Exact kinetic transport equation solutions in the particle propagation theory in the scattering medium

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Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1667–1684 www.elsevier.com/locate/jqsrt

Exact kinetic transport equation solutions in the particle propagation theory in the scattering medium B.A. Shakhova, M. Stehlikb, a

Main Astronomical Observatory, NAS, 252650 Kiev, Ukraine Institute of Experimental Physics, SAS, Watsonova 47, 04001 Kosˇice, Slovakia

b

Received 29 October 2007; received in revised form 14 January 2008; accepted 15 January 2008

Abstract Charged particle kinetics in an inhomogeneous medium (stochastic magnetic field) is investigated. Exact analytic expressions for Green function of kinetic equation in relaxation-time approximation are derived in one and three dimensions with arbitrary particle absorption. We separately consider the case of isotropic particle injection as well as the case of unidirectional instantaneous particle injection. The new way of solution makes it possible to get off any CauchyPrincipal Value integrals in some solutions which arise in the inverse Fourier–Laplace transform. Weak scattering regime and diffusion approximation is considered, and particle density is derived in three dimensions and arbitrary particle source. r 2008 Elsevier Ltd. All rights reserved. Keywords: Transport processes; Particle kinetics; Relaxation-time approximation

1. Introduction The most rigorous treatment of energetic charged particle propagation in stochastic media is based on kinetic equation [1–4]. The kinetic equation can describe transport of particles in the magnetic field, for example, which can be often presented as a superposition of mean regular field and magnetic irregularities of various scales. The regular magnetic field can cause particle focusing [5], and magnetic fluctuations usually produce scattering of charged particles [4,6,7]. Here we assume the particle gyroradius in the mean homogeneous magnetic field to be small in comparison with the characteristic length of magnetic irregularities. In that case the drift approximation of particle motion is applicable and we can start from the Chandrasekhar–Kaufman–Watson drift kinetic equation describing the propagation of ‘‘strongly magnetized’’ particles [6,8]. (In limit case it becomes to be the Boltzmann equation.) We do not take into consideration the processes of the particle cross field diffusion [9,10] and the particle diffusion on the Alfve`n waves [11], which can be described on the basis of Fokker–Planck equation [12–15]. Particle scattering on magnetic field irregularities can be treated as the light particle interaction with heavy ‘‘magnetic clouds’’ and can be studied by the linearized Boltzmann collision integral [6,13]. In some cases the Boltzmann kinetic Corresponding author. Tel.: +421 55 7922226; fax: +421 55 6336292.

E-mail addresses: [email protected] (B.A. Shakhov), [email protected] (M. Stehlik). 0022-4073/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2008.01.012

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equation can be solved analytically and exact expressions for a Green function can be obtained. They describe particle distribution after an impulsive injection of particles with a given velocity direction [16,17]. Using this method, a multiple scattering approach for kinetic equation solution was developed [18], which can be successfully applied to a number of problems in physical kinetics [19,20]. In papers [17,21] exact formulae have been derived for particle density under isotropic, instantaneous particle release and they have also considered particle transport in the case of anisotropic scattering. Similar expressions for the particle density have also been obtained in some other papers [22,23]. The diffusion approximation, widely used in the theory of the particle transport, proved to be deficient in a number of problems [6,24]. Recently, approximate methods of the kinetic equation solution based on the investigation of a set of equations for harmonics of distribution function have been developed [18,25]. They have also explored the accuracy of these methods in comparison with the exact analytical solutions to the Boltzmann equation and with numerical results of Monte Carlo simulations. The solving of kinetic equation has also been generalized to the case of a finite particle emission with given temporal dependence and angle distribution of particles in source [26,27]. In the present paper, we consider the propagation of energetic particles under their instantaneous injection into scattering medium, or, in other words, we consider Green functions of various kinetic equations. Starting from the linearized Boltzmann equation in the relaxation-time approximation, exact analytical expressions for the Green functions are derived in one- and three-dimensional spaces. Some solutions to kinetic equations, if these are ever known, contain discontinuities in the form of a principal value of integral. Because of our new mathematical manipulations, we can get off this stiffness and derive the solutions that are more comfortable for applications which use numerical calculations. Obtained solutions can be used in various fields, e.g., in the radiation transport theory, the neutron transport, the radio-wave scattering on random inhomogeneities, etc. This paper is organized as follows: In Section 2 the basic kinetic equations are presented. Section 3 contains solutions to the kinetic equation in one-dimensional case for both isotropic and uni-directional sources. The three-dimensional case is considered in Section 4. In both the sections new ‘‘alternative’’ solutions are suggested. The known solution forms involving the Cauchy-Principal Value (CPV) integrals (Sections 3.1 and 3.2) are followed by the new solutions with eliminated CPV-integrals (Sections 3.1.1 and 3.2.1). In Section 4 we give only the new solutions for simplicity. Some approximations of the kinetic equation are presented in Section 5. In Section 6 conclusions are given and derivation of one useful mathematical expression is added in the Appendix. 2. Kinetic equation Kinetic equation for particle distribution function f ðr; v; tÞ in phase space which describe propagation of charged particles in an inhomogeneous medium is usually expressed in the form ðqt þ v Or þ F Op Þf ðr; v; tÞ ¼ Col f ,

(2.1)

where F is the Lorentz force acting on particles of velocity v and of momentum p and qt  q=qt, Or  q=qr. Here and below we omit the sign of scalar production while it does not give rise to an ambiguity. Because here we are not interested in acceleration mechanisms, the third term on the left-hand side will be omitted in the paper. The collision integral Col f will be used in the relaxation-time approximation: Z ns d^v0 f ðr; v0 ; tÞ  ns f ðr; v; tÞ, Col f ¼ (2.2) 4p where d^v0 is the solid angle element in the velocity v0 space. It determines a change in the particle phase density caused by its collisions with scattering centers, which are supposed to be static, and the scattering is understood to be isotropic. The collision frequency ns ¼ 4pnvs0 is evidently proportional to the scattering center density n and the constant section s0 . The Green function Gðr; v; tÞ of the kinetic equation (2.1), (2.2) is a solution to the kinetic equation with d-like particle source in the right-hand side: Z ns c qt G þ v Or G þ n s G ¼ (2.3) d^v0 Gðr; v0 ; tÞ þ dðrÞdðtÞdð^v  v^ 0 Þ, 4p

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where dð^v  v^ 0 Þ ¼ sin1 y0 dðy  y0 Þdðj  j0 Þ ¼ dðm  m0 Þdðj  j0 Þ; m ¼ cos y; the unit vector v^ determines the particle injection direction; y; j and y0 ; j0 are spherical co-ordinates of the unit vectors v^ and v^ 0 , respectively (somewhere in literature the notation dð^v  v^ 0 Þ ¼ dðm  m0 Þdðj  j0 Þ  d2 ð^v  v^ 0 Þ is used [2]). The d-like source correspond physically to an instantaneous particle emission at the point of r ¼ 0. Notice that due to linearity of the kinetic equation (2.3), this is invariant with respect to a parallel shift in space. So, obtained solutions can be easily generalized to the case of source position at r0 . It is sufficient to substitute r ! ðr  r0 Þ in the solution. The introduced coefficient c 2 h0; 1i describes the particle absorption (or escape) [2]. Denoting the particle mean free path l ¼ v=ns , the dimensionless quantities q ¼ r=l ¼ rns =v and t ¼ vt=l ¼ ns t can be introduced. Consequently, Eq. (2.3) for Gðq; v^ ; v^ 0 ; tÞ is then Z c n3 d^v0 Gðq; v^ 0 ; v^ 0 ; tÞ þ s3 dðqÞdðtÞdð^v  v^ 0 Þ. (2.4) qt G þ v^ Oq G þ G ¼ 4p v In the one-dimensional case, say in the x-direction, Eq. (2.4) reads Z c 1 ns dm Gðx; m; tÞ þ dðxÞdðtÞdðm  m0 Þ qt G þ mqx G þ G ¼ 2 1 v

(2.5)

and m ¼ cos y is deducted from the x-axis. Here and below the x-co-ordinate has the meaning of dimensionless co-ordinate. Note, for completeness, that the last equation is equivalent to the Cauchy problem: Z c 1 qt G þ mqx G þ G ¼ dm Gðx; m; tÞ, 2 1 ns (2.6) Gðx; m; 0Þ ¼ dðxÞdðm  m0 Þ. v Let us note for convenience that in the following G denotes the Green function in real space unlike g, which means Fourier and/or Laplace transform of G. 2.1. Case of arbitrary absorbtion c 2 h0; 1i The third term on the left-hand side of (2.4) can be eliminated by the substitution G ¼ et F. Then Z c n3 (2.7) qt F þ v^ Oq F ¼ d^v0 Fðq; v^ 0 ; v^ 0 ; tÞ þ s3 dðqÞdðtÞdð^v  v^ 0 Þ. 4p v Introducing the variables T ¼ ct; R ¼ qc, Eq. (2.7) reads Z 1 c3 n3 d^v0 FðR; v^ 0 ; v^ 0 ; TÞ þ 3 s dðRÞdðTÞdð^v  v^ 0 Þ. qT F þ v^ OR F ¼ 4p v

(2.8)

The last equation is identical to Eq. (2.7) expressed for c ¼ 1, but replaced ns ! cns in the source. Therefore, the solution to (2.4) with ca1 can be immediately obtained from the solution to (2.7) with c ¼ 1 by interchanging the source constant in the result and then by relevant inverse coordinate substitution. So, most of the results are written for c ¼ 1. 3. Green function in the one-dimensional case 3.1. Isotropic particle source The kinetic equation (2.5) with the isotropic particle source in one-dimensional case acquires the form Z c 1 ns qt G þ mqx G þ G ¼ (3.1) dm Gðx; m; tÞ þ dðxÞdðtÞ. 2 1 2v

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Using the Fourier transform in the space variable x and the Laplace transform in the time t Z 1 Z dt dx expðst  ikxÞGðx; tÞ. gðk; m; sÞ ¼

(3.2)

0

Eq. (3.1) gives

 1 ns c k 1  arctan gðk; m; sÞ ¼ . k 1þs 2vð1 þ s þ ikmÞ

(3.3)

The main question in this investigation is to perform the inverse Fourier–Laplace transform for the Green function Gðk; sÞ. Expression (3.3) has been studied in [17] where the resulting Green function has been obtained for c ¼ 1 in the form h i ns t p p e zðmÞðxmtÞ=2 ln zðmÞ cos ðx  mtÞ  p sin ðx  mtÞ Gðx; m; tÞ ¼ 2 2 4v Z 1 ns t dZ ðxþZtÞ=2 Sðm; x  mtÞ þ e V:p: zðZÞ 4v 1 Z þ m h i p p  ln zðZÞ sin ðx þ ZtÞ þ p cos ðx þ ZtÞ SðZ; x þ ZtÞ 2 2 Z p=2 ns exp½t  k cot ktðk cot k  cos kx þ km sin kxÞ þ . (3.4) dk 2pv 0 ðcot2 k þ m2 Þsin2 k The arising sign function Sða; bÞ ¼ YðaÞYðbÞ  YðaÞYðbÞ

(3.5)

ensures that the Green function is zeroth at the position x, which cannot be reached in t by the particles of given m. Here YðxÞ is the Heaviside function, and ð1 þ mÞ zðmÞ ¼ . (3.6) ð1  mÞ Unfortunately the solution (3.4) is not suitable in applications due to hidden singularities. This problem is overcome in the next subsection. 3.1.1. The alternative solution for isotropic source The integrand in the second term of solution (3.4) possesses a pole in the integration area. This stiffness can be eliminated in the inverse transform of (3.3) using the following relations [23,28]: Z 1 ðo þ ikmÞ1 ¼ dx exp½xðo þ ikmÞ, (3.7) 0

          c k 1 k2 k k 1 k p k p þ  o  k cot sin2 1  arctan ¼1þ Y Y k o c c c 2 c 2 c  Z 1   1  1 k dZ c c 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ þ  1 . 2p 1 o  ikZ 2ik 2ik

(3.8)

The second relation is derived in the Appendix. Then the Fourier–Laplace transform (3.3) for arbitrary c acquires the form Z ns ns 1 gðk; m; sÞ ¼ þ dx exp½xð1 þ s þ ikmÞ 2vð1 þ s þ ikmÞ 2v 0 (         k2 k k 1 k p k p þ   1 þ s  k cot sin2 Y Y c c c 2 c 2 c )  Z 1 1  1  k dZ c c 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ þ  1 . (3.9) 2p 1 1 þ s  ikZ 2ik 2ik

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The inverse Laplace transform can be performed by using the known relation expðsxÞ 3 exp½aðt  xÞYðt  xÞ. sþa One obtains in the result the Fourier transform (c ¼ 1): Z 1 ns ns expðt þ ikZtÞ  expðt  ikZtÞ gðk; m; tÞ ¼ expðt  ikmtÞ þ dZ mþZ 2v 4piv 1 " 1  1 # 1 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ  1  1 2ik 2ik  ns k expðt þ k cot ktÞ  expðt  ikmtÞ h  p pi þ  Y k  . Y k þ 2v 2 2 ðcot k þ imÞsin2 k

(3.10)

(3.11)

Before performing the inverse Fourier transform, note some remarks. The factor h   p pi Y k Y kþ 2 2 causes that the integral in the third term in (3.11) has to be performed in the interval k 2 hp=2; p=2i. Moreover, the second term contains the factor  1  1 1 1 1 ðln zðZÞ þ ipÞ ¼ k k  ðln zðZÞ þ ipÞ , (3.12) 2ik 2i where zðZÞ is defined by (3.6). The inverse Fourier transform can be considered as the integral in the complex plane of k. The second term possesses the first-order poles. From (3.12) it follows that the pole for Zo0 or Z40 is situated in the upper semi-plane or lower one, respectively. Therefore, the integration contour has to be chosen along the real axis and semi-circle in the upper (for Zo0) or lower (for Z40) semi-plane. After integration one obtains the factor of the form SðZ; x þ ZtÞ ¼ ½YðZÞYðx þ ZtÞ  YðZÞYðx  ZtÞ. Using the known formulae: Z 1 2pdðx  mtÞ ¼ dk exp½ikðx  mtÞ, 1

Z 2pYðx  mtÞ ¼

1

dk exp½ikðx  mtÞ, 1 ik

one obtains the resulting Green function: Gðx; m; tÞ ¼

Z ns t ns t 1 dZ e dðx  mtÞ þ e 2v 4pv 1 Z þ m n h i p p ðxþZtÞ=2  zðZÞ ln zðZÞ sin ðx þ ZtÞ þ p cos ðx þ ZtÞ SðZ; x þ ZtÞ 2 2 h i o p p ðxZtÞ=2 zðZÞ ln zðZÞ sin ðx  mtÞ þ p cos ðx  mtÞ SðZ; x  mtÞ 2 2 Z p=2 ns t dk þ e ½k cot k  cos kx  ekt cot k 2 k þ m2 Þsin2 k 2pv ðcot 0  k cot k  cos kðx  mtÞ þ mk sin kx  ekt cot k  mk sin kðx  mtÞ.

(3.13)

Note that here the first term expresses the initial source particles moving without any scattering and the other terms express the scattered particles. This solution already does not contain any CPV-integral. In fact the singularity at Z ¼ 1 is here explicitly extracted in zðZÞ unlike in (3.4) where it is hidden and occurs in all terms with various parameter values. The space distribution of scattered particles predicted by solution (3.13) is demonstrated in Figs. 1–3. These show distribution of particles, which reach values of m ¼ 1, 0.5, or 0.2 during time t ¼ 0:5, 1, 2, 3, and 5. Note that the pictures are symmetric for m2  m because of the isotropic particle injection. Firstly, the particles are

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1 0.5

1 2 3

g

0.1

5

0.01

5

1E-3

3

2 1

µ=1

0.5

1E-4 -5

-4

-3

-2

-1

1

0 x

2

3

4

5

Fig. 1. Space distribution of scattered particles (3.13) with m ¼ 1 past the isotropic injection in one-dimensional case. The numbers near the curves denote the time past the injection.

µ = 0.5

1

1 2 3

g

0.1 5 0.01 5 3 1E-3 -5

-4

-3

2

-2

1 -1

0.5 0 x

1

2

3

4

5

Fig. 2. As in Fig. 1 for m ¼ 0:5.

absent in the space of jxjot. Secondly, the peaks at x ¼ mt correspond to the ‘‘origin’’ particles with the given value of m. See, for example, peaks at x ¼ 3  0:2 ¼ 0:6 for t ¼ 3 or x ¼ 5  0:2 ¼ 1 for t ¼ 5 in Fig. 3. The particles behind this point (x4mt) arise from scattered particles being emitted with a pitch angle cosine greater than that reached during the scattering (m ¼ 0:2 in Fig. 3) excepting Fig. 1, where m ¼ 1. Note that one can obtain an analogous picture in the case of unidirectional particle source in Section 3.2 and also in the threedimensional case (Section 4). 3.2. Unidirectional particle source Let us recoil to the kinetic equation (2.5) with the instantaneous unidirectional point source. As in the case of the isotropic source, see (3.3), one obtains the Fourier–Laplace transform  1 ns ns c k gðk; m; m0 ; sÞ ¼ dðm  m0 Þ þ 1  arctan . (3.14) k 1þs vð1 þ s þ ikm0 Þ 2vð1 þ s þ ikmÞð1 þ s þ ikm0 Þ

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1

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µ = 0.2 3

g

0.1 5 5

3

0.01

1E-3

0.5 -5

-4

-3

-2

-1

0 x

1

2

3

4

5

Fig. 3. As in Fig. 1 for m ¼ 0:2.

The inverse Fourier–Laplace transform for the Green function Gðk; sÞ can be obtained analogically as in the case of the isotropic source [17]: Gðx; m; m0 ; tÞ ¼

ns t ns expðtÞ p e dðx  mtÞdðm  m0 Þ þ zðm0 Þðxm0 tÞ=2 cos ðx  m0 tÞSðm0 ; x  m0 tÞ 2vðm  m0 Þ 2 v ns expðtÞ p zðmÞðxmtÞ=2 cos ðx  mtÞSðm; x  mtÞ  2vðm  m0 Þ 2 Z 1 ns expðtÞ dZ p þ V:p: zðZÞðxþZtÞ=2 sin ðx þ ZtÞSðZ; x þ ZtÞ 4pvðm  m0 Þ 2 1 Z þ m0 Z 1 ns expðtÞ dZ p  zðZÞðxþZtÞ=2 sin ðx þ ZtÞSðZ; x þ ZtÞ V:p: 4pvðm  m0 Þ 2 1 Z þ m   Z p=2 ns expðtÞ dk cot k  sin kx  m0 cos kx cot k  sin kx  m cos kx þ  . 4pvðm  m0 Þ 0 sin2 k cot2 k þ m2 cot2 k þ m20

(3.15)

The first term describes the direct (unscattered) particles. As in the case of isotropic source (see Eq. (3.4)), solution (3.15) possesses the CPV-integrals inconvenient in applications. 3.2.1. The alternative solution for unidirectional source Here the derivation of more comfortable solution without CPV-integrals is given. Using the simple identity   1 1 1 1 ¼  ð1 þ s þ ikmÞð1 þ s þ ikm0 Þ ikðm  m0 Þ ð1 þ s þ ikm0 Þ ð1 þ s þ ikmÞ together with relations (3.7), (3.8) one obtains from (3.14):  Z 1 ns dðm  m0 Þ ns 1 þ Gðk; m; m0 ; sÞ ¼ þ dx exp½xð1 þ s þ ikm0 Þ vð1 þ s þ ikmÞ 2ikvðm  m0 Þ 1 þ s þ ikm0 0         k2 k k 1 k p k p  þ  1 þ s  k cot sin2 Y Y c c c 2 c 2 c  Z 1   1  1 k dZ c c 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ þ  1 2p 1 1 þ s þ ikZ 2ik 2ik  Z 1 ns 1 þ  dx exp½xð1 þ s þ ikmÞ 2ikvðm  m0 Þ 1 þ s þ ikm 0

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  1      k2 k k p k p 2k þ  1 þ s  k cot sin  Y Y c c c 2 c 2 c  Z 1  1  k 1 dZ c c 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ þ  1 . 2p 1 1 þ s þ ikZ 2ik 2ik

(3.16)

At first, the inverse Laplace transform can be performed analogically to the one-dimensional case with the isotropic source. One obtains the Fourier transform in the form (c ¼ 1) Z 1 ns tikmt ns ns expðtÞ dZ tþikm0 t tikmt ðe þ e Þ ðeikZt  eikm0 t Þ gðk; m; m0 ; tÞ ¼ e 4pkvðm  m0 Þ 1 Z þ m0 v 2ikvðm  m0 Þ " 1  1 # 1 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ  1  1 2ik 2ik  ns k expðt þ kt cot kÞ  expðt  ikm0 tÞ h  p pi Y k þ þ  Y k  2v 2 2 ðm  m0 Þðcot k þ im0 Þsin2 k Z 1 ns expðtÞ dZ ðeikZt  eikmt Þ þ 4pkvðm  m0 Þ 1 Z þ m " 1  1 # 1 1 ðln zðZÞ þ ipÞ ðln zðZÞ  ipÞ  1  1 2ik 2ik  ns k expðt þ kt cot kÞ  expðt  ikmtÞ h  p pi þ Y kþ Y k . (3.17) 2 2v 2 2 ðm  m0 Þðcot k þ imÞsin k At last, performing the inverse Fourier transform, the Green function in the one-dimensional case of unidirectional source acquires the form Gðx; m; m0 ; tÞ ¼

ns t ns Yðx  m0 tÞ  Yðx  mtÞ e dðx  mtÞdðm  m0 Þ þ et m  m0 v 2v Z 1 h ns expðtÞ dZ p þ zðZÞðxþZtÞ=2 sin ðx þ ZtÞSðZ; x þ ZtÞ 4pvðm  m0 Þ 1 Z þ m0 2 Z i p ns expðtÞ 1 dZ zðZÞðxm0 tÞ=2 sin ðx  m0 tÞSðZ; x  m0 tÞ  2 4pvðm  m0 Þ 1 Z þ m h i p p ðxþZtÞ=2  zðZÞ sin ðx þ ZtÞSðZ; x þ ZtÞ  zðZÞðxmtÞ=2 sin ðx  mtÞSðZ; x  mtÞ 2 2 Z ns expðtÞ p=2 dk þ ½Cðm0 Þ  CðmÞ, (3.18) 2pvðm  m0 Þ 0 sin2 k

where CðmÞ ¼

expðkt cot kÞ cot k  sin kx  cot k  sin kðx  mtÞ m cos kðx  mtÞ  m cos kx  expðkt cot kÞ þ . cot2 k þ m2 cot2 k þ m2

This solution does not contain any CPV-integrals analogous to the case of isotropic source (3.13). 3.3. Stationary solution In stationary case the one-dimensional kinetic equation (2.5) for the Green function Gðx; m; m0 Þ acquires the form (c is arbitrary in this section) Z c 1 ns dm Gðx; m; m0 Þ þ dðxÞdðm  m0 Þ. (3.19) mqx G þ G ¼ 2 1 v

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The solution to (3.19) has been obtained in [2] in the series of singular case functions and their integrals, as an alternative to the Fourier method. The Fourier transform of the stationary equation is [1]  1 ns dðm  m0 Þ ns c þ 1  arctan k , (3.20) gðk; m; m0 Þ ¼ 2pvð1 þ ikmÞ 4pvð1 þ ikmÞð1 þ ikm0 Þ k which is analogous to (3.14). The inverse transform can be accomplished using (3.8) with the resulting expression for Gðx; m; m0 Þ [29]:        1 cns 1 1 x=m x=m x=m0 e dðm  m0 ÞS x; S x; S x; G ¼ ns e þ e m m m0 2vðm  m0 Þ   Z 1 2 x=m x=m0 c ns me Sðx; 1=mÞ m0 e Sðx; 1=m0 Þ  þ dZ ðm þ ZÞDðm; ZÞ ðm0 þ ZÞDðm0 ; ZÞ 4vðm  m0 Þ 1 Z Z c2 ns 1 Zex=Z Sðx; 1=ZÞ c3 ns 1 þ  dZ dZ zðZÞcx=2 4v 1 ðm þ ZÞðm0 þ ZÞDðZ; ZÞ 8pv 1 h Sðx; 1=ZÞ pcx pcxi þ Bðm; m0 ; ZÞ cos Aðm; m0 ; ZÞ sin  Dðm; ZÞDðm0 ; ZÞDðZ; ZÞ 2 2 2 2 2 3 3 Z p=2 c ns x ð1  x c mm0 Þ cosðcxxÞ þ cx ðm þ m0 Þ sinðcxxÞ dx þ , (3.21) 2pv 0 sin x2 ð1  cx cot xÞð1 þ c2 m2 x2 Þð1 þ c2 m20 x2 Þ where Sða; bÞ is the sign function (3.5) and  p2 mm0 c2 MðZ; ZÞKðZÞ Aðm; m0 ; ZÞ ¼ Mðm; ZÞMðm0 ; ZÞMðZ; ZÞKðZÞ  4  p2 ZczðZÞMðm; ZÞMðm0 ; ZÞ þ þ

p4 mm0 Zc3 zðZÞ 4

h i p2 c2 Zc ðm þ m0 þ mm0 zðZÞÞ 2zðZÞMðZ; ZÞ þ KðZÞ , 2 4

   p2 mm0 c2  Zc Mðm; ZÞMðm0 ; ZÞ  2zðZÞMðZ; ZÞ þ KðZÞ 2 4  c  ðm þ m0 þ mm0 czðZÞÞðKðZÞMðZ; ZÞ  p2 ZczðZÞÞ , 2

(3.22)

Bðm; m0 ; ZÞ ¼ p

Dða; bÞ ¼ M 2 ða; bÞ þ Mða; bÞ ¼ 1 þ

p2 c2 a2 , 4

ac zðbÞ, 2

KðZÞ ¼ z2 ðZÞ  p2 .

(3.23)

(3.24) (3.25) (3.26)

The solution contains the parameter cp1 explicitly. Only unscattered particles (the first term in (3.21)) remain if c ¼ 0 (the case of full absorption). The last integral diverges for c ¼ 1 (only elastic scattering without absorption), which means that the stationary solution to (3.20) does not exist in that case. 4. Green function in the three-dimensional case 4.1. Isotropic particle source The kinetic equation in three dimensions (2.4) with the isotropic particle source reads Z 1 n3 qt G þ v^ Oq G þ G ¼ d^v Gðq; v^ ; tÞ þ s 3 dðqÞdðtÞ. 4p 4pv

(4.1)

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Here c ¼ 1. Analogous to Section 3.1, we use the Fourier–Laplace transform Z 1 Z gðk; v^ ; sÞ ¼ dt dq expðst  ikqÞGðq; v^ ; tÞ,

(4.2)

0

and well-known equalities, Z d^v 4p k ¼ arctan ; 1 þ s þ ik^v k 1þs

Z

d^vdð^v  v^ 0 Þ 1 ¼ . 1 þ s þ ik^v 1 þ s þ ik^v0

In result,  1 n3s 1 k 1  arctan gðk; v^ ; sÞ ¼ . k 1þs 4pv3 ð1 þ s þ ik^vÞ

(4.3)

Using equalities (3.7) and (3.8), Eq. (4.2) can be rewritten in the form analogous to (3.9): gðk; v^ ; sÞ ¼

n3s 4pv3

Z

1

dx exp½xð1 þ s þ ik^vÞ 0

n3 þ s3 8pv

Z

Z

1

1

dZ 1 þ s  ikZ 0 1 Z 1 Z 1 3 n i dZ zðZÞ  s3 dx exp½xð1 þ s þ ik^vÞ 1 þ s  ikZ 4pv 2k 0 1 Z 1 Z 1 3 n k dZ Bðk; ZÞ  s3 dx exp½xð1 þ s þ ik^vÞ 4pv 2p 0 1 1 þ s þ ikZ Z 1 n3 k2 ½1  Yð2k  pÞ þ s3 . dx exp½xð1 þ s þ ik^vÞ 4pv 0 ð1 þ s  k cot kÞsin2 k dx exp½xð1 þ s þ ik^vÞ

(4.4)

The inverse Laplace transform of (4.4) can be obtained without any problems by using relation (3.10): Z t n3s n3s sin kðt  xÞ exp½x  ik^vxÞ gðk; v^ ; tÞ ¼ exp½t  ik^vt þ dx kðt  xÞ 4pv3 4pv3 0 Z 1 Z t n3 sin kðt  xÞ þ s3 exp½x  ik^vx dx dZ zðZÞ sin kZðt  xÞ kðt  xÞ 4pv 0 0 Z t Z 1 n3 þ s3 dx exp½x  ik^vx dZ Bðk; ZÞ exp½ikZðt  xÞ 4pv k 0 1 Z t k2 n3s ½1  Yð2k  pÞ þ dx exp½x  ik^vx þ kðt  xÞ cot k, 4pv3 sin2 k 0

(4.5)

where Bðk; ZÞ in the fourth term is

Bðk; ZÞ ¼

2k3  p½k  i ln zðZÞ½k  ðln zðZÞ þ ipÞ=ð2iÞ 2k3 þ p½k  i ln zðZÞ½k  ðln zðZÞ  ipÞ=ð2iÞ  . k  ðln zðZÞ þ ipÞ=ð2iÞ k  ðln zðZÞ  ipÞ=ð2iÞ (4.6)

R1 The third term in (4.5) was simplified due to R1 R 1the facts that 1 dZ zðZÞ cos kZðt  xÞ ¼ 0 because the integrand is an odd function of Z, and then 1 . . . ¼ 2 0 . . . for even function. The inverse Fourier transform of (4.5) gives

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the Green function in the case of isotropic particle source in three dimensions, Gðq; v^ ; tÞ [28]:   2    n3s et n3s et t  r2 ðq  v^ tÞ2 ^ dðq  vtÞ þ 2 3 Y G¼ Y  2ðt  q^vÞ 4pv3 2ðt  q^vÞ 8p v jq  v^ tj2 3 t Z b n e ðt  xÞ þ jq  v^ xj Yðt  x  jq  v^ xjÞ dx ln þ s 2 3 ðt  xÞ  jq  v^ xj ðt  xÞjq  v^ xj 16p v 0 Z 1 Z t n3s et dx þ daY½jq  v^ xj  aðt  xÞzðaÞð1=2Þjq^vxjþð1=2ÞaðtxÞ 64p3 v3 0 jq  v^ xj 0 n o p p  ½3p2 ln zðaÞ þ ln3 zðaÞ sin ½jq  v^ xj  aðt  xÞ þ ½p3  3p ln2 zðaÞ cos ½jq  v^ xj  aðt  xÞ 2 2 Z Z p=2 n3s et t k3 dk þ 3 3 , (4.7) dx 8p v 0 jq  v^ xjsin2 k 0 where the upper integration limit in the integral

Rb 0

dx . . . is b ¼ ðt2  r2 Þ=ð2ðt  q^vÞÞ.

4.1.1. Spherical symmetry The problem considered in three dimensions with the isotropic particle source possesses a spherical symmetry. Therefore, solution (4.7) depends on the absolute value of q (or r) and on the cosine of angle between q and v^ (or r and v). Let one return back to dimensional variables fr; v; tg and denote Z Z 2p Gðr; v; m; tÞ ¼ d^vr Gðr; v; tÞ ¼ dj Gðr; v; m; j; tÞ, (4.8) 0

where m ¼ rv=rv ¼ nq v^ . Performance corresponding substitution in (4.7) and integration over the angle j of the vector r yields pffiffiffiffiffiffiffiffiffiffiffiffiffi exp½ns t dðrm  vtÞdðr 1  m2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi Gðr; v; m; tÞ ¼ 4p r 1  m2   22   2  ns exp½ns t v t  r2 r  2rvmt þ v2 t2 Y þ  Y  4pvðr2  2rvmt þ v2 t2 Þ 2ðv2 t  rvmÞ 2ðv2 t  rvmÞ Z n2 exp½ns t b dx vðt  xÞ þ gðxÞ ln þ s 2 8pv vðt  xÞ  gðxÞ 0 ðt  xÞgðxÞ Z 1 Z t 3 n exp½ns t dx þ s daY½gðxÞ  avðt  xÞzðaÞð1=2vÞ½gðxÞavðtxÞ 32p2 v2 gðxÞ 0 0  pns ½gðxÞ  avðt  xÞ 3 2  ½3p ln zðaÞ þ ln zðaÞ sin 2v pn ½gðxÞ  avðt  xÞ s þ½p3  3p ln2 zðaÞ cos 2v Z t Z p=2 3 n exp½ns t dk exp½ns ðt  xÞk cot kk3 sin½kns gðxÞ=v þ s , (4.9) dx 2 2 4p v gðxÞsin2 k 0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where gðxÞ ¼ r2  2rvmx þ v2 x2 and the upper integration limit b ¼ ðv2 t  r2 Þ=ð2ðv2 t  rvmÞÞ. Of course, expression (4.9) is a solution to the equation pffiffiffiffiffiffiffiffiffiffiffiffiffi Z vð1  m2 Þ ns 1 dðrmÞdðr 1  m2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi , qm G þ n s G ¼ qt G þ vmqr G þ dm Gðr; v; m; tÞ þ (4.10) r 2 1 4pr 1  m2 which can be obtained from (2.3) in the case of spherical symmetry for the isotropic source and c ¼ 1.

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4.2. Unidirectional particle source Now we return to the original kinetic equation for Gðk; v^ ; v^ 0 ; sÞ in three dimensions (2.4) with the unidirectional particle source (and c ¼ 1). Using the Fourier–Laplace transform (4.2), one obtains  1 n3s 1 k 1  arctan , gðk; v^ ; v^ 0 ; sÞ ¼ k 1þs 4pv3 ð1 þ s þ ik^vÞð1 þ s þ ik^v0 Þ

(4.11)

and its analogical form obtained as in Section 4.1: gðk; v^ ; v^ 0 ; sÞ ¼

   Z x   Z 1 n3s dð^v  v^ 0 Þ n3s i i ^ ^ kð^ v  v kð^ v  v þ dx exp x 1 þ s þ Þ dZ exp Z Þ 0 0 v3 ð1 þ s þ ik^vÞ 16pv3 0 2 2 x    Z 1 2 da i k 2k ½1  Yð2k  pÞ 1  zðaÞ þ Bðk; aÞ þ  2þ , (4.12) 1 þ s  ika k p ð1 þ s  k cot kÞsin2 k 1

where Bðk; aÞ and zðaÞ are defined by (4.6) and (3.6), respectively. Similar to the case of isotropic source, the inverse Laplace transform of (4.12) can be obtained without problems:  Z t   n3s n3s i i exp t  kð^v  v^ 0 Þt dZ exp  kð^v  v^ 0 ÞZ gðk; v^ ; v^ 0 ; tÞ ¼ 3 exp½t  ik^vtdð^v  v^ 0 Þ þ 2 2 v 8pv3 t  Z x   3 Z t n i i þ s3 dx exp x  kð^v  v^ 0 Þx dZ exp  kð^v  v^ 0 ÞZ 2 2 8pv 0 x  sin kðt  xÞ k2 þ exp½kðt  xÞ cot k  kðt  xÞ 2p2 sin2 k Z 1 Z 1 k þ da zðaÞ sin kaðt  xÞ þ 2 da Bðk; aÞ exp½ikaðt  xÞ . (4.13) 2p 1 0 The inverse Fourier transform of (4.13) gives the Green function in the case of unidirectional particle source in three dimensions [30]:   Z n3 exp½t n3 exp½t t v^ þ v^ 0 v^  v^ 0 t  a Gðq; v^ ; v^ 0 ; tÞ ¼ s 3 dðq  v^ tÞdð^v  v^ 0 Þ þ s dad q  v 8pv3 2 2 t Z Z t x 3 n exp½t da þ s dððt  xÞ  CÞ dx 32p2 v3 ðt  xÞC 0 x Z Z x n3 exp½t t da txþC þ s Y½t  x  C ln dx 2 3 64p v txC 0 x ðt  xÞC Z 1 Z t Z x 3 n exp½t da þ s dx db zð1=2ÞCðb=2ÞðtxÞ Y½C  bðt  xÞ 128p3 v3 0 C x 0 n o p p 3 2  ½3p ln zðbÞ þ ln zðbÞ sin ðC  bðt  xÞÞ þ ½p3 þ 3p ln2 zðbÞ cos ðC  bðt  xÞÞ 2 2 Z t Z x Z p=2 3 3 n exp½t dk k sin kC þ s exp½ðt  xÞk cot k, (4.14) dx da 16p3 v3 C 0 x 0 where the denotation C  jq  12 ð^v þ v^ 0 Þx  12 ð^v  v^ 0 Þaj is used for the record simplicity. Note that particular terms of this solution express the expansion in series by collision frequency ns . The first term is proportional to n0s , and the last two terms are proportional to n4s . This can be seen explicitly past return to dimension variables fr; v; v0 ; tg [30]. Moreover, solution (4.14) can be easily generalized to the case of ca1 (co1) when particles are partially absorbed [31], see Section 2.1.

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4.3. Non-stationary solution in a moving medium The kinetic equation (2.3) can be rewritten in a more general form when the particle velocity is explicitly indicated in the source term (c ¼ 1): Z v v q t G þ v Or G þ G ¼ d^v0 Gðr; v0 ; v00 ; tÞ þ dðrÞdðv  v0 ÞdðtÞ, (4.15) l 4pl where v^ 0 ¼ v0 =v0 ; v0 ¼ jv0 j, and dðv  v0 Þ ¼ dð^v  v^ 0 Þdðv  v0 Þ=v20 . Here we return back to v=l ¼ ns . The Green function of this equation can be easily obtained using solution (4.14). It is sufficient to rewrite this solution back to dimensional variables fr; v; v0 ; tg and multiply the result by dðv  v0 Þ=v20 . In previous sections we deal with the case of static scattering centers. Let now the massive scattering centers move with the constant non-relativistic velocity u, and the instantaneous unidirectional point source emits light particles. Such a condition occurs when the light propagates in moving dust clouds past the light emission or in the problem of neutron propagation in flowing around the material. Then the mean free path l is the same in both the laboratory co-ordinate system and the system moving with velocity u of the scattering centers (where scattering centers are static). From [2], the Green function for equation with moving scattering centers can be obtained from the Green function for equation with static centers by ordinary Galilean transform of co-ordinates r and velocity v : r0 ¼ ðr  utÞ; v0 ¼ ðv  uÞ; v00 ¼ ðv0  uÞ. Here fr0 ; v0 ; v00 ; tg are connected with the system aligned with scattering centers. The kinetic equation in this system acquires the form that coincides with (4.15): Z v0 v0 d^v0 G 0 ðr0 ; v0 ; v00 ; tÞ þ dðr0 Þdðv0  v00 ÞdðtÞ. qt G 0 þ v 0 Or 0 G 0 þ G 0 ¼ (4.16) l 4pl Generally, the Green function in the laboratory co-ordinate system fr; v; tg fulfills the kinetic equation qt GL þ v Or G L ¼ Col G L þ dðrÞdðv  v0 ÞdðtÞ,

(4.17)

where Col G L is the collision integral in the laboratory system with moving scattering centers. It has a complicated structure; however, we do not need to point it here in an explicit form because by [2] G L ðr; v; v0 ; u; tÞ ¼ G0 ðr0 ; v0 ; v00 ; tÞ. Therefore, it is sufficient to perform the substitution r ! ðr  utÞ; v ! ðv  uÞ; v0 ! ðv0  uÞ; v ! jv  uj in the solution to (4.15) obtained by using (4.14), (see [32]). Note that analogically one can find the non-stationary solution in a medium with moving scattering centers in onedimensional case. 5. Approximations of the kinetic equation 5.1. The weak scattering Let particles take rare collisions with scatterers when considered time past injection is small in comparison with the inverse collision frequency. Then approximate solutions to (2.4) (or (2.5)) can be obtained by the perturbation theory, in which the collision frequency ns plays the role of a small parameter. Let us consider the kinetic equation in three dimensions with the isotropic source (4.1) for GðtÞ ¼ et FðtÞ. Expanding FðtÞ into series in ns , F ¼ F0 þ F1 þ   , one obtains a set [17,28] n3s dðqÞdðtÞ, 4pv3 Z c d^v Fn1 ðq; v^ ; tÞ. qt Fn þ v^ Oq Fn ¼ 4p qt F0 þ v^ Oq F0 ¼

(5.1) (5.2)

Eq. (5.1)has a trivial solution for unscattered particles F0 ðq; v^ ; tÞ ¼

n3s dðq  v^ tÞ or 4pv3

F0 ðr; v; tÞ ¼

1 dðr  vtÞ 4p

(5.3)

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in dimension variables. Eq. (5.2) for n ¼ 1 gives the next approximation      n3s ðq  v^ tÞ2 ðq  v^ tÞ2 F1 ðq; v^ ; tÞ ¼ 2 3 Y t Y  . 2ðt  q^vÞ 2ðt  q^vÞ 8p v jq  v^ tj2

(5.4)

The last expression rewritten in dimension variables is of the first order in ns . This procedure yields other terms of Fn in the nth order in ns . Notice that in (5.4) F1 ¼ 0 for jqj4j^vjt in connection with finite particle velocity. 5.2. The diffusion approximation If considered time t past particle emission considerably exceeds the inverse collision frequency, the particles undergo multiple scattering and the distribution function becomes close to isotropic. Then the diffusion approximation can be used. Usually the distribution Gðx; m; tÞ (in one dimension, for simplicity) is expanded into a series of the Legendre polynomials Pn ðmÞ, and by standard procedure [33] Rone obtains the equation R1 1 set for the density Nðx; tÞ ¼ 1 dm G and also for higher moments J n ðx; tÞ ¼ 1 dm Pn ðmÞG. This wellknown procedure has to be modified in the case of a unidirectional injection. Then the Green function consists of two parts: e m; m0 ; tÞ, Gðx; m; m0 ; tÞ ¼ expt dðm  m0 Þdðx  mtÞ þ Gðx;

(5.5)

e is to where the first term corresponds to unscattered particles moving in the direction of m0 . The second term G be expanded in the series. In result [17], the density Nðx; t; m0 Þ is ns N¼ v

rffiffiffi    2  Z ptffi     3 3m0 3m0 x 3m2 3ðx  m0 tÞ2 1  dy exp y2 1  0  exp t p 2 2 4 4y2 0

(5.6)

with well-known asymptotics ns Nðx; tÞ ¼ 2v

rffiffiffiffiffi   3 3x2 exp  pt 4t

(5.7)

relevant for tbx (when tb1, i.e., tbn1 s ). The obtained density (5.6) is the first-order solution, when J 2 ¼ 0 and qt J 1 ¼ 0. In the next approximation, when one substitutes J 2 ¼ 0 and qt J 1 ns J 1 , the equation set gives the telegraf equation for the density N [34,35]. 5.3. The particle density The above procedure for the diffusion approximation yields expression for the ‘‘diffusive’’ density. The technique how to obtain the particle density consists in the integration of Green function over all angle variables (^v or m). Another way consists in the integration of equation for g over all angle variables (^v or m) and then its inverse transform. Let us demonstrate this way on the example of three-dimensional case with the instantaneous unidirectional particle injection described by Eq. (2.4) for c ¼ 1, whose Fourier–Laplace transform is Z 1 n3 dð^v  v^ 0 Þ ^ ^ d^v0 gðk; v^ 0 ; v^ 0 ; sÞ þ s3 . (5.8) gðk; v; v0 ; sÞ ¼ 4pð1 þ s þ ik^vÞ v ð1 þ s þ ik^vÞ Integrating this equation over v^ , one easily obtains the algebraic equation for Nðk; v^ 0 ; sÞ, which has the simple solution   1 Z n3 1 k N ¼ d^v gðk; v^ ; v^ 0 ; sÞ ¼ s3 ð1 þ s þ ik^v0 Þ 1  arctan . (5.9) k 1þs v

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Using the alternative method of the inverse transform of Section 4.2, one obtains [23]      n3 n3 et jq  v^ 0 tj2 jq  v^ 0 tj2 Y t  Nðq; v^ 0 ; tÞ ¼ s3 et dðq  v^ 0 tÞ þ s 3  Y  v 2pv jq  v^ 0 tj2 2ðt  v^ 0 qÞ 2ðt  v^ 0 qÞ   Z t 3 t n e dx jq  v^ 0 xj t  x þ jq  v^ 0 xj Y 1 ln þ s 3 ðt  xÞ t  x  jq  v^ 0 xj 8pv 0 ðt  xÞjq  v^ 0 xj Z 1 3 t Z t n e dx þ s 2 3 da Y½jq  v^ 0 xj  aðt  xÞ 16p v 0 jq  v^ 0 xj 0  n jq  v^ 0 xj  aðt  xÞ LðaÞ ½3p2 LðaÞ  L3 ðaÞ  exp 2 o p p sin ½jq  v^ 0 xj  aðt  xÞ þ ½p3  3pL2 ðaÞ cos ½jq  v^ 0 xj  aðt  xÞ 2 2 Z Z p=2 ^ 0 xjÞ ðtxÞk cot k n3s et t dx 3 sinðkjq  v e þ 2 3 dk k . 2p v 0 jq  v^ 0 xj 0 sin2 k

1681

(5.10)

5.3.1. Cylindrical symmetry Due to the existence of the special direction of particle injection v^ 0 in the problem, it is convenient to choose the cylindrical coordinate system fz; R; jg with polar axis z along v^ 0 . Integrating (5.10) over j, it follows the resulting expression for Nðz; R; tÞ:      n3s t dðRÞ n3s et ðz  tÞ2 þ R2 ðz  tÞ2 þ R2 þ 3 Y t N ¼ 3 e dðz  tÞ Y  R v v ððz  tÞ2 þ R2 Þ 2ðt  zÞ 2ðt  zÞ  p 2 2  3 t Z t ððz  xÞ þ R Þ ne dx Y 1 þ s 3 p 2 2 tx 4v ðt  xÞ ððz  xÞ þ R Þ 0 p p 2 2 ½lnðt  x þ ððz  xÞ þ R ÞÞ  lnðt  x  ððz  xÞ2 þ R2 ÞÞ Z Z 1 p n3 et t dx þ s 3 da Y½ ððz  xÞ2 þ R2 Þ  aðt  xÞ p 2 2 8pv 0 ððz  xÞ þ R Þ 0 p ððz  xÞ2 þ R2 Þ  aðt  xÞ LðaÞ  exp 2 n p p  ½3p2 LðaÞ  L3 ðaÞ sin ½ ððz  xÞ2 þ R2 Þ  aðt  xÞ 2 o p p þ ½p3  3pL2 ðaÞ cos ½ ððz  xÞ2 þ R2 Þ  aðt  xÞ 2 p Z p=2 2 2 3 t Z t ne dx 3 sin k ððz  xÞ þ R Þ ðtxÞk coth k e þ s 3 dk k , (5.11) p 2 2 2 pv ððz  xÞ þ R Þ 0 sin k 0 where LðxÞ ¼ ln zðxÞ ¼ lnð1 þ xÞ  lnð1  xÞ. 6. Conclusions The paper deals with the exact analytic solution to the linearized Boltzmann equation (in the relaxation-time approximation) when light particles (or waves) undergo scattering on ‘‘heavy scatterers’’. It means that particles are elastically scattered on the magnetic inhomogeneities (in our case) as a billiard ball on the wall of infinity mass. Besides the known solutions to the kinetic equation, we have reviewed the new solutions in oneand three-dimensional cases for the instantaneous isotropic particle injection as well as for the anisotropic (unidirectional) source with the given vector of particle injection. The new solutions relate especially to such cases when known resulting expressions proved not to be convenient for a numerical application (when they contain a principal value of integrals, for example). We have also mentioned a way of finding approximate

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solutions, especially the case of weak scattering when particles undergo only a few collisions and also the opposite case of high scattering frequency and large considered time past the particle emission. The fact that our obtained solutions relate to the kinetic regime of particle distribution is emphasized here, so they are zeroth for time exceeding the coming of the first particles. Of course, this does not hold for the diffusion approximation. Our solutions are given in up to seven variables (the position vector, the particle velocity vector, and the time) irrespective of some other parameters (the collision frequency, the particle absorption). The existence of the Green functions allows us principally to obtain solutions in considerably complex problems, e.g., for arbitrary source dependence on time, space, and/or angle, by the integrating this source distribution with corresponding Green function (see [36, Chapter 7], for example). Relating to the particle density, note that it is usually obtained by integration of the resulting Green function over angle variables. Easier way consists in the integration of the Fourier–Laplace transform over the angles (see Section 5.3). It also concerns the search for the phase density (exact kinetic equation solution) with arbitrary particle source: this way—repeating the inverse transform on each occasion—seems to be easier than the integration of source function with given Green function. Furthermore, a suitable source function with sufficiently simple Fourier–Laplace transform can be often chosen and used. Therefore, the reviewed solutions can be used in various similar research fields which deal with a ray distribution in stochastic environment. Acknowledgements The authors thank E.G. Yanovitsky for the interest in this work. M.S. gratefully acknowledges the Cosmic Ray Physics Laboratory at MAO NASU, Kiev. This work was supported by the Science and Technology Assistance Agency under the contracts APVT-51-027904 and APVV-0538, and by SAS, project Nos. 2/6193 and 2/7063. Appendix A Here we give the brief proof of key expression (3.8). Let us consider the integral ( 1  1 ) Z k 1 dZ ic ic I¼ 1 þ ðLðZÞ þ ipÞ  1 þ ðLðZÞ  ipÞ . 2p 1 o  ikZ 2k 2k

(A.1)

In the new variable z ¼ LðZÞ=2 ¼ ½lnð1 þ ZÞ  lnð1  ZÞ=2, i.e., Z ¼ tanh z, and dZ ¼ dz=cosh2 z, is (      ) Z ik2 1 dz p k 1 p k 1 þ  zi I¼  zi . 2 c 2 c 2p 1 ðo  ik tanh zÞcosh2 z To carry out the z integration for k40, Re o41, we close the contour of integration by an infinite semi-circle with Im z40 on the complex z plane, see in Fig. 4. The integrand possesses the poles of the first order at zI ¼ iðk=c þ p=2Þ, zII ¼ iðk=c  p=2Þ and at zm ¼ ipm þ ½lnðik þ oÞ  lnðik  oÞ=2 and then the poles of the second order at zn ¼ iðn þ p=2Þ, for n; m ¼ 0; 1; 2; . . . : These points will belong to the upper half of the complex plane if the relations ðk=c þ p=2Þ40, ðk=c  p=2Þ40 and n; m ¼ 0; 1; 2; . . . hold. Then Imðlnðik þ oÞ  lnðik  oÞÞ=2o0 for k40, Re o41. The integral along the infinity semi-circle vanishes, and the integral I is equal to the sum of residue times 2ip. In result, "         # k2 k 1 2 k k p k p c c k 1 y þ  k  c arctan I¼  o  k cot sin y þ 2 . (A.2) c c c 2 c 2 k o c k In fact, due to a quasi-symmetry of terms in composed brackets in the first expression of I, the infinite sum of residues is trivial: except for n ¼ 0; m ¼ 1, all terms in the sum cancel each other. Thus, the first term in expression (A.2) comes from residues at zI and zII , and the second and the third terms come from residues at zn and zm , respectively. Expression (3.8) can be obtained by substituting I from Eq. (A.2) into the left-hand side of Eq. (A.1).

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Fig. 4. The location of integrand I poles in the plane of z ¼ x þ iy. Here z0 ¼ 12 ln½ðik þ oÞ=ðik  oÞ.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Davison B. Neutron transport theory. Oxford: Clarendon Press; 1958. Case KM, Zweifel PE. Linear transport theory. Reading, MA: Addison-Wesley; 1967. Dorman LI, Katz ME. Cosmic ray kinetics in space. Space Sci Rev 1977;20:529. Achatz U, Steinacker J, Schlickeiser R. Charged particle transport in a turbulent magnetized plasma: a reassessment and extension of quasilinear theory. Astron Astrophys 1991;250:266. Earl EA. The effect of adiabatic focusing upon charged-particle propagation in random magnetic fields. Astrophys J 1976;205:900. Toptygin IN. Cosmic rays in interplanetary magnetic fields. Dordrecht, Holland: D. Reider Publishing Company; 1985. Schlickeiser R, Dung R, Jackel U. Interplanetary transport of solar cosmic rays and dissipation of Alfve`n waves. Astron Astrophys 1991;242:L5. Chandrasekhar S, Kaufman A, Watson K. Ann Phys 1958;5:1. Dorman LI, Katz ME, Stehlik M. Cosmic-ray kinetics in a strong large-scale magnetic field. Bull Astron Inst Czech 1990;41:312. Chuvilgin LG, Ptuskin VS. Anomalus diffusion of cosmic rays across the magnetic field. Astron Astrophys 1993;279:278. Galperin BA, Toptygin IN, Fradkin AA. Particle scattering by magnetic inhomogeneities in strong magnetic field. Zh Teor Eksperim Fiz 1971;60:972. Jokipii JR. Cosmic ray propagation. I. Charged particles in a random magnetic field. Astrophys J 1966;146:480. Gleeson LJ, Axford WI. Cosmic rays in the interplanetary medium. Astrophys J 1967;149:1115. Earl EA. The flash phase of charged particles propagation in random magnetic field. Astrophys J 1996;460:794. Shakhov BA, Stehlik M. The Fokker–Planck equation in the second-order pitch angle approximation and its exact solution. JQSRT 2003;78:31. Fedorov YuI, Shakhov BA. Solar cosmic rays in homogeneous regular magnetic field. In: Proceedings of the 23th international cosmic ray conference, Calgary, vol.3, 1993. p. 215–8. Fedorov YuI, Shakhov BA. Solar cosmic ray propagation for isotropic scattering on the interplanetary magnetic field inhomogeneities. Geomag Aeron 1994;34:19 [in Russian]. Webb GM, Pantazopoulos M, Zank GP. Multiple scattering and the BGK Boltzmann equation. J Phys A 2000;33:3137. Zank GP, Lu Jy, Rice WKM, Webb GM. Transport of energetic charged particles in a radial magnetic field. Part 1. Large-angle scattering. J Plasma Phys 2000;64:507. Kaghashvili EKh, Zank GP, Lu JY, Dro¨ge W. Transport of energetic charged particles. Part 2. Small-angle scattering. J Plasma Phys 2004;70:505.

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B.A. Shakhov, M. Stehlik / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1667–1684

[21] Ko´ta J. Coherent pulses in the diffusive transport of charged particles. Astrophys J 1994;427:1035. [22] Fedorov YuI, Kyzyurov YuV, Nosov SF, Shakhov BA. Solution of the Boltzmann equation for nondiffusive solar cosmic ray propagation. Ann Geophysicae 1996;14:1016. [23] Shakhov BA, Stehlik M. 3-dimensional energetic particle distribution past an instantaneous uni-directional injection. Astrophys Space Sci 1997;250:75. [24] Earl EA. Diffusion of charged particles in a random magnetic field. Astrophys J 1973;180:227. [25] Earl EA. New description of charged particle propagation in random magnetic field. Astrophys J 1994;425:331. [26] Fedorov YuI. The angular distribution function of solar cosmic rays. Kinem Fiz Nebes Tel 1997;13:30 [in Russian]. [27] Fedorov YuI, Stehlik M. Description of anisotropic particle pulse transport based on the kinetic equation. Astrophys Space Sci 1997;253:55. [28] Shakhov BA. Nonstationary Green function of the transport kinetic equation for an isotropic source. Kinem Fiz Nebes Tel 1995;11:49 [in Russian]. [29] Shakhov BA. An alternative method for solving the kinetic transport equation in the stationary case. Kinem Fiz Nebes Tel 2000;16:188 [in Russian]. [30] Shakhov BA, Shakhova MB, Titov MP. The Green function of transport kinetic equation for impulsive point monodirectional source. Kinem Fiz Nebes Tel 1996;12:63 [in Russian]. [31] Shakov BA, Fedorov YuI, Kyzyurov YuV, Nosov SF. Solar cosmic ray propagation on the base of analytic solution of kinetic equation. Izvestiya AN SSSR, Fiz Ser 1995;59:48 [in Russian]. [32] Shakhov BA. The Green function of the kinetic transport equation in the medium with moving scatterers. Kinem Fiz Nebes Tel 1999;15:99 [in Russian]. [33] Shkarofski I, Johnston T, Bachynski M. The particle kinetics of plasmas. Massachusetts: Addison-Wesley Publishing Company; 1966. [34] Earl EA. Coherent propagation of charged-particle bunches in random magnetic fields. Astrophys J 1974;188:379. [35] Fedorov YuI, Shakhov BA. Description of non-diffusive solar cosmic ray propagation in a homogeneous regular magnetic field. Astron Astrophys 2003;402:805. [36] Morse PM, Feshbach H. Methods of theoretical physics, vol. 1. New York: MCGraw-Hill Book Company; 1953.