Automodel solutions for superdiffusive transport by the Levy walks

Transcription

1 Automodel solutions for superdiffusive transport by the Levy walks A.B. Kukushkin 1,2,3, A.A. Kulihenko 1 1 National Researh Center Kurhatov Institute, Mosow, , Russian Federation 2 National Researh Nulear University MEPhI, Mosow, , Russian Federation 3 Mosow Institute of Physis and Tehnology, Dolgoprudny, Mosow Region, , Russian Federation Abstrat The method of approximate automodel solution for the Green s funtion of the time-dependent superdiffusive (nonloal) transport equations (J. Phys. A: Math. Theor. 49 (2016) ) is extended to the ase of a finite veloity of arriers. This orresponds to extension om the Lévy flights-based transport to the transport of the type, whih belongs to the lass of Lévy walk + rests, to allow for the retardation effets in the Lévy flights. This problem overs the ases of the transport by the resonant photons in astrophysial gases and plasmas, heat transport by eletromagneti waves in plasmas, migration of predators, and other appliations. We treat a model ase of one-dimensional transport on a uniform bakground with a simple power-law step-length probability distribution funtion (PDF). A solution for arbitrary superdiffusive PDF is suggested, and the verifiation of solution for a partiular power law PDF, whih orresponds, e.g., to the Lorentzian wings of atomi spetral line shape for emission of photons, is arried out using the omputation of the exat solution. 1. Introdution A wide range of problems needs desribing the transport in the medium for a finite veloity of arriers. The proesses of nonloal transport, whih signifiantly differ om the onventional diffusion, are of speial interest (see, e.g., the survey [1] and [2]). The energy transfer by photons in spetral lines of atoms and ions in plasmas and gases in astrophysial objets, nonloal heat transport by eletromagneti waves in plasmas, migration of predators belong to suh proesses. These phenomena have superdiffusive harater and have to be desribed by an integral equation in spatial oordinates, irreduible to a diffusion differential equation. The latter makes the numerial simulation of superdiffusive transport a formidable task. The phenomenon of superdiffusion is losely related to the onept of Lévy flights [3-7]. The known example of suh phenomenon is the radiative transfer in plasmas and gases in the Biberman-Holstein model [8-11]. This model onsiders resonane photon sattering by an atom or ion with omplete redistribution over equeny in the at of absorption and re-emission. Here, rare distant flights of photons («jumps»), whih orrespond to emission/absorption in the «wings» of spetral line, dominate over ontribution of equent lose displaements, whih produe diffusive (Brownian) motion and orrespond to emission/absorption in the ore of spetral line. The distant flights aused by the long-tailed (e.g. power-law) wings of integral operator (i.e. of the step-length probability distribution funtion (PDF)) in the transport equation were shown [12] to be the Lévy flights. The dominant ontribution of long-ee-path photons to radiative transfer in spetral lines has been reognized in [13, 14]. The simple models based on this domination were developed for the quasisteady-state transport, now known as the Esape Probability methods [15-17]. For the time-dependent superdiffusive transport by the Lévy flights, reently a wide lass of the transport on a uniform bakground was shown [18-23] to possess an approximate automodel solution. The solutions for the Green s funtion were onstruted using the saling laws for the

2 propagation ont (i.e. time dependene of the relevant-to-superdiffusion average displaement of the arrier) and asymptoti solutions far beyond and far ahead the propagation ont. The validity of the suggested automodel solutions was proved by their omparison with exat numerial solutions, in the one-dimensional (1D) ase of the transport equation with a simple long-tailed PDF with various power-law exponents, and in the ase of the Biberman-Holstein equation of the 3D resonane radiative transfer for various (Doppler, Lorentz, Voigt and Holtsmark) spetral line shapes. The present work extends the method [18] to the ase of a finite veloity of arriers. This orresponds to extension om the Lévy flights-based transport to the transport of the type, whih belongs to the lass of Lévy walk + rests (see Fig. 1 in [1]), to allow for the retardation effets in the Lévy flights. Similarly to [18], we treat a model ase of one-dimensional transport on a uniform bakground with a simple power-law step-length PDF (setion 2). A solution for arbitrary superdiffusive PDF is suggested (setion 3), whih uses the asymptoti solutions far beyond and far ahead the propagation ont. The solution for partiular power law PDF, whih orresponds, e.g., to Lorentzian wings of atomi spetral line shapes for emission of photons and uses asymptoti solutions far beyond and far ahead the propagation ont [24, 25], is presented in setion 4. Its verifiation is performed in setion 5, using the numerial simulation of the exat solution [24] of the transport equation. Modifiation of the solution and its verifiation are presented in setions Basi equation and general solution The nonstationary equation for the Green s funtion f ( x, t) of the one-dimensional superdiffusive (nonloal) transport of exitation in a homogeneous medium, with allowane for a finite veloity of the motion of arriers, has the form (derivation of this equation may be found in [24]): f ( x, t) 1 1 x x ' x x ' f ( x, t) dx ' W ( x x ' ) f ( x ', t ) ( t ) ( x) ( t), t (1) where W() is the step-length PDF, whih desribes the probability density for the proess of arrier s start ( emission of the arrier by the medium) and subsequent stop ( absorption of the arrier by the medium) after passing the distane, W ( x x) dx1, (2) τ is the average waiting time, i.e. the time between the moments of stopping and starting the arrier (average lifetime of the medium s exitation), с is the (onstant) veloity of arriers, σ is the average inverse lifetime of the arrier with respet to arrier s annihilation (deexitation of medium); θ(x) is the Heaviside funtion; δ(x) is the Dira delta-funtion. Note that even for zero annihilation the volume-integrated exitation density is not onserved in time beause the onservation law holds true only for the sum of volume-integrated values of medium s exitation density and arriers density (if the transport problem is applied to the dynamis of the objets of the same type, for example, the searh for food by animals, the total number of animals in motion and at rest will be the onstant value). For the dimensionless PDF

3 1 W ( ) 0.5 (1 ), 0 2, (3) general solution of Eq. (1) was obtained in [24]: i st 1 ipx e ds dpe 2 (2 ) 0 su R i i e os( pu) s 1 1 (1 u) 0 f ( x, t, R ) lim, du (4) where time t is in the units of τ, spae oordinate x is in the units of harateristi ee path length 1/ 0 ( 0 is the harateristi value of absorption oeffiient), the retardation parameter R 0 is the ratio of the average waiting time to the average time of flight. In what follows we onsider the ase = 0. The asymptoti of the Green s funtion far ahead the propagation ont was derived in [24]: f ( tr 0, t, R) t W ( ) t, x R R. (5) In the ase of infinite veloity of arriers (R ), it oinides with the respetive asymptotis in [18] (see Eq. (6) therein). 3. Approximate automodel solution for arbitrary superdiffusive PDF Following the priniples of the method [18], we onstrut the following approximate automodel solution: ( t, R ) ( t, R ) ( t, R ) fauto( x, t, R ) t g W g t g, x, R R (6) where the automodel funtion g has the known asymptoti behavior, 1, s smin ( t, R ) ( Rt), gs () s, s smin, (7) where ( tr, ) is the propagation ont defined by the relation whih equates the exat solutions in the alternative limits (f. Eq. (25) in [23], instead of Eq. (5) in [18]), namely the asymptotis far ahead, Eq. (5), and the asymptotis far behind the propagation ont, whih is a plateau-like funtion, dependent on the time variable only: t W ( ) t f (0, t, R ). (8) R R It is easy to prove that the solution (6)-(8) tends to the exat solution in the both alternative limits.

4 Similarlу to [18], we introdue the funtion Q needed for determination of the automodel funtion g in the intermediate range of values of the automodel variable s: t Q(, t, R ) W Q(, t, R ) t Q(, t, R ) f ( x, t, R ). (9) R R To analyze the auray of the approximate automodel solution one has to show weak dependene of Q 1 and Q 2 funtions on, respetively, spae oordinate and time: Q(, t(, s, R ), R ) Q ( s,, R ) g( s, R ), (10) 1 Q( ( t, s, R ), t, R ) Q ( s, t, R ) g( s, R ), (11) 2 where the funtions t(, s) and ( ts, ) are determined by the relation ( tr, ) s. (12) Note, that the definition (8) of the propagation ont is a partial ase of the definition of ( ts, ) with (8) and (12): ( ), 1 t t s. The verifiation of the automodel solution (6)-(8) should be done in the following way: alulation of the exat solution (4) in the range of spae-time variable (not in the entire spae of these variables) whih will allow the determination of the automodel funtion in the range of values of s~1, where the automodel solution is expeted to be most sensitive to the interpolation between the limits (7) of high and low values of s, analysis of auray of self-similarity of the funtion Q, defined by Eq. (9), in the view of Eqs. (10) and/or (11), analysis of auray of the automodel solution (6)-(8) with respet to the exat solution (4). 4. Automodel solution for γ=0.5 power-law PDF In what follows we onsider the partial ase γ=0.5 whih orresponds, e.g., to the Lorentzian wings of atomi spetral line shape for emission of photons (see, e.g., the asymptotis of the Holstein funtion, Eq. (38) in [11]). The PDF (3) takes the form: 1 W ( ) 4(1 ) 32. (13) Solving Eq. (8), we obtain expliit expression for the propagation ont ( tr, ) :

5 1 1tR R f (0, t, R) 1 ( tr, ) os artg R f (0, t, R) (1 tr) 1 tr R f (0, t, R) (14) The automodel funtion (9) may be expressed expliitly in terms of the automodel variable s ( t, R ) : 2 Q ( s, t, R ) Q( ( t, R ) s, t, R ) 1 1 1tR 2 2 s R fexat ( ( t, R ) s, t, R ) 1 s os artg ( t, R ) R fexat ( ( t, R ) s, t, R ) ( tr, ) (1 tr) 1 tr R fexat ( ( t, R ) s, t, R ) The automodel solution has the form 2 (15) 1 fauto( x, t, R ) t Q(, t, R ), t Q(, t, R ). (16) R 4 1 (,, ) R Q x t R 32 The asymptotis far behind the propagation ont for large retardation parameter and even larger values of time ( t R ) was alulated in [25]: f ( x 0, t R, R ) arth R t 8 4. (17) R t Equation (17) gives the saling, whih oinides with that of Eq. (19) in [26], and speifies the numerial oeffiient. 5. Verifiation of automodel solution for γ=0.5 power-law PDF We start the verifiation of the automodel solution (6)-(8) with the alulation of the exat solution (4) to determine of the automodel funtion in the range of values of s~1, where the automodel solution is expeted to be most sensitive to the interpolation between the limits (7) of high and low values of s. Below are the results of numerial alulation of exat solution (4) for time moments t=100, 300, 1000 and retardation parameter R =1, 10.

6 (a) Figure 1. The result of the numerial alulation of the exat solution (4) for t=100 and R =1 in the entire spae oordinate range (a) and near the ballisti one. The asymptotis far ahead and far behind the propagation ont, and the position of the propagation ont are shown as well.

7 (a) Figure 2. The same as in figure 1 but for R =10.

8 (a) Figure 3. The same as in figure 1 but for t=300 and R =1.

9 (a) Figure 4. The same as in figure 1 but for t=300 and R =10.

10 (a) Figure 5. The same as in figure 1 but for t=1000 and R =1.

11 Figure 6. The same as in figure 1 but for t=1000 and R =10. Using the results of alulating the exat solution, we an determine the funtion Q (15) and analyze the auray of its self-similarity in the view of Eq. (11). The respetive results are presented for t=100, 300, 1000, and R =1 (Figures 7, 8) and R =10 (Figures 9, 10).

12 (a)

13 Figure 7. The funtion (15) of automodel variable s for R =1 and various time moments, t=100, 300, 1000, in various ranges of s. () Figure 8. Charaterization of self-similarity of the funtion (15) as a funtion of automodel variable s only, for R =1: relative deviation of (15) for t=100 and 300 om that for t=1000.

14 (a)

15 Figure 9. The same as in figure 7 but for R =10. () Figure 10. The same as in figure 8 but for R =10. It is seen om Figures 7-10 that the auray of the self-similarity of the funtion (15), as a funtion of automodel variable s only (i.e. the auray of appliability of relation (11) whih assumes independene of (15) on the time variable t), is high: relative deviation of (15) for t =100 and 300 om that for t=1000 does not exeed for R =1 and for R =10.

16 Now we are ready to make the next step: knowing the automodel funtion (15), we an onstrut the approximate automodel solution (6) and ompare it with the alulated exat solution (4). The automodel solution for the Green s funtion is not appliable for simultaneously very small values of time and spae oordinate: indeed, the automodel solution does not assume desription of the evolution of the system immediately after the ation of an instant point soure. Therefore, for selfsimilarity analysis we will take the exat solution for t=1000 as a referene solution. This means that the automodel funtion (15), determined om omparison of exat and automodel solutions for t=1000, is used in the automodel solutions for other values of time. The respetive omparison of automodel and exat solutions for t=100 and 300 is presented in Figures 11(a) and 11, for R =1, and in Figures 12(a) and 12, for R =10. (a) Figure 11. Comparison of automodel solution for t=100 (a) and t=300 with automodel funtion g(s)=q(s, t=1000, R =1), reovered om omparison of exat and automodel solutions for t=1000, with the respetive exat solution.

17 (a) Figure 12. The same as in figure 11 but for R = 10. It an be seen that, despite the small differene between the automodel funtions (15) at various time moments, the deviation of the proposed approximate automodel solution (6) om the exat one is as high as a fator of few units in the region of the propagation ont. Therefore, further modifiation of the approximate automodel solution (6) is neessary to improve the interpolation between the known asymptotis of the exat solution.

18 6. Modifiation of automodel solution for γ=0.5 power-law PDF To improve the interpolation between the known asymptotis of the exat solution, we introdue a ee parameter in the definition of the propagation ont (8): t W ( ) fexat (0, t, R ). (18) R For W(ρ) (13), Eq. (18) gives the modifiation of the expliit expression (14) for the dependene of the propagation ont on the time variable, whih, in turn, determined the maximum allowable value of the parameter, max ( tr, ) 1 R f (0, t, R ) 108(1 tr ) exat (19) and the respetive value of the propagation ont, 3 1 lim ( t, R, ) ( t, R, max ) tr. (20) max 4 4 It appears that the minimum of the logarithmi derivative 1 1 d ln f auto tr s 3 s s, t, R, 1 1 d ln Q1 ( t, R, ) Q1 ( s, t, R, ) 2 ( t, R, ) Q1 ( s, t, R, ) in the range s~1 is reahed at (, ) max tr., (21) The modifiation of the propagation ont definition leads to a modifiation of the approximate automodel solution: ( t, R, max ( t, R )) fauto( x, t, R, max ( t, R )) t g R 1 W g ( t, R, max ( t, R )) max ( t, R) (1 max ( t, R)) Rt ( t, R, max ( t, R )) t g. R (22) Where the asymptotis of g(s) are not hanged and defined by Eq. (7). The automodel funtion (15) is also modified to take the form

19 Q ( s, t, R, ) Q( ( t, R ) s, t, R, ( t, R )) 1 max s 36 ( t, R, ( t, R )) R f ( ( t, R, ( t, R )) s, t, R ) ( t, R ) (1 ( t, R )) s s 2 2 max exat max max max min 1 1 tr R fexat ( ( t, R, max ( t, R )) s, t, R ) max ( t, R) (1 max ( t, R)) smin s 1 os artg (1 tr) 1 tr R fexat ( ( t, R, max ( t, R )) s, t, R ) max ( t, R ) (1 max ( t, R )) smin s s. ( t, R, ( t, R )) max The dependene of the automodel funtion (23) on s for t=100, 300, 1000 and the haraterization of the self-similarity of the funtion (23), as a funtion of automodel variable s only, are shown for R =10 in Figures 13 and 14, respetively, and for R =10 in Figures 15 and (23)

20 (a)

21 Figure 13. The funtion (23) of automodel variable s for R =1 and various time moments, t=100, 300, 1000, in various ranges of s. () Figure 14. Charaterization of self-similarity of the funtion (23) as a funtion of automodel variable s only, for R =1: relative deviation of (23) for t=100 and 300 om that for t=1000.

22 (a)

23 () Figure 15. The same as in figure 13 but for R =10. Figure 11. The same as in figure 14 but for R =10. The final step of verifying the modified approximate automodel solution (22) is presented in Figures 17 and 18, quite similarly to Figures 11 and 12 for verifiation of the automodel solution (6). It appears that for R =10 we have substantial improvement of the auray of automodel solution, while for R =1 the auray is slightly worse. In general, the approximate automodel solution (22) seems to be better than that om Eq. (6).

24 (a) Figure 17. Comparison of exat solution (4) with two different approximate automodel solutions: automodel solution (6) with automodel funtion g(s)=q(s, t=1000, R =1), where Q is given by Eq. (15) and reovered om omparison of exat (4) and automodel (6) solutions for t=1000; modified automodel solution (22) with automodel funtion g(s) = Q(s, t=1000, R =1, max ), where Q is given by Eq. (23) and reovered om omparison of exat (4) and automodel (22) solutions for t=1000 (the propagation ont for automodel solution (22) is shown). Comparison is made for t=100 (a) and t=300.

25 (a) Figure 12. The same as in figure 17 but for R = Conlusions It is shown that the method [18] of approximate automodel solution for the Green s funtion of the time-dependent superdiffusive (nonloal) transport on a uniform bakground may be extended to the ase of a finite veloity of arriers. This orresponds to extension om the Lévy flights-based transport to the transport of the type, whih belongs to the lass of Lévy walk + rests [1] and is haraterized by trajetories with a finite average waiting time and a finite veloity of arriers. A solution for arbitrary superdiffusive step-length probability distribution funtion (PDF) is suggested,

26 and the verifiation of solution for a partiular power law PDF, whih orresponds, e.g., to the Lorentzian wings of atomi spetral line shape for emission of photons, is arried out using the omputation of the exat solution. The suess of identifying suh solutions is based on the identifiation of the dominant role the long-ee-path arriers in all three saling laws used to onstrut the automodel solution, namely, the saling laws for the propagation ont (i.e. relevant-tosuperdiffusion average displaement) and asymptoti solutions far beyond and far ahead the propagation ont. The detailed analysis of automodel solutions for two values of the harateristi retardation parameter R (the ratio of the average waiting time to the average time of flight) enabled us to evaluate the auray of defining the automodel funtion g (i.e. auray of its self-similarity), whih is not worse than few perent in a wide spae-time region, and the resulting auray of the improved automodel solution, whih is not worse than several tens of perent. The results of the modifiation of the simplest automodel solution via modifiation of the propagation ont in the amework of optimizing the parameters of interpolation between the known asymptotis of the exat solution show that there is a muh eedom for suh modifiations to ahieve the main goal of approximate automodel solutions for the Lévy flights-based transport and of various extensions of the priniples [18], namely the onstrution of approximate solutions of superdiffusive transport problems with high enough auray with essential savings of omputation time. Indeed, as shown in [22] and [23] for the ase of Lévy flights transport, obtaining automodel (self-similar) solutions in the entire spae of independent variables requires mass numerial simulations (distributed omputing), however, their total volume is signifiantly redued due to the self-similarity of the solution. Aknowledgements The work is partly supported by the Russian Foundation for Basi Researh (projet RFBR a). This work has been arried out using omputing resoures of the federal olletive usage enter Complex for Simulation and Data Proessing for Mega-siene Failities at NRC Kurhatov Institute, The authors are grateful to K.V. Chukbar for helpful disussion of the paper [26]. Referenes [1] Zaburdaev V, Denisov S and Klafter J 2015 Lévy walks Rev. Mod. Phys [2] Shlesinger M F, Klafter J, and Wong J 1982 J. Stat. Phys. 27, 499. [3] Mandelbrot B B 1982 The Fratal Geometry of Nature (New York: Freeman) [4] Shlesinger M, Zaslavsky G M and Frish U (ed) 1995 Lévy Flights and Related Topis in Physis (New York: Springer) [5] Dubkov A A, Spagnolo B and Uhaikin V V 2008 Lévy flight superdiffusion: an introdution Int. J. Bifuration Chaos [6] Klafter J and Sokolov I M 2005 Anomalous diffusion spreads its wings Physis World [7] Eliazar I I and Shlesinger M F 2013 Frational motions Phys. Rep [8] Biberman L M 1947 Zh. Eksper. Teor. Fiz ; Biberman L M 1949 Sov. Phys. JETP [9] Holstein T 1947 Phys. Rev [10] Biberman L M, Vorob ev V S and Yakubov I T 1987 Kinetis of Nonequilibrium Low Temperature Plasmas (New York: Consultants Bureau) [11] Abramov V A, Kogan V I and Lisitsa V S 1987 Reviews of Plasma Physis ed M A Leontovih and B B Kadomtsev vol 12 (New York: Consultants Bureau) p 151

27 [12] Pereira E, Martinho J and Berberan-Santos M 2004 Photon trajetories in inoherent atomi radiation trapping as Lévy flights Phys. Rev. Lett [13] Biberman L M 1948 Dokl. Akad. Nauk SSSR [14] Kogan V I 1968 Pro. ICPIG 67: A Survey of Phenomena in Ionized Gases (Invited Papers) (Russian: IAEA, Vienna) p 583 [15] Kalkofen W (ed) 1984 Methods in Radiative Transfer (Cambridge: Cambridge University Press) [16] Rybiki G B Ibid. h 1 [17] Napartovih A P 1971 Teplofiz. Vys. Temp [18] Kukushkin A B and Sdvizhenskii P A 2016 Automodel solutions for Lévy flight-based transport on a uniform bakground J. Phys. A: Math. Theor [19] Kukushkin A B and Sdvizhenskii P A 2014 Saling Laws for Non-Stationary Biberman- Holstein Radiative Transfer Proeedings of 41st EPS Conferene on Plasma Physis, Berlin, Germany, June 2014 ECA 38F P [20] Kukushkin A B, Sdvizhenskii P A, Voloshinov V V and Tarasov A S 2015 Saling laws of Biberman-Holstein equation Green funtion and impliations for superdiffusion transport algorithms International Review of Atomi and Moleular Physis (IRAMP) [21] Kukushkin A B and Sdvizhenskii P A 2017 Auray analysis of automodel solutions for Lévy flight-based transport: om resonane radiative transfer to a simple general model. J. Phys. Conf. Series [22] Kukushkin A B, Neverov V S, Sdvizhenskii P A and Voloshinov V V 2018 Numerial Analysis of Automodel Solutions for Superdiffusive Transport International Journal of Open Information Tehnologies (INJOIT) [23] Kukushkin A B, Neverov V S, Sdvizhenskii P A and Voloshinov V V 2018 Automodel Solutions of Biberman-Holstein Equation for Stark Broadening of Spetral Lines Atoms 6(3) 43 [24] Kulihenko A A and Kukushkin A B 2017 IRAMP 8(1) [25] Kulihenko A A and Kukushkin A B 2018 Pro. 45th EPS Conferene on Plasma Phys., Prague, Cheh Republi, 2018, ECA 42A P [26] Zaburdaev V Yu, Chukbar K V 2002 JETP 94(2) 252