A Reciprocal Formulation of Nonexponential
Radiative Transfer. 3: Binary Mixtures
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
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  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
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
Xu(s) = r+−˜
where the decay constants are
Σ±qhΣi − ˜
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
∂s2Xu(s) = r+−˜
We ﬁnd that the inverse of the normalization constant is
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
with the mean correlated mean free path
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)
pc(s)correlated free-path distribution pc(s) = Σt(s)e−Rs
∂sXc(s) = hsci∂2
Xc(s)correlated-origin transmittance Xc(s) = 1−Rs
hsicmean correlated free-path hsic=R∞
cimean squared correlated free-path hs2
medium-uncorrelated (deterministic) free path origins
sdistance since last surface/boundary or uncorrelated birth
Σtu(s)uncorrelated macroscopic cross section Σtu(s) = pu(s)
pu(s)uncorrelated (equilibrium) free-path distribution pu(s) = Σtu(s)e−Rs
∂sXu(s) = Xc(s)
Xu(s)uncorrelated-origin transmittance Xu(s) = 1−Rs
hsuimean uncorrelated free-path hsui=R∞
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
The non-classical macroscopic cross-section for ﬂights leaving a collision is thus the
ratio [Larsen and Vasques 2011]
Σtc (s) = pc(s)
+e−r+s+ ( ˜
Σ)r+e−r+s+ ( ˜
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
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,
p4β+ (hΣi − ˜
For a long path that avoids collision for large sthe cross-section approaches
Σtc (s) = r−.(15)
The uncorrelated-origin free-path distribution, used to enter the medium, is given sim-
pu(s) = −∂
∂sXu(s) = hΣiXc(s).(16)
The related cross section for uncorrelated origins is the ratio
Σtu (s) = pu(s)
Σ)r+e−r+s+ ( ˜
Σ)e−r+s+ ( ˜
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  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
Σ+1) + ˜
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
where dis the dimension of the space in which scattering occurs. This reduces to the
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
and for uncorrelated emission or scalar ﬂux about a correlated emitter is
Σeff =s(c−1)dhΣi(β− hΣi˜
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
Σ−β) + µiµo2β+˜
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  and Akcasu and Williams . 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−¯
with effective attenuation coefﬁcient
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
Equation 29 can be expressed as an exponential of s,
Xu(s) = e−¯
with effective attenuation coefﬁcient
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
Equation 32 can be expressed as an exponential of s,
Xu(s) = e−¯
with effective attenuation coefﬁcient
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 , 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  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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