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A Reciprocal Formulation of Nonexponential Radiative Transfer. 3: Binary Mixtures

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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 classical radiative transfer.
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A Reciprocal Formulation of Nonexponential
Radiative Transfer. 3: Binary Mixtures
Eugene d’Eon
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
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 − ˜
with parameters
β= (ΣAΣB)2pApB(5)
The medium is parameterized by the macroscopic cross-sections for the two phases,
Σi0, 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
s2Xu(s)ds =r++r˜
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
with the mean correlated mean free path
0pc(s)ds =1
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)eRs
sXc(s) = hsci2
Xc(s)correlated-origin transmittance Xc(s) = 1Rs
hsicmean correlated free-path hsic=R
0pc(s)s ds
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)eRs
sXu(s) = Xc(s)
Xu(s)uncorrelated-origin transmittance Xu(s) = 1Rs
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
The non-classical macroscopic cross-section for flights leaving a collision is thus the
ratio [Larsen and Vasques 2011]
Σtc (s) = pc(s)
+er+s+ ( ˜
Σ)r+er+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
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,
log r+
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-
ply by
pu(s) =
sXu(s) = hΣiXc(s).(16)
The related cross section for uncorrelated origins is the ratio
Σtu (s) = pu(s)
Σ)r+er+s+ ( ˜
Σ)er+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 [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
0pc(s)s2ds =1
0pc(s)s3ds =1
0pc(s)s4ds =1
24 β(2˜
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
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(1c)hΣi(hΣi˜
and for uncorrelated emission or scalar flux about a correlated emitter is
Σeff =s(c1)dhΣi(β− hΣi˜
Σ− 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
µodz =
4π(µi+µo) 1β(˜
o)( ˜
Σβ) + 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 β0, 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 Σ0σand Σ0+σ, with
equal probability (and σ<Σ0). The stationary interpretation of the attenuation law
Xu(s)is [Williams 1997]
Xu(s) = esΣ0cosh(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) = 2sed1σ1+d2σ2+1se1
2s(d1(Σ01+σ1)d2(Σ02 +σ2)).(29)
Equation 29 can be expressed as an exponential of s,
Xu(s) = e¯
with effective attenuation coefficient
22 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+αibeing 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) = esΣ0cosh s
δ)) (32)
Equation 32 can be expressed as an exponential of s,
Xu(s) = e¯
with effective attenuation coefficient
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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We demonstrate the computational advantage gained by introducing non-exponential transmission laws into radiative transfer theory for two specific situations. One is the problem of spatial integration over a large domain where the scattering particles cluster randomly in a medium uniformly filled with an absorbing gas, and only a probabilistic description of the variability is available. The increasingly important application here is passive atmospheric profiling using oxygen absorption in the visible/near-IR spectrum. The other scenario is spectral integration over a region where the absorption cross-section of a spatially uniform gas varies rapidly and widely and, moreover, there are scattering particles embedded in the gas that are distributed uniformly, or not. This comes up in many applications, O2 A-band profiling being just one instance. We bring a common framework to solve these problems both efficiently and accurately that is grounded in the recently developed theory of Generalized Radiative Transfer (GRT). In GRT, the classic exponential law of transmission is replaced by one with a slower power-law decay that accounts for the unresolved spectral or spatial variability. Analytical results are derived in the single-scattering limit that applies to optically thin aerosol layers. In spectral integration, a modest gain in accuracy is obtained. As for spatial integration of near-monochromatic radiance, we find that, although both continuum and in-band radiances are affected by moderate levels of sub-pixel variability, only extreme variability will affect in-band/continuum ratios.
The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore-Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning, and extended by Brantley. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.
Particle transport in random media obeying a given mixing statistics is key in several applications in nuclear reactor physics and more generally in diffusion phenomena emerging in optics and life sciences. Exact solutions for the ensemble-averaged physical observables are hardly available, and several approximate models have been thus developed, providing a compromise between the accurate treatment of the disorder-induced spatial correlations and the computational time. In order to validate these models, it is mandatory to resort to reference solutions in benchmark configurations, typically obtained by explicitly generating by Monte Carlo methods several realizations of random media, simulating particle transport in each realization, and finally taking the ensemble averages for the quantities of interest. In this context, intense research efforts have been devoted to Poisson (Markov) mixing statistics, where benchmark solutions have been derived for transport in one-dimensional geometries. In a recent work, we have generalized these solutions to two and three-dimensional configurations, and shown how dimension affects the simulation results. In this paper we will examine the impact of mixing statistics: to this aim, we will compare the reflection and transmission probabilities, as well as the particle flux, for three-dimensional random media obtained by resorting to Poisson, Voronoi and Box stochastic tessellations. For each tessellation, we will furthermore discuss the effects of varying the fragmentation of the stochastic geometry, the material compositions, and the cross sections of the transported particles.
Linear particle transport in stochastic media is key to such relevant applications as neutron diffusion in randomly mixed immiscible materials, light propagation through engineered optical materials, and inertial confinement fusion, only to name a few. We extend the pioneering work by Adams, Larsen and Pomraning \cite{benchmark_adams} (recently revisited by Brantley \cite{brantley_benchmark}) by considering a series of benchmark configurations for mono-energetic and isotropic transport through Markov binary mixtures in dimension $d$. The stochastic media are generated by resorting to Poisson random tessellations in $1d$ slab, $2d$ extruded, and full $3d$ geometry. For each realization, particle transport is performed by resorting to the Monte Carlo simulation. The distributions of the transmission and reflection coefficients on the free surfaces of the geometry are subsequently estimated, and the average values over the ensemble of realizations are computed. Reference solutions for the benchmark have never been provided before for two- and three-dimensional Poisson tessellations, and the results presented in this paper might thus be useful in order to validate fast but approximated models for particle transport in Markov stochastic media, such as the celebrated Chord Length Sampling algorithm.
Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering and life sciences. In this work, we investigate the statistical properties of $d$-dimensional isotropic Poisson geometries by resorting to Monte Carlo simulation, with special emphasis on the case $d=3$. We first analyse the behaviour of the key features of these stochastic geometries as a function of the dimension $d$ and the linear size $L$ of the domain. Then, we consider the case of Poisson binary mixtures, where the polyhedra are assigned two `labels' with complementary probabilities. For this latter class of random geometries, we numerically characterize the percolation threshold, the strength of the percolating cluster and the average cluster size.