Russia, 2000; Volume 1, p. 481. (In Russian) [Google Scholar]
- Abramov, V.A.; Kogan, V.I.; Lisitsa, V.S. Radiative transfer in plasmas. In Reviews of Plasma Physics; Leontovich, M.A., Kadomtsev, B.B., Eds.; Consultants Bureau: New York, NY, USA, 1987; Volume 12, p. 151. [Google Scholar]
- Kalkofen, W. (Ed.) Methods in Radiative Transfer; Cambridge University Press: Cambridge, UK, 1984; ISBN 0-521-25620-8. [Google Scholar]
- Rybicki, G.B. Escape Probability Methods. In Methods in Radiative Transfer; Kalkofen, W., Ed.; Cambridge University Press: Cambridge, UK, 1984; Chapter 1; ISBN 0-521-25620-8. [Google Scholar]
- Napartovich, A.P. On the τ
_{eff}method in the radiative transfer theory. High. Temp.**1971**, 9, 23–26. [Google Scholar] - Biberman, L.M.; Vorob’ev, V.S.; Lagar’kov, A.N. Radiative transfer in ionization continuum. Opt. Spectrosc.
**1965**, 19, 326. (In Russian) [Google Scholar] - Kukushkin, A.B.; Lisitsa, V.S.; Savel’ev, Y.A. Nonlocal transport of thermal perturbations in a plasma. JETP Lett.
**1987**, 46, 448–451. Available online: http://www.jetpletters.ac.ru/ps/1232/article_18610.pdf (accessed on 26 February 2021). - Rosenbluth, M.N.; Liu, C.S. Cross-field energy transport by plasma waves. Phys. Fluids
**1976**, 19, 815–818. [Google Scholar] [CrossRef] - Kukushkin, A.B. Analytic description of energy loss by a bounded inhomogeneous hot plasma due to the emission of electromagnetic waves. JETP Lett.
**1992**, 56, 487–491. Available online: http://www.jetpletters.ac.ru/ps/1293/article_19528.pdf (accessed on 26 February 2021). - Kukushkin, A.B. Heat transport by cyclotron waves in plasmas with strong magnetic field and highly reflecting walls. In Proceedings of the 14th IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research, Wuerzburg, Germany, 30 September–7 October 1992; International Atomic Energy Agency: Vienna, Austria, 1993; Volume 2, p. 35. [Google Scholar]
- Kukushkin, A.B. Generalized Escape-Probability Method in the Theory of High-Intensity Radiative Transfer in Continuous Spectra. In Proceedings of the AIP Conference Proceedings 299, Dense Z-pinches 3rd International Conference, London, UK, 19–23 April 1993; Haines, M., Knight, A., Eds.; AIP Press: New York, NY, USA, 1994; p. 519. [Google Scholar] [CrossRef]
- Tamor, S. Calculation of Energy Transport by Cyclotron Radiation in Fusion Plasmas. Fusion. Technol.
**1983**, 3, 293–303. [Google Scholar] [CrossRef] - Tamor, S. Synchrotron radiation loss from hot plasma. Nucl. Instr. Meth. Phys. Res.
**1988**, A271, 37–40. [Google Scholar] [CrossRef] - Tamor, S. A Simple Fast Routine for Computation of Energy Transport by Synchrotron Radiation in Tokamaks and Similar Geometries; Science Applications, Inc.: La Jolla, CA, USA, 1981; Lab. for Applied Plasma Studies Report SAI-023-81-189 LJ0LAPS-72, Science Applications. [Google Scholar]
- Abramov, Y.Y.; Napartovich, A.P. Transfer of resonance line radiation from a point source in the half-space. Astrofizika
**1969**, 5, 187–202. (In Russian) [Google Scholar] - Abramov, Y.Y.; Napartovich, A.P. The excitation wave caused by a light flare. Astrofizika
**1968**, 4, 195–206. (In Russian) [Google Scholar] - Levinson, I.B. Resonant-radiation transfer and nonequilibrium phonons in ruby. Zh. Eksp. Teor. Fiz.
**1978**, 75, 234–248. [Google Scholar] - Subashiev, A.V.; Semyonov, O.; Chen, Z.; Luryi, S. Temperature controlled Lévy flights of minority carriers in photoexcited bulk n-InP. Phys. Lett. A
**2014**, 378, 266–269. [Google Scholar] [CrossRef] [Green Version] - Luryi, S.; Semyonov, O.; Subashiev, A.V.; Chen, Z. Direct observation of Lévy flights of holes in bulk n-doped InP. Phys. Rev. B
**2012**, 86, 201201(R). [Google Scholar] [CrossRef] [Green Version] - Ivanov, V. Transfer of Radiation in Spectral Lines; NBS Special Publication no 385; US Govt Printing Office: Washington, DC, USA, 1973.
- Mihalas, D. Stellar Atmospheres; Freeman: San Francisco, CA, USA, 1970; p. 399. [Google Scholar]
- Barthelemy, P.; Bertolotti, J.; Wiersma, D.S. A Lévy flight for light. Nature
**2008**, 453, 495–498. [Google Scholar] [CrossRef] [PubMed] - Mercadier, N.; Chevrollier, M.; Guerin, W.; Kaiser, R. Microscopic characterization of Lévy flights of light in atomic vapors. Phys. Rev. A
**2013**, 87, 063837. [Google Scholar] [CrossRef] [Green Version] - Pereira, E.; Martinho, J.; Berberan-Santos, M. Photon Trajectories in Incoherent Atomic Radiation Trapping as Lévy Flights. Phys. Rev. Lett.
**2004**, 93, 120201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Uchaikin, V.V. Self–similar anomalous diffusion and Lévy–stable laws. Phys. Usp.
**2003**, 46, 821–849. [Google Scholar] [CrossRef] - Chukbar, K.V. Stochastic transport and fractional derivatives. JETP
**1995**, 81, 1025. [Google Scholar] - Kukushkin, A.B.; Sdvizhenskii, P.A. Automodel solutions for Lévy flight-based transport on a uniform background. J. Phys. A Math. Theor.
**2016**, 49, 255002. [Google Scholar] [CrossRef] [Green Version] - Kukushkin, A.B.; Sdvizhenskii, P.A. Accuracy analysis of automodel solutions for Lévy flight-based transport: From resonance radiative transfer to a simple general model. J. Phys. Conf. Series
**2017**, 941, 012050. [Google Scholar] [CrossRef] - Kukushkin, A.B.; Sdvizhenskii, P.A. Scaling Laws for Non-Stationary Biberman-Holstein Radiative Transfer. In Proceedings of the 2014 41st EPS Conference on Plasma Physics, Berlin, Germany, 23–27 June 2014; European Conference Abstracts. Volume 38F. P4.133. Available online: http://ocs.ciemat.es/EPS2014PAP/pdf/P4.133.pdf (accessed on 26 February 2021).
- Kukushkin, A.B.; Sdvizhenskii, P.A.; Voloshinov, V.V.; Tarasov, A.S. Scaling laws of Biberman-Holstein equation Green function and implications for superdiffusion transport algorithms. Int. Rev. Atom. Mol. Phys.
**2015**, 6, 31–41. [Google Scholar] - Kukushkin, A.B.; Neverov, V.S.; Sdvizhenskii, P.A.; Voloshinov, V.V. 2018 Automodel Solutions of Biberman-Holstein Equation for Stark Broadening of Spectral Lines. Atoms
**2018**, 6, 43. [Google Scholar] [CrossRef] [Green Version] - Kulichenko, A.A.; Kukushkin, A.B. Superdiffusive Transport of Biberman-Holstein Type for a Finite Velocity of Carriers: General Solution and the Problem of Automodel Solutions. Int. Rev. Atom. Mol. Phys.
**2017**, 8, 5–14. Available online: http://www.auburn.edu/cosam/departments/physics/iramp/8_1/Kulichenko_Kukushkin.pdf (accessed on 26 February 2021). - Kukushkin, A.B.; Neverov, V.S.; Sdvizhenskii, P.A.; Voloshinov, V.V. Numerical Analysis of Automodel Solutions for Superdiffusive Transport. Int. J. Open Inf. Technol.
**2018**, 6, 38–42. [Google Scholar] - Kukushkin, A.B.; Kulichenko, A.A. Automodel solutions for superdiffusive transport by Lévy walks. Phys. Scripta
**2019**, 94, 115009. [Google Scholar] [CrossRef] [Green Version] - Kulichenko, A.A.; Kukushkin, A.B. Superdiffusive Transport Based on Lévy Walks in a Homogeneous Medium: General and Approximate Self-Similar Solutions. J. Exp. Theor. Phys.
**2020**, 130, 873–885. [Google Scholar] [CrossRef] - Kukushkin, A.B.; Kulichenko, A.A.; Sokolov, A.V. Optimization identification of superdiffusion processes in biology: An algorithm for processing observational data and a self-similar solution of the kinetic equation. arXiv
**2020**, arXiv:2007.06064. [Google Scholar] - Frish, S.E. Optical Spectra of Atoms; Fizmatgiz: Moscow-Leningrad, Russia, 1963. (In Russian) [Google Scholar]
- Griem, H.R. Principles of Plasma Spectroscopy; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Sobel’man, I.I. Introduction to the Theory of Atomic Spectra; Pergamon Press: Oxford, UK, 1972. [Google Scholar]
- Kogan, V.I.; Lisitsa, V.S.; Sholin, G.V. Broadening of spectral lines in plasmas. In Reviews of Plasma Physics; Kadomtsev, B.B., Ed.; Consultants Bureau: New York, NY, USA, 1987; Volume 13, pp. 261–334. [Google Scholar]
- Bureyeva, L.A.; Lisitsa, V.S. A Perturbed Atom.; CRC Press: Boca Raton, FL, USA, 2000; ISBN 978-9058231383. [Google Scholar]
- Oks, E. Diagnostics Of Laboratory And Astrophysical Plasmas Using Spectral Lines Of One-, Two-, and Three-Electron. Systems; World Scientific: Hackensack, NJ, USA, 2017; ISBN 978-981-4699-07-5. [Google Scholar] [CrossRef] [Green Version]
- Demura, A.V. Beyond the Linear Stark Effect: A Retrospective. Atoms
**2018**, 6, 33. [Google Scholar] [CrossRef] [Green Version] - Sukhoroslov, O.; Volkov, S.; Afanasiev, A.A. Web-Based Platform for Publication and Distributed Execution of Computing Applications. In Proceedings of the 14th International Symposium on Parallel and Distributed Computing (ISPDC), Limassol, Cyprus, 29 June–2 July 2015; pp. 175–184. [Google Scholar]
- Volkov, S.; Sukhoroslov, O. A Generic Web Service for Running Parameter Sweep Experiments in Distributed Computing Environment. Procedia Comput. Sci.
**2015**, 66, 477–486. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Comparison of Veklenko’s [12] solution (5) (solid red curve) with asymptotics far in advance of the propagation front (10) (dashed blue curve) and far behind the propagation front (11) (dashed green curve) and with ordinary diffusion Green’s function (dotted magenta curve).

**Figure 2.**Typical trajectories of migrants, standing at the origin at the zero point in time, for a period of time 1000, $\gamma =0.5$, ${R}_{c}=1.$ (

**a**) Four trajectories in the dimensionless $x,y$ coordinates (i.e., multiplied by ${\kappa}_{0}$). The stops are shown as dots. Long runs (Lévy flights) connect sections with short (Brownian-like) walks (see inset in the figure). (

**b**) Twelve trajectories in $r,t$ coordinates $\left(r=\sqrt{{x}^{2}+{y}^{2}}\right)$.

**Figure 3.**The values of time, t

_{10%}, for which the condition 0.9 ≤ f

_{auto}(ρ,t > t

_{min})/f

_{exact}(ρ,t > t

_{min}) ≤ 1.1 is met for various values of γ.

**Figure 4.**Functions Q

_{W}(s,t) (33) for γ = 0.5 and different values of t in the range from t

_{10%}= 33.66 to t

_{max}= 10

^{8}(

**a**). Normalized functions Q

_{W}(s,t)/{Q

_{W}}

_{av}(s) for the same range of t (

**b**). The relative errors of the self-similar solution, f

_{auto}(ρ,t)/f

_{exact}(ρ,t), for the same range of t (

**c**).

**Figure 5.**The same plots as in Figure 4 but for γ = 1.0 and the values of t in the range from t

_{10%}= 46.47 to t

_{max}= 10

^{8}(

**a**). Normalized functions Q

_{W}(s,t)/{Q

_{W}}

_{av}(s) for the same range of t (

**b**). The relative errors of the self-similar solution, f

_{auto}(ρ,t)/f

_{exact}(ρ,t), for the same range of t (

**c**).

**Figure 6.**The same plots as in Figure 4 but for γ = 1.5 and the values of t in the range from t

_{10%}= 1853.15 to t

_{max}= 10

^{8}(

**a**). Normalized functions Q

_{W}(s,t)/{Q

_{W}}

_{av}(s) for the same range of t (

**b**). The relative errors of the self-similar solution, f

_{auto}(ρ,t)/f

_{exact}(ρ,t), for the same range of t (

**c**).

**Figure 7.**The self-similar function g in Equation (12), reconstructed from comparison with the exact solution with the help of Equations (15) and (17) for Lorentz and Doppler spectral line shapes. Function Q

_{G}

_{2}for ρ > ρ

_{min}, t > t

_{min}for (

**a**) dispersional (Lorentz) and (

**c**) Doppler spectral line shapes. Projection of the ratio Q

_{G}

_{2}(s, t)/Q

_{G}

_{2}(s, t*) onto the {Q

_{G}

_{2}, t} plane for (

**b**) dispersional (Lorentz), t* = 50, and (

**d**) Doppler, t* = 1500, spectral line shapes. The dependence of the error, which is given by the deviation from unity, on the t

_{min}value is shown in the right-hand-side figures. The range of variable s is limited by the conditions ρ < ρ

_{max}.

**Figure 8.**The result of accuracy analysis of self-similar solution for various values of the Voigt parameter and the propagation front ρ = ρ

_{fr}taken in the form of Equation (7), for different values of t in the range from t

_{min}= 30 to t

_{max}= 10

^{6}: (

**left**) functions Q

_{G2}(s,t) (17); (

**center**) normalized functions Q

_{G2}(s,t)/{Q

_{G2}}

_{av}(s), where subscript av denotes averaging over time from t

_{min}= 30 to t

_{max}= 10

^{8}; (

**right**) relative errors of the self-similar solution f

_{auto}(ρ,t)/f

_{exact}(ρ,t).

**Figure 9.**The level lines of the relative deviation of the self-similar solution from the exact one, for the results from Figure 8. The propagation front (7) is shown with a red dashed line.

**Figure 11.**The same as in Figure 9 but for the propagation front ρ = ρ

_{fr}(red dashed line) taken in the form (8).

**Figure 12.**The result of accuracy analysis of self-similar solution for the propagation front ρ = ρ

_{fr}taken in the form (7), for different values of t in the range from t

_{min}= 40 to t

_{max}= 10

^{3}with the time step equal to 20, if 40 ≤ t ≤ 500, and 50, if 500 < t ≤ 1000: (

**a**) functions Q

_{G2}(s,t) (17); (

**b**) relative error of the self-similar solution, f

_{auto}(ρ,t)/f

_{exact}(ρ,t).

**Figure 13.**The solution for the Green’s function (35) by the Monte Carlo method for two-dimensional transport (blue curve); the asymptotics of the Green’s function near the ballistic cone (42) (orange dashed curve); the front law (48) (vertical black dotted line) for (

**a**) ${R}_{c}=5,$ (

**b**) ${R}_{c}=10,$ (

**c**) ${R}_{c}=100$, $\gamma =1$ and $t=100$.

**Figure 15.**Self-similarity function ${Q}_{1}\left(s,t,{R}_{c},\gamma \right)$ (50) for ${R}_{c}=0.1;1;5;10;100$ and $\gamma =1$ versus $s$ at different times in the (

**a**,

**b**) three- and (

**c**–

**g**) two-dimensional cases.

**Figure 16.**Relative deviation of the self-similarity function ${Q}_{1}\left(s,t,{R}_{c},\gamma \right)$ (50) for ${R}_{c}=0.1;1;5;10;100$ and $\gamma =1$ versus $s$ at different times in the (

**a**,

**b**) three- and (

**c**–

**g**) two-dimensional cases.

**Figure 17.**Self-similar solutions (46) constructed from the self-similarity function ${Q}_{1}\left(s,t=3000,{R}_{c},\gamma \right)$ at the maximum time (dotted lines); general solutions (40) (solid lines). (

**a**,

**b**) Direct calculation of the Green’s function (40) in the three-dimensional case; (

**c**–

**g**) Monte Carlo simulation of trajectories in accordance with the transport Equation (35) in the two-dimensional case. Results are presented for ${R}_{c}=0.1;1;5;10;100$ and $\gamma =1$.

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**MDPI and ACS Style**

Kukushkin, A.B.; Kulichenko, A.A.; Neverov, V.S.; Sdvizhenskii, P.A.; Sokolov, A.V.; Voloshinov, V.V.
Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases. *Symmetry* **2021**, *13*, 394.
https://doi.org/10.3390/sym13030394

**AMA Style**

Kukushkin AB, Kulichenko AA, Neverov VS, Sdvizhenskii PA, Sokolov AV, Voloshinov VV.
Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases. *Symmetry*. 2021; 13(3):394.
https://doi.org/10.3390/sym13030394

**Chicago/Turabian Style**

Kukushkin, Alexander B., Andrei A. Kulichenko, Vladislav S. Neverov, Petr A. Sdvizhenskii, Alexander V. Sokolov, and Vladimir V. Voloshinov.
2021. "Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases" *Symmetry* 13, no. 3: 394.
https://doi.org/10.3390/sym13030394