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A Reciprocal Formulation of Nonexponential

Radiative Transfer. 3: Binary Mixtures

Eugene d’Eon

8i

Abstract

We derive the form of reciprocal generalized radiative transfer (RGRT) that includes

the Levermore-Pomraning attenuation law for paths leaving a deterministic origin. The

resulting model describes linear transport within multi-dimensional stochastic binary

mixtures with Markovian mixing statistics and nonstochastic albedo and phase function.

The derivation includes a new attenuation law and related free-path distribution between

collisions, which was previously estimated using a Monte Carlo approach. We also show

that previous stationary descriptions of binary mixtures with binomial mixing statistics

reduce to exponential attenuations and, thus, have an exact homogenization within clas-

sical radiative transfer.

Keywords: Generalized Radiative Transfer, Levermore-Pomraning, Boltzmann, non-

exponential, non-Beerian, binary mixture

1 Intro

In the previous two papers in this series [d’Eon 2018; d’Eon 2019] we presented moti-

vation and details of a formulation of reciprocal generalized radiative transfer (RGRT)

that accounts for correlation between scattering events in piecewise homogeneous

media. When scatterers in random media are spatially correlated, the chord-length

distributions between them are nonexponential. Classical transport theory is extended

to account for the nonexponential free-path lengths between scatterers by using a

semi-Markov nonexponential random ﬂight model with a two-point memory [Grosjean

1951]. The nonclassical Boltzmann equation that describes such transport has recently

been derived [Larsen and Vasques 2011].

Audic and Frisch [1993] proposed the ﬁrst reciprocal form of two-point nonclassical

transport in the context of stochastic binary mixtures. They noted the necessary dis-

tinction between free-path length statistics for a path beginning at the boundary and

the free-path lengths between collisions. They proposed accumulating histograms of

the intercollision free-path lengths within a given class of binary mixtures by a prepro-

cess of tracking Monte Carlo histories in explicit realizations of that class. We show

that RGRT naturally predicts the intercollision free path distribution directly from the

Pomraning-Levermore attenuation law, and we ﬁnd this prediction to agree with Monte

Carlo simulation in a rod with forward scattering.

The resulting form of RGRT should improve the utility of Audic and Frisch’s accel-

erated Monte Carlo approach to binary mixtures, which should be more efﬁcient than

arXiv:submit/2620001 [math-ph] 20 Mar 2019

the chord-length sampling (CLS) method [Zimmerman and Adams 1991; Donovan

and Danon 2003; Larmier et al. 2018] and should be useful for predicting transport

in molecular clouds [Boissé 1990], shielding materials [Becker et al. 2014], and other

applications of clumpy stochastic media [Pomraning 1998; Sanchez and Pomraning

1991].

2 Markovian Binary Mixtures

In this section we derive the statistical functions required by RGRT to describe trans-

port in Markovian binary mixtures with nonstochastic albedo and nonstochastic phase

function. This includes attenuation laws Xc,Xu, and related free-path distributions

pc, and pu, where “c” refers to free paths with an origin that is correlated spatially

to scatterers in the medium. Deterministic origins, such as boundary interfaces and

imbedded objects use the label “u” to denote their uncorrelated relationship to the

scatterers. These quantities and their relationships are summarized in Table 1.

The exact, uncorrelated-origin attenuation law for binary mixtures with Markovian

mixing statistics is [Avaste and Vaynikko 1974; Levermore et al. 1986; Vanderhaegen

1986]

Xu(s) = r+−˜

Σ

r+−r−

e−r+s+˜

Σ−r−

r+−r−

e−r−s(1)

where the decay constants are

2r±=hΣi+˜

Σ±qhΣi − ˜

Σ2+4β(2)

with parameters

hΣi=pAΣA+pBΣB(3)

˜

Σ=pBΣA+pAΣB+λ−1

A+λ−1

B(4)

β= (ΣA−ΣB)2pApB(5)

pi=λi

λA+λB

.(6)

The medium is parameterized by the macroscopic cross-sections for the two phases,

Σi≥0, and the mean chord lengths within each phase, λi>0. Using the volume

fractions pifor each phase yields the atomic mix total macroscopic cross-section hΣi.

To produce reciprocal transport under this attenuation law in bounded media with

scattering and nonstochastic single-scattering albedo crequires [d’Eon 2018] that the

distribution of free-path lengths between collision events is the normalization of

∂2

∂s2Xu(s) = r+−˜

Σ

r+−r−

r2

+e−r+s+˜

Σ−r−

r+−r−

r2

−e−r−s.(7)

We ﬁnd that the inverse of the normalization constant is

Z∞

0

∂2

∂s2Xu(s)ds =r++r−−˜

Σ=hΣi(8)

02468

0.0

0.2

0.4

0.6

0.8

s

pc(s)

Intercollision statistics:

ΣA=0.26, ΣB=1.08, λA=1.33, λB=0.7

Figure 1: Distribution of intercollision free-path lengths pc(s)for a binary stochastic

medium with Markovian mixing statistics. Monte Carlo (ﬁlled) vs Eq.(9) (continuous).

The known exact free-path distribution of distance-to-ﬁrst-collision from entry at the

boundary (pu(s)) is shown (dashed) for reference. The Monte Carlo simulation aver-

ages 60,000 histories in 60,000 unique realizations, each with one ﬁrst collision and

one second collision (assuming forward scattering in a rod).

yielding the correlated-origin free-path distribution

pc(s) = 1

hΣir+−˜

Σ

r+−r−

r2

+e−r+s+˜

Σ−r−

r+−r−

r2

−e−r−s(9)

with the mean correlated mean free path

hsci=Z∞

0pc(s)ds =1

hΣi.(10)

This probability distribution function pc(s)is the ensemble-averaged distribution of

distances between collisions over arbitrarily many phase transitions. We are not aware

of this result being published previously and note that a simple Monte Carlo experi-

ment similar to [Adams et al. 1989] shows excellent agreement with Eq.(9) (Figure 1).

In this experiment, we sampled explicit realizations of 1D rod mixtures, entering at

the boundary, scattering forward with no absorption and accumulating the depths of

collisions 1, 2, and 3. We found excellent agreement between pu(s)and the location of

the ﬁrst collision. Likewise, for the statistics of distances between collisions 1 and 2,

and also between 2 and 3, we found excellent agreement with pc(s).

Eq.(10) predicts that the mean free path between collisions in a Markovian binary mix-

ture is invariant to the mixing parameters and is simply the classical homogeneous

mean free path produced by the atomic mix assumption.

Symbol Description Relations

medium-correlated (stochastic) free path origins

sdistance since last medium collision or correlated birth

Σtc(s)correlated macroscopic cross section Σtc(s) = pc(s)

Xc(s)

pc(s)correlated free-path distribution pc(s) = Σt(s)e−Rs

0Σt(s0)ds0=−∂

∂sXc(s) = hsci∂2

∂s2Xu(s)

Xc(s)correlated-origin transmittance Xc(s) = 1−Rs

0pc(s0)ds0

hsicmean correlated free-path hsic=R∞

0pc(s)s ds

hs2

cimean squared correlated free-path hs2

ci=R∞

0pc(s)s2ds

medium-uncorrelated (deterministic) free path origins

sdistance since last surface/boundary or uncorrelated birth

Σtu(s)uncorrelated macroscopic cross section Σtu(s) = pu(s)

Xu(s)

pu(s)uncorrelated (equilibrium) free-path distribution pu(s) = Σtu(s)e−Rs

0Σtu(s0)ds0=−∂

∂sXu(s) = Xc(s)

hsic

Xu(s)uncorrelated-origin transmittance Xu(s) = 1−Rs

0pu(s0)ds0

hsuimean uncorrelated free-path hsui=R∞

0pu(s)s ds

Table 1: Summary of our notation and relationships between quantities in RGRT.

Computing the extinction of pc(s)as normal [Larsen and Vasques 2011], we next ﬁnd

that the attenuation law between collisions is

Xc(s) = 1

hΣir+−˜

Σ

r+−r−

r+e−r+s+˜

Σ−r−

r+−r−

r−e−r−s.(11)

The non-classical macroscopic cross-section for ﬂights leaving a collision is thus the

ratio [Larsen and Vasques 2011]

Σtc (s) = pc(s)

Xc(s)=(r+−˜

Σ)r2

+e−r+s+ ( ˜

Σ−r−)r2

−e−r−s

(r+−˜

Σ)r+e−r+s+ ( ˜

Σ−r−)r−e−r−s(12)

From Σtc (s)we ﬁnd that the collision probability per incremental path length at the

beginning of a path that follows a collision is higher than in the case of no correlation,

Σtc (0) = hΣi+β

hΣi≥ hΣi.(13)

This is intuitively satisfying in the case that one phase is much less dense than the other.

It is more likely to end a collision in the denser phase and so it is more likely to leave

a collision from a location in the denser phase, and therefore, the scattering centers are

clustered around your current location, producing a short-range collision probability

that exceeds the homogeneous prediction. The atomic mix collision probability density

is only encountered at one distance from the collision,

Σtc

log r+

r−

p4β+ (hΣi − ˜

Σ)2

=hΣi(14)

For a long path that avoids collision for large sthe cross-section approaches

lim

s→∞

Σtc (s) = r−.(15)

The uncorrelated-origin free-path distribution, used to enter the medium, is given sim-

ply by

pu(s) = −∂

∂sXu(s) = hΣiXc(s).(16)

The related cross section for uncorrelated origins is the ratio

Σtu (s) = pu(s)

Xu(s)=(r+−˜

Σ)r+e−r+s+ ( ˜

Σ−r−)r−e−r−s

(r+−˜

Σ)e−r+s+ ( ˜

Σ−r−)e−r−s.(17)

The collision probability density beginning an uncorrelated walk uses

Σtu (0) = hΣi.(18)

This provides additional support for our conjecture [d’Eon 2019] that the macroscopic

cross-section for beginning an uncorrelated walk with all possible realizations in equi-

librium has the homogeneous atomic-mix value and is invariant to the spatial correla-

tion imposed on the scattering centers.

Concurrently, Frankel [2019] has derived a Σt(s)cross-section that corresponds to our

Σtu (s), proposing to use this for intercollision statistics, which will lead to nonrecipro-

cal transport and, as we show in Figure 1, does not accurately estimate the free-path

lengths between collisions.

2.1 Diffusion Approximations

We can compute the low-order moments of pc(s)to determine the various moment-

preserving diffusion approximations in the binary mixture in various dimensional

spaces (assuming isotropic random media as opposed to layered anisotropic random

media of alternating slabs of the two phases). We easily ﬁnd

hs2

ci=Z∞

0pc(s)s2ds =1

hΣi

2˜

Σ

˜

Σ−β(19)

hs3

ci=Z∞

0pc(s)s3ds =1

hΣi

6β+˜

Σ2

(β−˜

Σ)2(20)

hs4

ci=Z∞

0pc(s)s4ds =1

hΣi

24 β(2˜

Σ+1) + ˜

Σ3

(˜

Σ−β)3(21)

from which the Green’s function approximations follow directly [d’Eon 2019]. We

brieﬂy note that the diffusion coefﬁcient for collision rate density about an isotropic

point source that emits from the scattering centers in the medium is

DCc=hs2

ci

2d=1

hΣi

˜

Σ

d(˜

Σ−β)(22)

where dis the dimension of the space in which scattering occurs. This reduces to the

classical P

1result of D=1/(dhΣi)when the cross sections in both phase match and

β=0. The effective attenuation coefﬁcient for scalar collision rate density about a

correlated-emission point source is

Σeff =sd(1−c)hΣi(hΣi˜

Σ−β)

˜

Σ(23)

and for uncorrelated emission or scalar ﬂux about a correlated emitter is

Σeff =s(c−1)dhΣi(β− hΣi˜

Σ)2

β(c−1)hΣi+βc˜

Σ− hΣi˜

Σ2.(24)

2.2 Half-Space Single-Scattering Reﬂection Law

We now consider a plane-parallel problem in a 3D half-space with isotropic scatter-

ing and index-matched smooth boundary. The BRDF f1for single scattering is given

by [d’Eon 2018]

f1(µi,µo) = 1

µiµo

c

4π

1

hsicZ∞

0Xcz

µiXcz

µodz =

c

4π(µi+µo) 1−β(˜

Σ+1)µiµo

(µ2

i+µ2

o)( ˜

Σ−β) + µiµo2β+˜

Σ2+1!(25)

where µiis the cosine of the inclination to the normal vector of the half space for the

incident illumination, and µois the outgoing cosine. We see the classical result scaled

by a factor that decreases the reﬂectance, and when the cross sections of both phase

match and β→0, the classical result is recovered, as desired. This results appears to

be new. Note, again, that this derivation assumes an isotropic multidimensional random

mixture [Larmier et al. 2016; Larmier et al. 2017a], not a 3D volume with layered slabs

separated by exponential chord lengths in the direction of the normal, for which results

such as these are known [Pomraning 1988].

3 Binomial Binary Mixtures

Three distinct models for transport in one-dimensional spatially random media with

densities described by the binomial binary random process have been previous by

Williams [1997] and Akcasu and Williams [2004]. We set out to derive the form of

RGRT that corresponds to each of the three models but, much to our surprise, found

that the three attenuation laws are, in fact, exponential laws in disguise. We brieﬂy

present the homogenization relations for the three models, which are exact.

3.1 Model 1

This model considers stacks of slabs, each of uniform thickness d>0, where the total

cross sections are chosen independently and randomly from Σ0−σand Σ0+σ, with

equal probability (and σ<Σ0). The stationary interpretation of the attenuation law

Xu(s)is [Williams 1997]

Xu(s) = e−sΣ0cosh(s/d)(dσ).(26)

Equation 26 can be expressed as an exponential of s,

Xu(s) = e−¯

Σs(27)

with effective attenuation coefﬁcient

¯

Σ=Σ0+log(sech(dσ))

d.(28)

3.2 Model 2

This model considers two different materials in alternating slabs. The cross sections are

Σ1=Σ01 +β σ1and Σ2=Σ02 +ασ2, where αand βare random variables which take

on values ±1. The slab thicknesses are also allowed to differ, d1and d2, respectively.

The stationary interpretation of the attenuation law Xu(s)is [Williams 1997]

Xu(s) = 2−sed1σ1+d2σ2+1se1

2s(−d1(Σ01+σ1)−d2(Σ02 +σ2)).(29)

Equation 29 can be expressed as an exponential of s,

Xu(s) = e−¯

Σs(30)

with effective attenuation coefﬁcient

¯

Σ=1

2−2 log ed1σ1+d2σ2+1+d1(Σ01 +σ1) + d2(Σ02 +σ2) + log(4).(31)

3.3 Model 3

Model 3 includes variable thickness of the slabs, δi=¯

δ+αi∆being the thickness of

each slab, with αirandom variables taking on ±1, and the same cross-section in each

slab Σ0. The proposed attenuation law is [Williams 1997]

Xu(s) = e−sΣ0cosh s

d(σ(∆+¯

δ)) (32)

Equation 32 can be expressed as an exponential of s,

Xu(s) = e−¯

Σs(33)

with effective attenuation coefﬁcient

¯

Σ=Σ0−log(cosh(σ(∆+¯

δ)))

d.(34)

4 Conclusion

We have taken the Levermore-Pomraning attenuation law for Markovian stochastic

binary mixtures and derived the related form of RGRT that encompasses this law. The

derivation has exhibited the free-path distribution between collisions, a new result that

appears to be exact for forward scattering in a rod. The complete RGRT formulation

deterministically uniﬁes the general theories of GRT [Rybicki 1965; Peltoniemi 1993;

Davis 2006; Moon et al. 2007; Taine et al. 2010; Davis and Xu 2014; Davis et al. 2018]

and nonclassical Boltzmann transport [Larsen and Vasques 2011] with the Levermore-

Pomraning theory [Pomraning 1998] and the related nonexponential random ﬂight

Monte Carlo acceleration scheme of Audic and Frisch [1993], while avoiding the need

for any accumulation of free-path statistics in a prepass.

In our derivation we took the Levermore-Pomraning law as a black-box input with

abstract parameters. No knowledge of the derivation of the law or the meaning of the

parameters took any role in the subsequent analysis, yet known properties about the

microstructure naturally fell out of the process, such as the atomic mix macroscopic

cross section, which appears at the beginning of a free-path with a deterministic origin,

Σtu (0) = hΣi,(35)

providing additional motivation for a conjecture in the previous paper in this se-

ries [d’Eon 2019]. This exercise illustrates how the transport distributions in GRT,

which can be arrived at in ways that are quite disconnected from knowledge of the

microstructure (or may correspond to laws for which no microstructure could ever

exhibit), contain within them additional information about the microstructure.

One possible interpretation of our derivations is that we have transformed the binary

mixture of the two phases, each with particles in them, into a two phase medium

with one phase void and the other corresponding to the scatterers alone. The double

heterogeneity is collapsed, and the transmission law relates to the lineal path func-

tion [Torquato and Lu 1993] in the void phase, from which we immediately produce

the chord-length distribution, which becomes the inter-collision free-path distribution

in the limit that the particles shrink to zero size and inﬁnite density to produce classical

exponential attenuation in each phase. The power of this relationship should also prove

useful in more complex mixtures and layers of heterogeneity where, by reciprocity, we

expect this relationship to hold.

Important future work includes expanding upon the benchmarks of Audic and

Frisch [1993] to compare the accuracy of RGRT for binary mixtures, using recent

benchmark solutions for multidimensional mixtures [Larmier et al. 2017b; Larmier

et al. 2018], and also to measure efﬁciency relative to the CLS algorithm.

While GRT holds promise for binary mixtures, the current presentation is limited to

nonstochastic single-scattering albedo and phase function, assumptions long known to

greatly simplify the solutions. It is not possible to support different absorption levels

in the two phases. Nor can stochastic reaction rates be estimated (such as the collision

rate in phase 2, for .e.g, or, by reciprocity, a volume source in the medium that emits

from the scatterers in phase 2). These restrictions were not acknowledged by Audic and

Frisch. Future work is required to generalize reciprocal GRT to path-length dependent

single-scattering albedo c(s)and derive this function for Markovian binary mixtures.

It is also worth mentioning that, given pc(s)for Markovian binary mixtures, we can

relate transport in such media to nonexponential random ﬂights to apply the Cauchy-

like formulas for mean total track-length and collision rate inside of ﬁnite volumes

under uniform illumination, that have been generalized to provide results at arbitrary

positions in phase space [De Mulatier et al. 2014].

In the next paper, where plane-parallel transport in RGRT is explored under a variety of

free-path distributions, we will expand upon the single-scattering BRDF in this paper

to produce complete solutions for half space transport with isotropic scattering.

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