Green's functions for non-classical transport with general anisotropic scattering

Abstract

In non-classical linear transport the chord length distribution between collisions is non-exponential and attenuation does not respect Beer's law. Generalized radiative transfer (GRT) extends the classical theory to account for such two-point correlation between collisions and neglects all higher order correlations. For this form of transport, we derive the exact time-independent Green's functions for the isotropic point source in infinite 3D homogeneous media with general anisotropic scattering. Green's functions for both collision rate density, which characterizes absorption and reaction rates in the system, and radiance/flux, which characterizes displacement of radiation/particles, are solved in Fourier space. We validate the derivations using gamma random flights to produce the first anisotropic scattering benchmark solutions for the generalized linear Boltzmann equation. For gamma-4 flights with linearly anisotropic scattering and gamma-6 flights with Rayleigh scattering the collision rate density is found explicitly in real space as a sum of diffusion modes.

Full text

Green’s functions for non-classical transport with general
anisotropic scattering
Eugene d’Eon
NVIDIA
Abstract
In non-classical linear transport the chord length distribution between collisions is non-exponential and attenuation
does not respect Beer’s law. Generalized radiative transfer (GRT) extends the classical theory to account for such two-
point correlation between collisions and neglects all higher order correlations. For this form of transport, we derive the
exact time-independent Green’s functions for the isotropic point source in infinite 3D homogeneous media with general
anisotropic scattering. Green’s functions for both collision rate density, which characterizes absorption and reaction rates
in the system, and radiance/flux, which characterizes displacement of radiation/particles, are solved in Fourier space. We
validate the derivations using gamma random flights to produce the first anisotropic scattering benchmark solutions for the
generalized linear Boltzmann equation. For gamma-4 flights with linearly anisotropic scattering and gamma-6 flights with
Rayleigh scattering the collision rate density is found explicitly in real space as a sum of diffusion modes.
Keywords: generalized radiative transfer, non-classical Boltzmann, Green’s function, point source, random flight,
diffusion
1 Introduction
Linear transport of monoenergetic neutral particles in stochatic geometries can significantly deviate from the predic-
tions of classical transport theory [Chandrasekhar 1960; Davison 1957] when there are significant spatial correlations
in the geometries. This has motivated the use of non-exponential random flights to form generalized radiative transfer
(GRT) theories that exactly exhibit some desired non-exponential distribution of chord lengths between collisions [Bur-
rus 1960; Doub 1961; Randall 1964; Rybicki 1965; Alt 1980; Sahni 1989; Audic and Frisch 1993; Kostinski 2001;
Davis and Marshak 2004; Davis 2006; Moon et al. 2007; Taine et al. 2010; Frank et al. 2010; Larsen and Vasques
2011; Zarrouati et al. 2013; Vasques and Larsen 2014; Davis and Xu 2014; Frank et al. 2015; Xu et al. 2016; Rukolaine
2016; Liemert and Kienle 2017; Binzoni et al. 2018; Frank and Sun 2018; d’Eon 2018; Jarabo et al. 2018; Bitterli
et al. 2018]. We derive new green’s functions for these theories in the case of anisotropic scattering.
Point source Green’s functions for monoenergetic transport in infinite homogeneous medium with a completely general
symmetric phase function are known exactly in classical exponential media via Fourier transform [Davison 2000;
Wallace 1948; Grosjean 1951; Vanmassenhove and Grosjean 1967; Williams 1977; Paasschens 1997; Ganapol 2003;
Zoia et al. 2011] and have been studied using other approaches, such as path integrals [Tessendorf 2011]. These
functions have a number of direct applications [Narasimhan and Nayar 2003], they provide important benchmarks for
checking correctness of more advanced codes [Ganapol 2008] and they play a role in bounded media via Placzek’s
lemma [Case et al. 1953]. The goal of this paper is to derive such Green’s functions for GRT.
In the case of isotropic scattering, the Fourier transform approach has been extended for GRT and the Green’s functions
are known [d’Eon 2013; d’Eon 2019]. In this paper, we show how to use the derivations of Grosjean [1951] to derive
the exact forms for general anisotropic scattering. To the best of our knowledge, Grosjean’s 1951 derivation has
never been validated numerically in the non-exponential case with anisotropic scattering. We perform Monte Carlo
simulation of gamma random flights in 3D and find good agreement. We omit the lengthy details of Grosjean’s full
derivation and refer the interested reader to his monograph. Our present goal is to briefly highlight only those details
that are required to form benchmark solutions in GRT and discuss their role in these theories.
2 General Theory
2.1 The random flights of GRT
Let us consider a unit isotropic point source at the origin of a homogeneous infinite 3D medium emitting particles that
travel at constant speed valong piecewise straight paths (Figure 1). Collisions between the particle and the medium

p1(s)
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pc(s)
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pc(s)
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pc(s)
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(a)
r
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C(r, µ)
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✓
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µ= cos ✓
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(b)
Figure 1: (a) We consider the random flight where a particle leaves an isotropic point source in 3D (black circle) and
travels at constant speed along straight paths between collision events (white cirlces). The free-path lengths between
collisions are random variates drawn from the distribution pc(s). The initial free-path length p1(s)may be different
from pc(s). At each collision the particle scatters according to a given phase function P(Ωi→Ωo)with probability c
and is absorbed otherwise, terminating the flight. (b) We seek the rate density C(r,µ)that particles enter collisions at
some distance r from the point source restricted to a cone of directions indexed by cosine µto the position vector.
result either in absorption, with probability 1 −c, or scattering into a new direction using a given phase function with
probability 0 <c≤1. We consider a general symmetric phase function with Legendre expansion
P(Ωi→Ωo) = 1
4π
∞
∑
l=0
(2l+1)flP
l(Ωi·Ωo) = 1
4π
∞
∑
l=0
AlP
l(Ωi·Ωo)(1)
where Ωiis the direction before collision and Ωois the scattered direction. The phase function is normalized over the
unit sphere, A0=1.
Free path-lengths between collisions are drawn from a given normalized distribution pc(s), which can be estimated
by Monte Carlo tracking in specific realizations of the stochastic geometry [Audic and Frisch 1993; Moon et al.
2007; Larsen and Vasques 2011] or derived from a known attenuation law in the system [Torquato and Lu 1993;
d’Eon 2018]. We denote the initial free-path length distribution from the point source p1(s), which is either pc(s)
or a related distribution, pu(s). If emission is spatially correlated to the first scattering event in the same way that
spatial correlation arises between collisions then p1(s) = pc(s)and the full random flight is homogeneous. However,
emission from a deterministic location in the stochastic geometry must consider statistics to the next collision using
all realizations of the geometry and not only those that have a particle at the origin from which to begin the flight. For
this reason, emission from a so called uncorrelated origin has free-path length statistics pu(s)that are distinct from
the intercollision statistics pc(s)[Audic and Frisch 1993; d’Eon 2018]. An equivalent distinction has been recognized
for continuous-time random walks [Feller 1971; Tunaley 1974; Tunaley 1976; Weiss and Rubin 1983] where it was
proposed that1
pc(s) = −p0
u(s)/hsci,hsci ≡ Z∞
0pc(s)sds (2)
where hsciis the mean free path length between collisions. Eq.(2) was also found to be a necessary condition for
achieving Helmholtz reciprocity in GRT [d’Eon 2018], which is desired given that reciprocity is observed in any
realization of the stochastic geometry and therefore expected of the mean transport.
2.1.1 Collision-rate density
Now consider a single particle leaving the isotropic point source. Our primary quantity of interest in GRT is the
collision rate density, since most reaction rates in a given system will be proportional to this quantity. The angular
1Specifically, this occurs in the case of tightly-coupled walks where the wait time is proportional to the displacement, which corresponds to the
constant speed model.

collision-rate density C(r,µ)is defined such that 4πr2C(r,µ)drdµis the mean number of collisions (either scattering
or absorbing) experienced by the particle within a shell of radii [r,r+dr]about the point source who directions before
collision have cosines in [µ,µ+dµ]. Direction cosines µare measured with respect to the normalized position vector
and index cones of directions with µ=1 pointing away from the point source (Figure 1). We denote the scalar collision
rate density by
C(r) = Z1
−1C(r,µ)dµ.(3)
To distinguish between correlated and uncorrelated emission we also write Cc(r)and Cu(r), respectively for the scalar
collision rate densities. With non-stochastic single-scattering albedo cat every collision, the mean number of collisions
is 1/(1−c)and therefore the 0th moment of Cis [Ivanov 1994]
Z∞
04πr2C(r)dr =1
1−c,(4)
for all phase functions and free-path length distributions. In a system with spherical symmetry, which will always be
the case in this paper, the angular collision rate density C(x,Ω)at position xand direction Ωis related to C(r,µ)by
C(x,Ω) = C(r,µ)/(2π).
The phase function and single-scattering albedo immediately give the distribution of particles in the system as they
leave collisions. The in-scattering rate density B(x,Ω)is related to the collision rate density by [Larsen and Vasques
2011]
B(x,Ω) = cZ4π
P(Ω0·Ω)C(x,Ω0)dΩ0(5)
and satisfies the generalized Peierls integral equation,
B(x,Ω) = cZZZ PΩ·x−x0
|x−x0|Bx0,x−x0
|x−x0|+Qc(x0)
4πpc(|x−x0|)
|x−x0|2+Qu(x0)
4π
pu(|x−x0|)
|x−x0|2dV 0.(6)
This is an integral equation of non-homogeneous non-exponential random flights [Grosjean 1951; Weiss and Rubin
1983] and is the basis of our derivation for Green’s functions in GRT.
2.1.2 Radiance and fluence
We are also interested in finding Green’s functions for the radiance and fluence in the system. In GRT, the semi-
Markov nature of the transport breaks the classical proportionality between collision rate and radiance [d’Eon 2019],
and the two are no longer related by the inverse mean free path, requiring separate Green’s function derivations and
separate Monte Carlo estimators. The radiance in GRT satisfies a more complicated equation, the generalized linear
Boltzmann equation (GLBE). However, the GLBE has been proven equivalent to Eq.(7) [Larsen and Vasques 2011],
and the radiance I(x,Ω)(specific intensity/angular flux) is simply related to Bby [Larsen and Vasques 2011]
I(x,Ω) = Z∞
0B(x−sΩ,Ω) + Qc(x−sΩ)
4πXc(s) + Qu(x−sΩ)
4πXu(s)ds (7)
where
Xc(s) = Z∞
s
pc(s0)ds0(8)
is the attenuation law leaving a collision [Larsen and Vasques 2011] and
Xu(s) = Z∞
s
pu(s0)ds0(9)
is the uncorrelated-origin attenuation law [d’Eon 2018]. Finally, the fluence (scalar flux) is
φ(x) = Z4π
I(x,Ω)dΩ.(10)
For a single point source, where either one of Qcor Quis a unit dirac delta at the origin and the other is zero, it suffices
to solve Eq.(6) for B(r,µ), and then the radiance and fluence are determined after by Eq.(7). We solve for these Green’s
functions in the next section using Fourier transforms.
2.1.3 Relationship to previous GRT models
This formulation of random flights is an extension of the generalized models of Larsen [2011] and Davis [2006] to
support reciprocal transport in bounded domains by including pu(s), following [Audic and Frisch 1993], who proposed
such an extension for Markovian binary mixtures. Boundary conditions for this extension were given in an earlier

paper [d’Eon 2019]. The extended integral equations Eq.(6) and (7) are also analagous to Sahni’s [1989], where he
considered a different class of Markovian binary mixtures than Audic and Frisch in what could be considered the first
proposal of a non-exponential two-point transport theory. Our extended formulation also supports several proposed
forms of non-exponential random flights where all free paths use pu(s)[Davis 2006; Taine et al. 2010; Davis and Xu
2014; Wrenninge et al. 2017; Liemert and Kienle 2017; Binzoni et al. 2018]. However, application of these models
and related Green’s functions to bounded domains are known to result in non-reciprocal transport.
2.2 Grosjean’s solution
In his thesis, Grosjean [1951] solved a fully general random flight in an infinite 3D medium with an isotropic point
source. He solved for C(r,µ)and B(r,µ), given in spherical harmonic expansions whose coefficient functions are
given by Fourier inversion (requiring numerical inversion in most cases). In the most general form of his work, Gros-
jean permitted free-path-length distributions between collisions pn(s), single-scattering albedos cn(s), and scattering
kernels (phase functions) P
n(Ωi→Ωo)that are chosen independently for each collision order n>0, to build fully
heterogeneous random flights. He also presented simplifications for the case of a completely homogeneous random
flight where the intercollision free-path length distribution pc(s)and phase function P(Ωi→Ωo)are identical for all
collision orders nand the single-scattering albedo 0 <c≤1 at every collision is the same constant.
For our homogeneous random flight (p1(s) = pc(s)), Grosjean showed that the angular collision rate density has a
spherical harmonic expansion given by
C(r,µ) =
∞
∑
l=0
Cl(r)(2l+1)P
l(µ)(11)
where
Cl(r) = 1
8πcZ∞
0h(l)(u)u jl(r u)du (12)
in terms of expansion functions h(l)(u)that are the solution of the linear system ([Grosjean 1951], p. 77)
h(l)(u) = 2
πuF(l,0)(u) +
∞
∑
m=0
Amh(m)(u)F(l,m)(u).(13)
The functions Fare given by ([Grosjean 1951], p. 70)
F(l,m)(u) = c
2im−lZ∞
0Z1
−1pc(z)P
l(µ)P
m(µ)eiµuz dµdz (14)
=cZ∞
0pc(y)dy imP
md
idz (jl(z))z≡yu
(15)
where jl(z)is the spherical Bessel function
jl(z) = pπ
2Jl+1/2(z)
√z.(16)
The notation
P
md
idz (jl(z)) (17)
is understood to mean the differential operator formed from replacing znin the expansion of Legendre polynomial
P
m(z)with
∂n
in∂zn(18)
applied to jl(z). The functions Fobey a symmetry F(l,m)(u)=(−1)(l+m)F(m,l)(u).
The Fourier integrals for Cl(r)may only be convergent in the sense of Cesaro summability, prompting the separation
of the density of first collisions to express the total collision-rate density as ([Grosjean 1951], p.75)
C(r,µ) = pc(r)
4πr2δ(1−µ) +
∞
∑
l=0
C+
l(r)(2l+1)P
l(µ)(19)
with
C+
l(r) = 1
8πcZ∞
0h(l)(u)−2u
πF(l,0)u jl(r u)du.(20)

From these derivations we immediately have the scalar collision-rate density C(r) = 2C0(r)for correlated emission.
For the case of uncorrelated emission, p1(s) = pu(s), and we refer to Grosjean’s more general derivation. For this
almost homogeneous random flight, we find a modified system of equations for the expansion functions hin Eq.(23)
h(l)(u) = 2
πuF(l,0)
1(u) +
∞
∑
m=0
Amh(m)(u)F(l,m)(u)(21)
where the functions F1arise from appropriately modifying Eq.(14) to include the free-path length distribution p1(s)
instead of always pc(s), giving
F(l,m)
1(u) = cZ∞
0p1(y)dy imP
md
idz (jl(z))z≡yu
.(22)
For the radiance and fluence, we use Grosjean’s solutions for the Neumann series of a heterogeneous flight ([Grosjean
1951], Eqs.(259,259’)). The rate density for the particle to enter its nth collision at r with cosine µis
C(r,µ|n) =
∞
∑
l=0
Cl(r|n)(2l+1)P
l(µ)(23)
where
Cl(r|n) = 1
8πcZ∞
0h(l)
n(u)u jl(r u)du (24)
and the Neumann series hfunctions are
h(l)
n(u) =
∞
∑
m=0
Amhn−1F(l,m)
n,(n=2,3,4...)(25)
h(l)
1(u) = 2uF(l,0)
1(26)
where F(l,m)
nare defined using the free-path length distribution for the nth free path pn(s). The homogeneous system
of equations in Eq.(13) follows from Eqs.(25) using the definition h(l)(u) = ∑∞
n=1h(l)
n(u)and that F(l,m)
n=F(l,m)for
the homogeneous flight. We now add an additional segment to each term in this Neumann series using a free-path
distribution that is proportional to the attenuation law for leaving a collision. This creates a ficticious collision density
that essentially applies Eq.(7) to the in-scattering rate density, which follows our previous approach for the case of
isotropic scattering [d’Eon 2019]. From Eq.(25) it is clear how a given hnrelates to the previous order hn−1, and we
find
h(l)
φ(u) =
∞
∑
m=0
Amh(m)F(l,m)
X(27)
where the Ffunctions use the attenuation law for leaving a collision (Eq.(8)) instead of a free path distribution,
F(l,m)
X(u) = cZ∞
0Xc(y)dy imP
md
idz (jl(z))z≡yu
.(28)
This accounts for all flux that arises from collisions in the system. Adding the uncollided flux from the source we find
the total fluence
φ(r) = X0(s)
4πr2+1
4πcZ∞
0h(0)
φ(u)u jk(ru)du (29)
where
X0(s) = Z∞
s
p1(s0)ds0.(30)
The full radiance integrated around a given cone with cosine µis
I(r,µ) = X0(s)
4πr2δ(1−µ) +
∞
∑
l=0
Il(r)(2l+1)P
l(µ)(31)
with
Il(r) = 1
8πcZ∞
0h(l)
φ(u)u jl(r u)du.(32)
Radiance at some position xand direction Ωin the system is given by I(x,Ω) = I(r,µ)/(2π).

2.2.1 The case of classical exponential random flights
For clarity, we briefly examine classical radiative transfer under the present formalism.
In classical linear transport with no spatial correlation between collisions in homogeneous media, the above derivation
is equivalent to alternatives involving Kuscer/Chandrasekhar polynomials that satisfy a two-term recurrence [Davison
2000; Kušˇ
cer 1955; Grosjean 1963; Ganapol 2003]. With pc(s) = e−s, the Ffunctions reduce to
F(l,m)(x) = cim−l
2Z1
−1
P
l(µ)P
m(µ)
1−ixµdµ(33)
with known general solutions in terms of Legendre Q functions [Grosjean 1963; Vanmassenhove and Grosjean 1967],
the first few low order terms being
F(0,0)(u) = ctan−1(u)
u,F(0,1)(u) = ctan−1(u)−u
u2,F(1,1)(u) = cu−tan−1(u)
u3.(34)
3 Gamma random flights in 3D
To test Grosjean’s derivations for the case of non-exponential random flights we chose gamma random flights [Beghin
and Orsingher 2010; Le Caër 2011; Pogorui and Rodríguez-Dagnino 2011; d’Eon 2013], which admit explicit solu-
tions in some cases and have a number of interesting properties with respect to diffusion theory. Intercollision free-path
lengths are distributed according the normalized gamma distribution
pc(s) = essa−1
Γ(a),a>0,(35)
which includes classical exponential transport when a=1. For Monte Carlo validation, random free-path lengths are
easily sampled from
s=−log ξ1ξ2...ξa,(36)
where ξn∈[0,1]are aindependent random uniform variates. We used the generalized collision estimator for collision-
rate density and radiance [d’Eon 2019] to compute the Monte Carlo reference solutions below.
Combining Eq.(35) with (14) we require the integrals
F(l,m)(u) = c
2Z∞
0Z1
−1
ezza−1
Γ(a)im−lP
l(µ)P
m(µ)eiµuzdµdz =c
2im−lZ1
−1
P
l(µ)P
m(µ)
(1−iµu)adµ(37)
in the general case a>0. For the case m=0, we found a general solution
F(l,0)(u) = √πc2−l−1ulΓ(a+l)
Γ(a)2˜
F1a+l
2,1
2(a+l+1);l+3
2;−u2(38)
using the regularized hypergeometric function 2˜
F1. We suspect a completely general solution is possible using a
two-term recurrence, similar to the exponential case, but we did not find it.
Isotropic Scattering For the case of isotropic scattering, the expansion coefficients in Eq.(1) are
A0=1,Al>0=0.(39)
The general solution of the linear system for correlated emission (13) with expansion coefficients (39) yields
h(l)=2uF(l,0)
π−πF(0,0)(40)
and, for the uncorrelated point source, solution of Eq.(21) is
h(l)=2u
π F(l,0)F(0,0)
1
1−F(0,0)+F(l,0)
1!.(41)
Linearly-Anisotropic Scattering With linearly-anistropic scattering with parameter −1<b<1, we find
A0=1,A1=b,Al>1=0,(42)
yielding expansion functions for correlated emission
h(l)=−2u((bF(1,1)(u)−1)F(l,0)(u) + bF(0,1)(u)F(l,1)(u))
πb(F(0,1)(u))2−F(1,1)(u)+F(0,0)(u)(bF(1,1)(u)−1) + 1.(43)

0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
r
4πr2Cc(r)
0 5 10 15 20
10-4
0.001
0.010
0.100
r
4πr2Cc(r)
Infinite 3D, isotropic point source, linearly-anisotropic scattering, Gamma-2 random flight -correlated emission
Collision-rate density Cc[r], c =0.7, b =0.7
Figure 2: Scalar collision-rate density Cc(r)about an isotropic point source in 3D with linearly-anisotropic scattering
and intercollision free-path lengths drawn from e−ss. Validation of Eq.(48) (continuous) with respect to Monte Carlo
(dots).
The bulky expressions for the uncorrelated case are omitted.
Rayleigh Scattering We also consider a simple three-term phase function due to Rayleigh that has application in
light scattering [Chandrasekhar 1960], with
A0=1,A1=0,A2=1
2,Al>2=0 (44)
yielding correlated expansion functions
h(l)=2u((2−F(2,2)(u))F(l,0)(u) + F(0,2)(u)F(l,2)(u))
π(−(F(0,2)(u))2+F(0,0)(u)(F(2,2)(u)−2)−F(2,2)(u) + 2).(45)
The bulky expressions for the uncorrelated case are omitted.
3.1 Gamma-2 random flight in 3D
With a=2 and an intercollision FPD pc(s) = e−ss, we find, using Eq.(14),
F(0,0)(u) = c
1+u2,F(0,1)(u) = c1
u3+u−tan−1(u)
u2,F(1,1)(u) = c2 tan−1(u)−u(u2+2)
u2+1
u3
F(0,2)(u) = cu1
u2+1+2−3 tan−1(u)
u3,F(1,2)(u) = cu2+1u2+9tan−1(u)−u7u2+9
2(u6+u4)(46)
F(2,2)(u) = cuu2+1
u2+1+8−3u2+3tan−1(u)
u5.
3.1.1 Linearly-anisotropic scattering
Combining Eq.(46) with Eq.(43) we find
h(0)=2cu bcu2−bc u2+1tan−1(u)2+u4
π(bc2(u2+1)tan−1(u)2+bcu2(−c+u2+2)−2bc (u2+1)utan−1(u) + u4(−c+u2+1)) (47)
yielding scalar collision-rate density
Cc(r) = 1
2π2rZ∞
0
ubcu2−bc u2+1tan−1(u)2+u4
(u4((b−1)c+1)−b(c−2)cu2+bc (u2+1)tan−1(u) (ctan−1(u)−2u) + u6)sin(ru)du.
(48)
A comparison of this result to Monte Carlo reference is provided in Figure 2. While diffusion is an exact result for
collision-rate density in 3D with gamma-2 flights and isotropic scattering [d’Eon 2013], we see that this does not
extend to more general phase functions.

0.0 0.5 1.0 1.5 2.0 2.5 3.0
1.0
1.1
1.2
1.3
1.4
1.5
r
4πr2ϕc(r)
0 5 10 15 20 25
0.001
0.010
0.100
1
r
4πr2ϕc(r)
Infinite 3D, isotropic point source, linearly-anisotropic scattering, Gamma-2 random flight -correlated emission
Fluence ϕc[r], c =0.8, b =0.7
Figure 3: Scalar flux / fluence φc(r)about a correlated isotropic point source in 3D with linearly-anisotropic scattering
and intercollision free-path lengths drawn from e−ss. Validation of Eq.(50) (continuous) with respect to Monte Carlo
(dots) and comparison of the diffusion approximation, Eq.(51).
For the fluence, we find
F(0,0)
X=1
u2+1+tan−1(u)
u,F(0,1)
X=−u
u2+1(49)
and
h(0)
φ=−
2c2−u2+1u2tan−1(u)bc−u2+u2+bc u2+1utan−1(u)2+bc u2+12tan−1(u)3+u3−bc+u2+u2
π(u2+1) (bc2(u2+1)tan−1(u)2+bcu2(−c+u2+2)−2bc (u2+1)utan−1(u) + u4(−c+u2+1))
(50)
The fluence then follows from Eq.(29). Taking a (0,2)order Pade approximant of the Fourier-transformed density
πh(0)
φ/(2cu)we find the diffusion appromxation for the fluence [d’Eon 2013]
φ(r)≈e−r(r+1)
4πr2−
3c(bc −3)exp 
−√3r
rb(2c2−4c+3)−6c+15
(c−1)(bc−3)

2πr(b(2c2−4c+3)−6c+15)(51)
A comparison of these result to Monte Carlo reference is provided in Figure 3.
In Figure 4 we compare the angular distributions C(r,µ)and I(r,µ)/hsciusing 4 term Legendre expansions about a
correlated point source at a radius r=11.4437. At this distance from the point source this low order expansion seems
reasonably accurate with respect to Monte Carlo and illustrates how collision rate and flux are not proportional in
GRT.
For the uncorrelated source, we find (22)
F(0,0)
1(u) = c
2u2+2+ctan−1(u)
2u,F(0,1)
1(u) = −cu
2(u2+1)(52)
giving (21)
h(0)=cu 2bcu2−2bc u2+1tan−1(u)2+u4+u5+u3tan−1(u)
π(bc2(u2+1)tan−1(u)2+bcu2(−c+u2+2)−2bc (u2+1)utan−1(u) + u4(−c+u2+1)) (53)
for the scalar collision rate density
Cu(r) = e−r(r+1)
8πr2
+Z∞
0
csin(ru)−u2+1tan−1(u)−bcu2+bc tan−1(u)u2+1tan−1(u) + u+ (b−1)u4+bu3c+u2+u5
4π2r(u2+1) (u4((b−1)c+1)−b(c−2)cu2+bc (u2+1)tan−1(u) (ctan−1(u)−2u) + u6)du.
(54)

-1.0 -0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
μ
2πr2C[r,μ]
c=0.99, b =0.7, r =11.4437
Figure 4: Comparison of the angular collision rate density C(r,µ)(continuous) and the classically scaled radiance
I(r,µ)/hsci(dashed) for gamma-2 flights and linearly-anisotropic scattering showing agreement with Monte Carlo
(dots) and how the two densities are not proportional in GRT.
0.0 0.5 1.0 1.5 2.0 2.5
0.50
0.52
0.54
0.56
0.58
r
4πr2Cc(r)
0 5 10 15 20
10-4
0.001
0.010
0.100
r
4πr2Cc(r)
Infinite 3D, isotropic point source, linearly-anisotropic scattering, Gamma-2 random flight -uncorrelated emission
Collision-rate density Cu[r], c =0.7, b =0.7
Figure 5: Scalar collision-rate density Cu(r)about an uncorrelated-emission isotropic point source in 3D with
linearly-anisotropic scattering and intercollision free-path lengths drawn from e−ss. Validation of Eq.(54) (continuous)
with respect to Monte Carlo (dots).
A comparison of this result to Monte Carlo reference is provided in Figure 5.
3.2 Gamma-3 random flight in 3D
With intercollision FPD pc(s) = 1
2e−ss2(a=3) we find
F(0,0)(u) = c
(u2+1)2,F(0,1)(u) = −cu
(u2+1)2,F(1,1)(u) =
c2u3+u
(u2+1)2−tan−1(u)
u3
F(0,2)(u) = 1
2c 3 tan−1(u)
u3+−5u2−3
(u3+u)2!,F(1,2)(u) = 3cu1
u2+1+2−3 tan−1(u)
2u4−cu
(u2+1)2(55)
F(2,2)(u) =
c3u2+9tan−1(u)−u(19u4+48u2+27)
(u2+1)2
2u5.
3.2.1 Linearly-anisotropic scattering
Combining Eq.(55) with Eq.(43) we find
h(0)=2cu −bcu +bc tan−1(u) + u3
πbcu (c−2u2−1) + bc (u2+1)2−ctan−1(u) + u3(u2+1)2−c (56)

0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
r
4πr2Cc(r)
0 10 20 30 40
10-6
10-5
10-4
0.001
0.010
0.100
1
r
4πr2Cc(r)
Infinite 3D, isotropic point source, linearly-anisotropic scattering, Gamma-3 random flight -correlated emission
Collision-rate density Cc[r], c =0.8, b = -0.9
Figure 6: Scalar collision-rate density Cc(r)about an isotropic point source in 3D with linearly-anisotropic scattering
and intercollision free-path lengths drawn from e−ss2/2. Validation of Eq.(57) (continuous) with respect to Monte
Carlo (dots).
yielding scalar collision-rate density
Cc(r) = 1
2π2rZ∞
0
u−bcu +bc tan−1(u) + u3
bcu (c−2u2−1) + bc (u2+1)2−ctan−1(u) + u3(u2+1)2−csin(ru)du.(57)
A comparison of this result to Monte Carlo reference is provided in Figure 6.
3.3 Gamma-4 random flight in 3D
With a=4 we find that the exact fluence about the point source with linearly-anisotropic scattering can be expressed
explicitly as a sum of diffusion modes. With the intercollision FPD pc(s) = 1
6e−ss3we find
F(0,0)(u) = −cu2−3
3(u2+1)3,F(0,1)(u) = −4cu
3(u2+1)3,F(1,1)(u) = c−3cu2
3(u2+1)3,F(0,2)(u) = 4cu2
3(u2+1)3
F(1,2)(u) =
c9 tan−1(u)−u(23u4+24u2+9)
(u2+1)3
6u4,F(2,2)(u) =
cu(11u6+60u4+72u2+27)
(u2+1)3−27 tan−1(u)
3u5.(58)
Combining Eq.(58) with Eq.(43) we find
h(0)=−2cu bc +u4−2u2−3
πbc2+c(u2+1) (b(3u2−1) + u2−3) + 3(u2+1)4(59)
yielding scalar collision-rate density
Cc(r) = 1
2π2rZ∞
0−ubc +u4−2u2−3
bc2+c(u2+1) (b(3u2−1) + u2−3) + 3(u2+1)4sin(ru)du.(60)
A comparison of this result to Monte Carlo reference is provided in Figure 7. The complete scalar collision rate density
can be solved by standard contour integration ([Grosjean 1963], pp.73–75), yielding
Cc(r) = ∑
v∈v+
e−rv
4πr1−v2bc +v4+2v2−3
2c(b(2c+3v4−3) + v4+4v2−5)(61)
where v+is the set of roots with positive real part of the dispersion equation
c2v4+10v2+1−c11v4+26v2+111−v23+10 1−v28=0 (62)
for which we found two real and two complex roots in v+. Figure 7 also includes comparisons of this exact result to
two forms of moment-preserving diffusion approximation found using the methods in [d’Eon 2019]. For the Classical

0.0 0.5 1.0 1.5 2.0
0.00
0.05
0.10
0.15
0.20
0.25
r
4πr2Cc(r)
0 5 10 15 20
10-5
10-4
0.001
0.010
0.100
r
4πr2Cc(r)
Infinite 3D, isotropic point source, linearly-anisotropic scattering, Gamma-4 random flight -correlated emission
Collision-rate density Cc[r], c =0.5, b =0.7
Figure 7: Scalar collision-rate density Cc(r)about an isotropic point source in 3D with linearly-anisotropic scat-
tering and intercollision free-path lengths drawn from 1
6e−ss3. Validation of Eq.(60) (continuous) with respect to
Monte Carlo (dots). Comparisons to a classical diffusion approximation (Eq.(63), dot-dashed) and modified-diffusion
approximation (Eq.(64), dashed) are also shown.
diffusion approximation, we find
Cc(r)≈(3−bc)e−r
√2√bc+5
(c−1)(bc−3)
8πr(bc +5).(63)
and removing the first-collided portion, we find the Grosjean-form diffusion approximation
Cc(r)≈e−rr
24π+c
1−c
e−r
v
4πrv2,v=2b(5(c−2)c+8)−30(c−2)
3(c−1)(bc −3).(64)
3.4 Gamma-6 flights in 3D
With intercollision FPD pc(s) = 1
120 e−ss5we find
F(0,0)(u) = cu4−10u2+5
5(u2+1)5,F(0,1)(u) = 2cu 3u2−5
5(u2+1)5,F(1,1)(u) = c5u4−38u2+5
15 (u2+1)5
F(0,2)(u) = −2cu2u2−7
5(u2+1)5,F(1,2)(u) = 4cu 3u2−1
5(u2+1)5,F(2,2)(u) = c1−3u22
5(u2+1)5.(65)
3.4.1 Rayleigh scattering
Similar to the gamma-4 case with linearly-anisotropic scattering, with Rayleigh scattering and gamma-6 flights we
find a solvable scalar collision rate density. Combining Eq.(65) with Eq.(45) we find
h(0)=−
2cu cu4−10u2+1−2u2+13u4−10u2+5
πc2(u4−10u2+1)−c(11u4−26u2+11) (u2+1)3+10 (u2+1)8(66)
yielding scalar collision-rate density
Cc(r) = 1
2π2rZ∞
0−
u2cu4−10u2+1−2u2+13u4−10u2+5
uc2(u4−10u2+1)−c(11u4−26u2+11) (u2+1)3+10 (u2+1)8sin(ru)du.(67)
A comparison of this result to Monte Carlo reference is provided in Figure 8.

0 5 10 15
0.0
0.5
1.0
1.5
2.0
r
4πr2C0(r)
0 20 40 60 80 100 120
0.001
0.010
0.100
1
r
4πr2C0(r)
Infinite 3D, isotropic point source, Rayleigh scattering, Gamma -6 random flight -correlated emission
Collision-rate density C0[r], c =0.999
Figure 8: Scalar collision-rate density C0(r)about an isotropic point source in 3D with Rayleigh scattering and
intercollision free-path lengths drawn from 1
120 e−ss5. Validation of Eq.(67) (continuous) with respect to Monte Carlo
(dots).
The complete scalar collision rate density is then solved by standard contour integration ([Grosjean 1963], pp.73–75)
Cc(r) = ∑
v∈v+1−v2cv4+10v2+1+2v4+10v2+5v2−13
3c2c(v4+12v2+3) + (v2+3) (11v2+9) (v2−1)3
e−rv
4πr(68)
where v+is the set of roots with positive real part of the dispersion equation
c2v4+10v2+1−c11v4+26v2+111−v23+10 1−v28=0.(69)
In this case we always noted two real roots and six complex roots for a variety of absorption levels c.
4 Conclusion
We have derived the point source Green’s functions for infinite media with anisotropic scattering in non-classical linear
transport where the free-path distributions between collisions and attenuation laws are non-exponential. The general
solutions are expressed as Fourier inversions, which were validated numerically using gamma random flights in 3D.
Distinct solutions for both collision rate, and fluence and their angular counterparts were derived and tested using
Monte Carlo. For low integer-order gamma flights and low Legendre orders we found the solutions to be numerically
straightforward to manage. Higher order angular expansions and more challenging free-path distribution such as
power-law flights [Davis 2006] will require more numerical care when dealing with the oscillatory Fourier inversions.
Nevertheless, the provided gamma flight solutions provide important benchmarks for GRT and for validating more
efficient approximations, such as SPN [Palmer and Vasques 2020].
5 Acknowledgements
We thank M.M.R. Williams for helpful feedback on the manuscript and Forrest Brown for bringing several older
works [Randall 1964; Doub 1961] to our attention.
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