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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 PΩ·x−x0

|x−x0|Bx0,x−x0

|x−x0|+Qc(x0)

4πpc(|x−x0|)

|x−x0|2+Qu(x0)

4π

pu(|x−x0|)

|x−x0|2dV 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∞

0B(x−sΩ,Ω) + Qc(x−sΩ)

4πXc(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

md

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

md

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∞

0h(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

md

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

md

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) = cu−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˜

F1a+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))

πb(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) = c1

u3+u−tan−1(u)

u2,F(1,1)(u) = c2 tan−1(u)−u(u2+2)

u2+1

u3

F(0,2)(u) = cu1

u2+1+2−3 tan−1(u)

u3,F(1,2)(u) = cu2+1u2+9tan−1(u)−u7u2+9

2(u6+u4)(46)

F(2,2)(u) = cuu2+1

u2+1+8−3u2+3tan−1(u)

u5.

3.1.1 Linearly-anisotropic scattering

Combining Eq.(46) with Eq.(43) we ﬁnd

h(0)=2cu bcu2−bc u2+1tan−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

ubcu2−bc u2+1tan−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+1u2tan−1(u)bc−u2+u2+bc u2+1utan−1(u)2+bc u2+12tan−1(u)3+u3−bc+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+1tan−1(u)2+u4+u5+u3tan−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+1tan−1(u)−bcu2+bc tan−1(u)u2+1tan−1(u) + u+ (b−1)u4+bu3c+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) =

c2u3+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) = 3cu1

u2+1+2−3 tan−1(u)

2u4−cu

(u2+1)2(55)

F(2,2)(u) =

c3u2+9tan−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

πbcu (c−2u2−1) + bc (u2+1)2−ctan−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−ctan−1(u) + u3(u2+1)2−csin(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) = −cu2−3

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) =

c9 tan−1(u)−u(23u4+24u2+9)

(u2+1)3

6u4,F(2,2)(u) =

cu(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

πbc2+c(u2+1) (b(3u2−1) + u2−3) + 3(u2+1)4(59)

yielding scalar collision-rate density

Cc(r) = 1

2π2rZ∞

0−ubc +u4−2u2−3

bc2+c(u2+1) (b(3u2−1) + u2−3) + 3(u2+1)4sin(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

4πr1−v2bc +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

c2v4+10v2+1−c11v4+26v2+111−v23+10 1−v28=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) = cu4−10u2+5

5(u2+1)5,F(0,1)(u) = 2cu 3u2−5

5(u2+1)5,F(1,1)(u) = c5u4−38u2+5

15 (u2+1)5

F(0,2)(u) = −2cu2u2−7

5(u2+1)5,F(1,2)(u) = 4cu 3u2−1

5(u2+1)5,F(2,2)(u) = c1−3u22

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 cu4−10u2+1−2u2+13u4−10u2+5

πc2(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−

u2cu4−10u2+1−2u2+13u4−10u2+5

uc2(u4−10u2+1)−c(11u4−26u2+11) (u2+1)3+10 (u2+1)8sin(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−v2cv4+10v2+1+2v4+10v2+5v2−13

3c2c(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

c2v4+10v2+1−c11v4+26v2+111−v23+10 1−v28=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.

References

ALT, W. 1980. Biased random walk models for chemotaxis and related diffusion approximations. Journal of mathe-

matical biology 9, 2, 147–177. https://doi.org/10.1007/BF00275919.

AUDIC, S., AN D FRISCH, H. 1993. Monte-Carlo simulation of a radiative transfer problem in a random medium:

Application to a binary mixture. Journal of Quantitative Spectroscopy and Radiative Transfer 50, 2, 127–147.

https://doi.org/10.1016/0022-4073(93)90113-V.

BEGHIN, L., AND ORSINGHER, E. 2010. Moving randomly amid scattered obstacles. Stochastics: An International

Journal of Probability and Stochastics Processes 82, 2, 201–229. https://doi.org/10.1080/17442500903359163.

BINZONI, T., MARTELLI, F., AND KOZUBOWSKI, T. J. 2018. Generalized time-independent correlation transport

equation with static background: inﬂuence of anomalous transport on the ﬁeld autocorrelation function. JOSA A

35, 6, 895–902. https://doi.org/10.1364/JOSAA.35.000895.

BITTERLI, B., RAVICHANDRAN, S., MÜLLER, T., WRENNINGE, M., NOVÁK, J., MARSCHNER, S., AN D JAROSZ,

W. 2018. A radiative transfer framework for non-exponential media. ACM Transactions on Graphics 37, 6.

https://doi.org/10.1145/3272127.3275103.

BURRUS, W. 1960. Radiation transmission through boral and similar heterogeneous materials consisting of randomly

distributed absorbing chunks. Tech. rep., Oak Ridge National Lab., Tenn. https://doi.org/10.2172/4196641.

CASE, K. M., DE HOFFMAN, F., AND PLACZEK, G. 1953. Introduction to the Theory of Neutron Diffusion, vol. 1.

US Government Printing Ofﬁce.

CHANDRASEKHAR, S. 1960. Radiative Transfer. Dover.

DAVIS, A. B., AN D MARS HAK, A. 2004. Photon propagation in heterogeneous optical media with spatial cor-

relations: enhanced mean-free-paths and wider-than-exponential free-path distributions. Journal of Quantitative

Spectroscopy and Radiative Transfer 84, 1, 3–34. https://doi.org/10.1016/S0022-4073(03)00114-6.

DAVIS, A. B., AND XU, F. 2014. A generalized linear transport model for spatially correlated stochastic media. Jour-

nal of Computational and Theoretical Transport 43, 1-7, 474–514. https://doi.org/10.1080/23324309.2014.978083.

DAVIS, A. B. 2006. Effective propagation kernels in structured media with broad spatial correlations, illustration

with large-scale transport of solar photons through cloudy atmospheres. In Computational Methods in Transport.

Springer, 85–140. https://doi.org/10.1007/3-540-28125-8_5.

DAVISON, B. 1957. Neutron Transport Theory. Oxford University Press.

DAVISON, B. 2000. Angular distribution due to an isotropic point source and spherically symmetrical eigensolutions

of the transport equation (MT-112). Progress in Nuclear Energy 36, 3, 323 – 365. Nuclear Reactor Theory in

Canada 1943-1946. https://doi.org/10.1016/S0149-1970(00)00012-3.

D’EON, E. 2013. Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Lin-

ear Boltzmann Transport in Arbitrary Dimension. Transport Theory and Statistical Physics 42, 6-7, 237–297.

https://doi.org/10.1080/00411450.2014.910231.

D’EON, E. 2018. A reciprocal formulation of nonexponential radiative transfer. 1: Sketch and motivation. Journal of

Computational and Theoretical Transport.https://doi.org/10.1080/23324309.2018.1481433.

D’EON, E. 2019. A reciprocal formulation of nonexponential radiative transfer. 2: Monte-Carlo Estima-

tion and Diffusion Approximation. Journal of Computational and Theoretical Transport 48, 6, 201–262.

https://doi.org/10.1080/23324309.2019.1677717.

DOUB, W. 1961. Particle self-shielding in plates loaded with spherical poison particles. Nuclear Science and

Engineering 10, 4, 299–307. https://doi.org/10.13182/NSE61-A15371.

FELLER, W. 1971. An Introduction to Probability theory and its application Vol II. John Wiley and Sons.

FRANK, M., AN D SUN, W. 2018. Fractional diffusion limits of non-classical transport equations. Kinetic & Related

Models 11, 6, 1503–1526. https://doi.org/10.3934/krm.2018059.

FRANK, M., GOUDON, T., ET AL. 2010. On a generalized Boltzmann equation for non-classical particle transport.

Kinetic and Related Models 3, 395–407. https://doi.org/10.3934/krm.2010.3.395.

FRANK, M., KRY CKI , K., LARSEN, E. W., AND VASQUES , R. 2015. The nonclassical Boltzmann equation and

diffusion-based approximations to the Boltzmann equation. SIAM Journal on Applied Mathematics 75, 3, 1329–

1345. https://doi.org/10.1137/140999451.

GANA POL, B. D. 2003. Fourier transform transport solutions in spherical geometry. Transport Theory and Statistical

Physics 32, 5, 587 – 605. https://doi.org/10.1081/TT-120025067.

GANA POL, B., 2008. Analytical Benchmarks for Nuclear Engineering Applications.

GROS JEA N, C. 1951. The Exact Mathematical Theory of Multiple Scattering of Particles in an Inﬁnite Medium.

Memoirs Kon. Vl. Ac. Wetensch. 13, 36.

GROS JEA N, C. C. 1963. A new approximate one-velocity theory for treating both isotropic and anisotropic multiple

scattering problems. Part I. Inﬁnite homogeneous scattering media. Tech. rep., Universiteit, Ghent.

IVANOV, V. 1994. Resolvent method: exact solutions of half-space transport problems by elementary means. Astron-

omy and Astrophysics 286, 328–337.

JARABO, A., ALIAGA, C., A ND GUTIERREZ, D. 2018. A radiative transfer framework for spatially-correlated

materials. ACM Transactions on Graphics 37, 4, 14. https://doi.org/10.1145/3197517.3201282.

KOSTINSKI, A. B. 2001. On the extinction of radiation by a homogeneous but spatially correlated random medium.

JOSA A 18, 8, 1929–1933. https://doi.org/10.1364/JOSAA.18.001929.

KUŠ ˇ

CER, I. 1955. Milne’s problem for anisotropic scattering. Journal of Mathematics and Physics 34, 1-4, 256–266.

https://doi.org/10.1002/sapm1955341256.

LARSEN, E. W., AND VASQUES , R. 2011. A generalized linear Boltzmann equation for non-

classical particle transport. Journal of Quantitative Spectroscopy and Radiative Transfer 112, 4, 619–631.

https://doi.org/10.1016/j.jqsrt.2010.07.003.

LECAËR, G. 2011. A new family of solvable pearson-dirichlet random walks. Journal of Statistical Physics 144, 1,

23–45. https://doi.org/10.1007/s10955-011-0245-4.

LIEMERT, A., AND KIENLE, A. 2017. Radiative transport equation for the Mittag-Lefﬂer path length distribution.

Journal of Mathematical Physics 58, 5, 053511. https://doi.org/10.1063/1.4983682.

MOON, J., WALTE R, B., AND MARSCHNER, S. 2007. Rendering discrete random media using precomputed scatter-

ing solutions. Rendering Techniques 2007, 231–242. https://doi.org/10.2312/EGWR/EGSR07/231-242.

NARASIMHAN, S. G., AND NAYAR, S. K. 2003. Shedding light on the weather. In 2003 IEEE Computer Society

Conference on Computer Vision and Pattern Recognition, 2003. Proceedings., vol. 1, IEEE, I–I.

PAASSCHENS, J. C. J. 1997. Solution of the time-dependent boltzmann equation. Phys. Rev. E 56, 1 (Jul), 1135–1141.

https://doi.org/10.1103/PhysRevE.56.1135.

PALM ER, R. K., AND VASQ UES , R. 2020. Asymptotic Derivation of the Simpliﬁed P

NEquations for Nonclassical

Transport with Anisotropic Scattering. https://arxiv.org/abs/2001.05890.

POGORUI , A. A., AND RODRÍGUEZ-DAGNINO, R. M. 2011. Isotropic random motion at ﬁnite speed with K-Erlang

distributed direction alternations. Journal of Statistical Physics 145, 1, 102.

https://doi.org/10.1007/s10955-011-0328-2.

RANDALL , C. 1964. Generalized treatment of particle self-shielding. In The Naval Reactors Handbook Vol. 1:

Selected Basic Techniques, A. Radkowsky, Ed. United States Atomic Energy Comission, 553.

RUKO LAINE, S. A. 2016. Generalized linear boltzmann equation, describing non-classical particle transport, and

related asymptotic solutions for small mean free paths. Physica A: Statistical Mechanics and its Applications 450,

205–216. https://doi.org/10.1016/j.physa.2015.12.105.

RYBICKI, G. B. 1965. Transfer of radiation in stochastic media. Tech. Rep. 180, Smithsonian Astrophysical Obser-

vatory, June.

SAHNI, D. 1989. Equivalence of generic equation method and the phenomenological model for linear trans-

port problems in a two-state random scattering medium. Journal of mathematical physics 30, 7, 1554–1559.

https://doi.org/10.1063/1.528288.

TAINE, J., BELL ET, F., LEROY, V., AN D IACONA, E. 2010. Generalized radiative transfer equation for porous

medium upscaling: Application to the radiative fourier law. International Journal of Heat and Mass Transfer 53,

19-20, 4071–4081. https://doi.org/10.1016/j.ijheatmasstransfer.2010.05.027.

TESSENDORF, J. 2011. Angular smoothing and spatial diffusion from the Feynman path integral represen-

tation of radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer 112, 4, 751–760.

https://doi.org/10.1016/j.jqsrt.2010.11.004.

TORQUATO, S., AND LU, B. 1993. Chord-length distribution function for two-phase random media. Physical Review

E 47, 4, 2950.

https://doi.org/10.1016/0306-4549(92)90013-2.

TUNA LEY, J. 1974. Theory of ac conductivity based on random walks. Physical Review Letters 33, 17, 1037.

https://doi.org/10.1103/PhysRevLett.33.1037.

TUNA LEY, J. 1976. Moments of the montroll-weiss continuous-time random walk for arbitrary starting time. Journal

of Statistical Physics 14, 5, 461–463. https://doi.org/10.1007/BF01040704.

VANMASSENHOVE, F., AND GROSJEAN, C. 1967. Electromagnetic Scattering. In Proc. Second Interdisciplinary

Conf. Electromagnetic Scattering, 721–763.

VASQU ES, R., AND LARSEN, E. W. 2014. Non-classical particle transport with angular-dependent path-length

distributions. i: Theory. Annals of Nuclear Energy 70, 292–300. https://doi.org/10.1016/j.anucene.2013.12.021.

WALLACE, P. 1948. Angular distribution of neutrons inside a scattering and absorbing medium. Canadian journal of

research 26, 2, 99–114. https://doi.org/10.1139/cjr48a-011.

WEISS, G. H., AND RUBIN, R. J. 1983. Random walks: theory and selected applications. Adv. Chem. Phys 52,

363–505. https://doi.org/10.1002/9780470142769.ch5.

WILLIAMS, M. 1977. On the role of the adjoint boltzmann equation in the calculation of energy deposition. Journal

of Physics D: Applied Physics 10, 17, 2343. https://doi.org/10.1088/0022-3727/10/17/006.

WRENNINGE, M., VILLEM IN, R., AND HERY, C. 2017. Path traced subsurface scattering

using anisotropic phase functions and non-exponential free ﬂights. Tech. Rep. 17-07, Pixar.

https://graphics.pixar.com/library/PathTracedSubsurface.

XU, F., DAVIS , A. B., AND DINER, D. J. 2016. Markov chain formalism for generalized radiative transfer in a

plane-parallel medium, accounting for polarization. Journal of Quantitative Spectroscopy and Radiative Transfer

184, 14–26. https://doi.org/10.1016/j.jqsrt.2016.06.004.

ZARROUATI, M., ENGUEHARD, F., AND TAINE, J. 2013. Statistical characterization of near-wall radiative properties

of a statistically non-homogeneous and anisotropic porous medium. International journal of heat and mass transfer

67, 776–783. https://doi.org/10.1016/j.ijheatmasstransfer.2013.08.021.

ZOIA, A., DUMONTEIL, E., AND MA ZZO LO, A. 2011. Collision densities and mean residence times for d-

dimensional exponential ﬂights. Physical Review E 83, 4, 041137. https://doi.org/10.1103/PhysRevE.83.041137.