A critique of the stochastic transition matrix formalism

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Annals of Nuclear Energy 35 (2008) 458–471

annals of

NUCLEAR ENERGY

A critique of the stochastic transition matrix formalism

Richard Sanchez

Commissariat a l’Energie Atomique, DEN/DANS/DM2S, Service d’Etudes de Reacteurs et de Mathematiques Appliquees,

CEA de Saclay, 91191 Gif-sur-Yvette Cedex, France

Received 18 May 2007; accepted 5 July 2007Available online 17 September 2007

Abstract

The Stochastic Transition Matrix (STM) formalism has been introduced by Akcasu to compute the ensemble average flux of bima-terial stochastic density statistics with deterministic sources. The method uses an integral flux representation based on a stochastic prop-agator and as boundary conditions the entering and exiting fluxes at the left of a finite slab. In this paper we generalize the STMformalism to the energy-dependent continuous transport equation, analyze and discuss basic issues of the formalism and compare itto the classical Levermore–Pomranig model and to reference calculations for simplified rod problems. We also extend the formalismin two different ways to consistently compute the ensemble and the material averages flux.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Transport in stochastic media can be described by a setof realizations X = {x,p(x)}, where p(x) P 0 is the densityof probability for realization x and

RX pðxÞdx ¼ 1. Each

realization represents a well-defined transport problemthat, for a finite slab system, reads

ðlox þAxÞwx ¼ Sx; L < x < R;

wxðx; vÞ ¼ winL ðvÞ; x ¼ L; l > 0;

wxðx; vÞ ¼ winR ðvÞ; x ¼ R; l < 0;

8><>: ð1Þ

where v = (l,E), Ax = Rx � Hx, Rx(x,v) is the total crosssection and Hx is the scattering operator

ðHxwxÞðx; vÞ ¼Z

dv0Rs;xðx; v0 ! vÞwxðx; v0Þ:

The aim of transport in stochastic media is to evaluate thestatistical behavior of the flux wx(x,v), as characterized bythe ensemble average flux, hwðx; vÞi ¼

RX wxðx; vÞpðxÞdx,

and the flux variance, for example. A practical problemof interest is that of stochastic material mixtures where

0306-4549/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2007.07.018

E-mail address: [email protected]

one is brought to analyze realizations that describe randomdistributions of materials in a domain. More precisely, oneintroduces a finite set of materials XM = {k, k = 1, . . . M},where each material has prescribed deterministic propertiesXM’k! {Rk,Hk,Sk} (Sanchez, 1989), and view each reali-zation as a map x: [L,R]! XM that gives the materialk = x(x) at location x for the realization. An immediateadvantage of this material statistics description is that therealizations are locally finite. A second advantage is thepossibility of introducing and evaluating the material

ensemble fluxes

hwðx; vÞik ¼Z

XkðxÞwxðx; vÞpðxÞdx=pkðxÞ;

where Xk(x) = {x 2 X, x(x) = k} denotes the set of realiza-tions that have material k at x and pkðxÞ ¼

RXkðxÞ pðxÞdx is

the probability to find material k at x. The material ensem-ble flux Æw(x,v)æk is the average flux when material k is at x.With the help of the ensemble material fluxes one can cal-culate the associated reaction rates RkÆwæk which are notaccessible from the sole knowledge of the ensemble averageflux Æwæ. A third advantage of material statistics is that thestatistical process can be described in terms of the distribu-tions of material chord lengths along each trajectory. For

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R. Sanchez / Annals of Nuclear Energy 35 (2008) 458–471 459

the slab problem the statistics are given by the process x(x)for x 2 [L,R]. Similar simplifications apply to the typicalone-dimensional geometries.

A particular case that has been intensively studied in theliterature is that of bimaterial Markovian statistics, forwhich the process x(x) is Markov or, equivalently, the den-sity of probability for a chord length for material k (k = 1,2) is exponential

fkðlÞ ¼1

kkðlÞe�R l

0dz=kkðzÞ;

where dx/kk(x) is the probability for a realization that hasmaterial k at x to jump to material k 0 = 3 � k in (x,x + dx).

Recently, Akcasu (2006) introduced a set of equations,the Modified Levermore–Pomraning (MLP) equations,that give the exact ensemble and material average fluxesin a half-space for bimaterial Markov density statistics.This statistics correspond to a material with stochastic den-sity for which the spatial dependence of all the cross sec-tions is the same for each realization:

Rxðx; vÞ ¼ NxðxÞrðvÞ;Rs;xðx; v0 ! vÞ ¼ NxðxÞrsðv0 ! vÞ;

ð2Þ

where r(v) > 0 and rs(v0 ! v) P 0 are deterministic. This

problem was earlier defined by Pomraning (1991), who alsointroduced a density source,

Sxðx; vÞ ¼ NxðxÞsðvÞ; ð3Þwhere s(v) P 0 is deterministic, and gave a complete impli-cit solution for the exact ensemble and material averagesfluxes in terms of quadratures. The half-space problemswith statistics as in (2) and (3) will be called the HSSDproblem.

The conditions in (2) give

Ax ¼ NxðxÞA; ð4Þ

where A ¼ r� h is now a deterministic operator and h is ascattering operator like H but with kernel rs(v

0 ! v). More-over, for a bimaterial mixture this implies that Nx(x) =Nk(x) for x(x) = k.

We shall call the statistics defined by conditions (2) den-

sity statistics and reserve the term density statistics and

sources when condition (3) is also satisfied.In a companion paper (Sanchez, submitted for publica-

tion) we have analyzed Akcasu’s MLP equations and, inparticular, studied a MLP heuristic model for the calcula-tion of the ensemble and material average fluxes in finiteslabs. The model was based on the use of the MLP equa-tions with the natural boundary conditions, that is, the leftand right entering fluxes. However, Akcasu’s originalapproach for the calculation of the ensemble average fluxin a finite slab was done by introducing a method basedin an integral equation for the flux. In this Stochastic Tran-sition Matrix (STM) formalism (Akcasu, 2006, submittedfor publication) a propagator operator is introduced towrite an integral equation for the flux in terms of the angu-lar fluxes wL, entering and exiting at the left boundary, and

of the sources. In the STM approach the statistical descrip-tion is completely contained in the ensemble average prop-agator and the equation for the ensemble average flux isclosed by neglecting the correlations between the leftboundary fluxes wL and the propagator. For the HSSDand related problems (Pomraning, 1991) with no sourcesthe STM formalism is exact and equivalent to the MLPequations. For finite slabs for which the left boundaryfluxes are not deterministic, the STM formalism gives amodel solution for the MLP equations.

The aim of this paper is to clarify some of the issues con-cerning the STM model and to generalize the formalism tothe calculation of the material ensemble fluxes, which toour opinion are the quantities of interest, while providingcomparative numerical results for the simplified rod (two-streams) model. As in our companion paper, we haveadopted the broader setup defined by Pomraning andworked with the continuous form of the transport equationwith energy dependence and density sources.

The physical interpretation of the propagator defiesintuition and it has an impact on the assumptions on whichthe STM model is based. In the next section we introduceand discuss the propagator operator defined by Akcasuand give a physical interpretation based on the Green func-tions for the slab problem. In the following section we ana-lyze the STM approach using the continuous form of thetransport equation with energy dependence and sourcesand discuss the approximations that have to be made forthe boundary fluxes and for the sources. We outline Akca-su’s technique to compute the ensemble average propaga-tor for the case of continuous transport and propose twodifferent techniques to evaluate the material and ensembleaverage fluxes Æwæk and Æwæ. The section ends with a deriva-tion of the equivalent differential forms for the two STMtechniques introduced for the calculation of the materialfluxes. Numerical examples for a simplified rod problemare given and discussed in Section 4. In the conclusionswe summarize our views of the STM formalism. TheSTM approach is based on the existence of the inverse ofa Green operator, an issue related to the existence of thesolution of a boundary inverse problem. A short technicaldiscussion concerning this last problem is given in Appen-dix A. In Appendix B we give the formulation of the STMcomputer program used for our calculations in Section 4.This program can be obtained by contacting the author.

2. The propagator

In our companion paper (Sanchez, submitted for publi-cation) we have discussed the straightforward applicationof the MLP equations to finite slab problems based onthe use of the natural boundary conditions. As the numer-ical examples illustrated, this MLP model yields unphysicalresults in the presence of sources, a fact that is related tothe lack of symmetry and positiveness of the model. Amore promising approach has been proposed by Akcasufor the direct calculation of the ensemble average flux

460 R. Sanchez / Annals of Nuclear Energy 35 (2008) 458–471

Æw(x,v)æ. This method, called the Stochastic TransitionMatrix (STM) formalism, is based on the use of a propaga-tor operator. The purpose of this section is an analysis ofthe propagator approach.

Following Akcasu (submitted for publication) we writethe solution of the transport equation for a realization x,Eq. (1), using as boundary conditions the entering and exit-

ing angular fluxes on the left of the slab:

wxðx; vÞ ¼ ðU L;xwL;xÞðx; vÞ þZ x

LðUy;xl�1SxÞðx; vÞdy; ð5Þ

where the propagator Uy,x(x) is an integral operator,

ðUy;xf Þðx; vÞ ¼Z

uxðy; v0 ! x; vÞf ðy; v0Þdv0;

that satisfies the operator equation

ðlox þ AxÞUy;x ¼ 0; y < x < R;

Uy;xðxÞ ¼ I; x ¼ y;

�ð6Þ

where I is the identity.Having established out notation, in the following we

shall omit lower index x except when introducing newquantities. To complete the solution in (5) one has toobtain the left exiting flux wout

L;x. This is done by evaluationEq. (5) at x = R for negative l directions:

winR ðvÞ ¼ ðUþL win

L ÞðR; vÞ þ ðU�L woutL ÞðR; vÞ

þZ R

LðUyl

�1SÞðR; vÞdy; l < 0:

In this equation UþL;x and U�L;x indicates the action of oper-ator UL,x on functions defined for l 2 (0, 1) and forl 2 (�1,0), respectively. Under the conditions that opera-tor U�L ðRÞ admits an inverse T x ¼ U�L;xðRÞ

�1, one can ob-tain from this equation wout

L in terms of winL , win

R and S.Finally, by replacing the expression for wout

L in Eq. (5) weobtain

wðx; vÞ ¼ ð½UþL � U�L TUþL ðRÞ�winL Þðx; vÞ þ ðU�L Twin

R Þðx; vÞ

þZ R

Lð½hðL;xÞðyÞUy � U�L TU yðRÞ�l�1SÞðx; vÞdy;

ð7Þ

where h(L,x) is the characteristic function of interval (L,x).

2.1. Physical interpretation of the propagator

To interpret the physical meaning of the propagator wewrite it in terms of the volume and surface Green’s func-tions for transport equation (1). In terms of these Greenfunctions, the solution of the transport equation reads:

wxðx; vÞ ¼ ðGþL;xwinL;xÞðx; vÞ þ ðG�R;xwin

R;xÞðx; vÞ

þZ R

LðGy;xSxÞðx; vÞdy; ð8Þ

where we have introduced the integral operators

ðGy;xf Þðx; vÞ ¼Z

gxðy; v0 ! x; vÞf ðy; v0Þdv0;

ðGþL;xf Þðx; vÞ ¼Z

l0>0

jl0jgxðL; v0 ! x; vÞf ðv0Þdv0;

ðG�R;xf Þðx; vÞ ¼Z

l0<0

jl0jgxðR; v0 ! x; vÞf ðv0Þdv0

and gx(y,v 0 ! x,v) is the Green’s function solution of

ðlox þ AxÞgx ¼ dðx� yÞdðv� v0Þ; L 6 x 6 R;

gxðy; v0 ! L; vÞ ¼ 0; l > 0;

gxðx; v0 ! R; vÞ ¼ 0; l < 0

8><>:for L 6 y 6 R (Sanchez, 1998).

Consider now Eq. (5). In this equation the exiting fluxwout

L can be written as the sum of contributions fromwin

L ;winR and S, wout

L ðvÞ ¼ P�½ðGþL winL ÞðL; vÞ þ ðG�R win

R ÞðL; vÞ þ

R RL ðGySÞðL; vÞdy� for l < 0, where P� is the projec-

tor (P�f)(v) = f(v) for l < 0 and (P�f)(v) = 0 for l > 0.Replacing this expression for wout

L in STM Eq. (5) yieldsthe following relations between the propagator and theGreen’s operators:

UþL ðxÞ þ U�L ðxÞP�GþL ðLÞ ¼ GþL ðxÞ;U�L ðxÞP�G�R ðLÞ ¼ G�R ðxÞ;hðL;xÞðyÞU yðxÞl�1 þ U�L ðxÞP�GyðLÞ ¼ GyðxÞ:

ð9Þ

Under the condition that P�G�R ðLÞ has an inverse, from thesecond relation we get

U�L ðxÞ ¼ G�R ðxÞ½P�G�R ðLÞ��1: ð10Þ

Hence, the action of U�L ðxÞ on woutL is to compute the equiv-

alent flux winR ðw

outL Þ ¼ ½P�G�R ðLÞ�

�1woutL entering the right

boundary, that results in the flux woutL exiting the left

boundary, and then compute the flux at x produced by thisentering flux. In other words, U�L ðxÞw

outL is the flux at x due

to the equivalent flux winR ðw

outL Þ entering the right boundary.

It is important to note that in the actual problem woutL is the

sum of the contributions from the incoming fluxes and thesources, wout

L ¼ woutL ðw

inL Þ þ wout

L ðwinR Þ þ wout

L ðSÞ, so theequivalent flux win

R ðwoutL Þ accounts for trajectories that do

not exit via the right boundary and is ‘equivalent’ only inthe sense that it produces the same left exiting flux.

The first equation in (9) gives

UþL ðxÞ ¼ GþL ðxÞ � U�L ðxÞP�GþL ðLÞ:Note that wout

L ðwinL Þ ¼ P�GþL ðLÞw

inL is the flux exiting the left

boundary due to the entering flux winL . That is, wout

L ðwinL Þ re-

sults from the particles that neither are absorbed nor leavevia the right boundary. Therefore, UþL ðxÞw

inL is the flux pro-

duced at x by the incoming flux winL minus the flux pro-

duced by the right entering flux equivalent to woutL ðw

inL Þ,

where the meaning of ‘equivalent’ is the one discussedearlier.

We consider now the third equation in (9) for the sourcecontribution. For y > x one gets U�L ðxÞ ¼ GyðxÞP�GyðLÞ�1.Note that Sðwout

L Þ ¼ ½P�GyðLÞ��1woutL is the source at y that


produces the left exiting flux woutL . Therefore, this second

definition of U�L ðxÞ coincides, at it should be, with the ear-lier one in (10).

Next, for y < x one has

U yðxÞl�1 ¼ GyðxÞ � U�L ðxÞP�GyðLÞ:This relation can be read in a similar fashion as we did forUþL ðxÞ: Uy(x)l�1S is the flux at x produced by the particlesemitted by the source at y minus the flux produced by theright entering flux equivalent to wout

L ðSÞ.Finally, inverse relations can be obtained by identifying

the different operators in (7) with GþL ðxÞ, G�R ðxÞ and Gy(x) in(8), respectively:

GþL ðxÞ ¼ UþL ðxÞ � U�L ðxÞTUþL ðRÞ;G�R ðxÞ ¼ U�L ðxÞT ;GyðxÞ ¼ ½hðL;xÞðyÞU y � U�L ðxÞTU yðRÞ�l�1:

The operator T ¼ U�L ðRÞ�1 can be calculated, for example,

by evaluating Eq. (10) at x = R: T ¼ P�G�R ðLÞG�R ðRÞ�1 ¼

P�G�R ðLÞ. This operator acts on a flux entering the rightside of the slab and yields the flux leaving the left side.

We see that the physical interpretation of the propaga-tor is not simple. The center piece of the interpretation isEq. (10), that tells us that ðU�L wout

L Þðx; vÞ is the flux at(x,v) due to an equivalent flux win

R ðwoutL Þ entering the right

surface. The equivalent flux is the one that produces thesame left exiting flux wout

L . The interpretation depends onwhere the exiting particles in win

L came from. For a slabproblem with only a right entering flux win

R ; ðU�L woutL Þðx; vÞ

is exactly the flux at (x,v) produced by the entering fluxwin

R . However, for a problem with only a left entering fluxwin

L ; ðU�L woutL Þðx; vÞ is the flux at (x,v) produced by the

equivalent right entering flux winR ðw

inL Þ, where equivalent

has the meaning discussed above, and, similarly for a prob-lem with an internal source. Fig. 1 illustrates the case forwout

L ðwinR Þ and wout

L ðwinL Þ. In the actual problem, with volume

sources and left and right entering fluxes, all modes arepresent and that is why ðUþL win

L Þðx; vÞ and (Uyl�1S)(x,v)

have to subtract the extra contributions ‘artificially’ added

:

Fig. 1. Action of operator U�L ðxÞ. Left: for a flux winR entering the right

surface and resulting on the flux woutL leaving the left surface, U�L wout

L

computes winR from wout

L and then uses winR to compute the flux in the slab.

Right: for a flux winL entering the left surface and resulting on the flux wout

L

leaving the left surface, U�L woutL computes the ‘equivalent’ win

R from woutL

and then uses winR to compute the flux in the slab. The figure shows one

‘equivalent’ trajectory for each case. For the second case the ‘equivalent’trajectory win

R ! woutL is never equal to the physical trajectory win

L ! woutL .

by U�L when acting on escaping particles (woutL ) produced by

entering particles or internal sources.Which comes out of our physical analysis of the propa-

gator is that the basic issue has to do with writing the trans-port problem for a slab using as boundary conditions theentering and exiting fluxes on the left. This problem, whichis equivalent to the reconstruction of the equivalent fluxwout

R ðvÞ, is discussed next.

2.2. A comment on the existence of the inverse operator

Formulation (5) is based on the assumption that the leftboundary flux defines a unique solution of the transportequation or, equivalently, that operator U�L ðRÞ admits aninverse. We have calculated the inverse of the later opera-tor, but only on the assumption that operator P�G�R ðLÞ isinvertible. Thus, one has to demonstrate the uniqueness ofthe following inverse problem: given the left boundaryfluxes and the sources, find the right entering flux. Theproof of this fact is out of the scope of the present work,but we shall make some comments. The first observation isthat P�G�R ðLÞ is, clearly, invertible for purely absorbingtransport and also for ‘rod’ transport with scattering andabsorption. For the general transport case one has toprove that the equation ½P�G�R ðLÞw

outR �ðvÞ ¼ f ðvÞ has a

unique solution for woutR ðvÞ. In Appendix A we give a par-

tial proof.

3. The STM model

Akcasu (2006, submitted for publication) analyzed theboundary conditions to be used with the MLP model fora finite slab and introduced a stochastic propagator to pro-pose a possible application of the MLP equations to finiteslabs. The STM model is equivalent to the LMP equationsand, like the latter, has the advantage of given the exactsolution with scattering for the HSSD problem. Akcasu’sidea is to obtain an approximate equation for the ensembleaverage flux by ensemble averaging Eq. (5). The result isthe STM model:

hwðx; vÞi � ðhU LihwLiÞðx; vÞ þ hQðx; vÞi; ð11Þ

where Qxðx; vÞ ¼R x

L ðUy;xl�1SxÞðx; vÞdy. In this equationthe correlation between the propagator and the left bound-ary flux has been neglected. We shall assume that the leftentering flux is deterministic, so the correlation exists onlybetween the propagator and the left exiting flux. For a fi-nite slab the precedent equation is closed by using Eq.(11) at x = R to evaluate the left exiting flux in terms ofthe incoming fluxes and the sources, much as we did inthe case of a single realization,

hwinR ðvÞi ¼ ðhUþL ihw

inL iÞðR; vÞ þ ðhU�L ihw

outL iÞðR; vÞ þ hQðR; vÞi

ð12Þ

The result is an equation similar to (7), but now in terms ofensemble average propagators:


hwðx; vÞi ¼ ð½hUþL i � hU�L ihT ihUþL ðRÞi�hwinL iÞðx; vÞ

þ ðhU�L ihT ihwinR iÞðx; vÞ þ hQðx; vÞi

� ðhU�L ihT ihQðR; :ÞiÞðx; vÞdy;

where hT i ¼ hU�L ðRÞi�1.

The final form of Eq. (11) depends on the approxima-tion adopted for the source contribution. The simplestapproximation is to neglect correlations,

hUyl�1Si � hU yil�1hSi; ð13Þ

to obtain

hQðx; vÞi �Z x

LðhU yil�1hSiÞðx; vÞdy; ð14Þ

Note that, in the case of a deterministic source, the sourcecontribution is exact. However, for this type of source thereis no known statistical problem for which the left exitingflux is deterministic, so Eq. (14) remains an approximation.

There is another case for which one can obtain an exactformula for the source contribution in (11), and the readerfamiliar with reference (Sanchez, submitted for publica-tion) should not be surprised to learn that this happensfor density statistics and sources.

To analyze this case we need first to obtain an explicitformula for the ensemble average of the propagator. Forevery realization x the solution of evolution Eq. (6) forthe propagator can be formally written in terms of an infi-nite sum of operators:

Uy;xðxÞ ¼XnP0

ð�ÞnAy;xðxÞ�n;

where operators Ay,x(x)*n are recursively defined as

Ay;xðxÞ�n ¼1

l

Z x

yAxðzÞAy;xðzÞ�ðn�1Þdz; n > 0 ð15Þ

with the initial value Ay;xðxÞ�0 ¼ I.Recursion (15) shows that oyAy,x(x)*n = � Ay,x(x)*(n�1)

l�1Ax(y) for n > 0. Hence, oyUy,x(x) = Uy,x(x)l�1Ax(y)and one can write the source term in (5) in the equivalentform:

Qxðx;vÞ ¼ ½ð1�U L;xÞA�1x Sx�ðx;vÞ�

Z x

L½U y;xoyðA�1

x SxÞ�ðx;vÞdy:

From this formula we see that the ensemble averaging ofthe source terms will be exact if A�1

x Sx is deterministic. Thisis possible for any source of the type

Sxðx; vÞ ¼ ðAxf Þðx; vÞ;

where f(x,v) is a deterministic function. It is difficult tothink of a solution of this equation without appealing tocondition (2) for density statistics. For density statistics,where Ax ¼ NxA, a solution to the previous equation isa source of the form

Sxðx; vÞ ¼ NxðxÞsðx; vÞ ð16Þfor arbitrary s(x,v).

Then, after ensemble averaging, one gets

hQðx; vÞi ¼ ½ð1� hULiÞA�1s�ðx; vÞ �Z x

LðhUyiA�1oysÞðx; vÞdy:

ð17Þ

The source in (16) is more general than a density source be-cause it allows for the spatial dependence in s(x,v). Hence,we see that one can include more general sources in theSTM formulation with an exact treatment of the correla-tion between the source and the propagator. An approxi-mation appears, however, because the left exiting fluxesare non deterministic.

When s(x,v)! s(v) is independent of x the above equa-tion reduces to

hQðx; vÞi ¼ ½ð1� hU LiÞA�1s�ðx; vÞ:

We recognize that this type of source is a density sourceSx(x,v) = Nx(x)s(v), a source for which the left exiting fluxis deterministic (Sanchez, submitted for publication). Oneconcludes that the STM model for density statistics andsources is exact for a half-space.

An interesting property of density sources is that thecorresponding STM equation (11) accepts a uniform solu-tion, Æw(x,v)æ = w(v), under the condition sðvÞ ¼ ðAwÞðvÞ(Sanchez, submitted for publication), as it is easily checkedfrom Eq. (11) with the source term in (17). This is not sosurprising because, for Sxðx; vÞ ¼ NxðxÞðAwÞðvÞ and deter-ministic left and right incoming fluxes equal to w(v), everyrealization has the same uniform flux wx(v) = w(v). Nosuch uniform solutions exist for a deterministic source.

3.1. Computation of ÆUyæ

The practical use of the STM technique depends on thepossibility of an exact calculation of the ensemble averageof the propagator, ÆUyæ. This ensemble average can beexactly calculated for certain stochastic models and, in par-ticular, for Markov statistics (Akcasu, submitted for publi-cation). In the general case, calculation of ÆUyæ will requireto evaluate the ensemble average of expression (15), involv-ing cross correlation between the two operators. However,a closed solution can be obtained when the A(z) satisfy thecommutation relation

AxðzÞl�1Axðz0Þ ¼ Axðz0Þl�1AxðzÞ; y 6 z; z0 6 x: ð18Þ

In this case one has Ay,x(x)*n = Ay,x(x)n/n!, where Ay;xðxÞ ¼l�1

R xy AxðzÞdz, and the propagator can be written in an

exponential form

Uy;xðxÞ ¼ e�Ay;xðxÞ: ð19Þ

Hence, operators Uy,x(x) have the structure of asemigroup.

It is important to note that commutation relation (18)holds only for density statistics, as can be checked by clo-sely inspecting this relation. Therefore, in the followingwe shall assume density statistics.


According to (4), one has Ay;xðxÞ ¼ N y;xðxÞl�1A, where

Ny;xðxÞ ¼R x

y NxðzÞdz and cA ¼ l�1A. Next, by assuming

that operator cA diagonalizes,1 we can write cA ¼ M bA DbAM�1bA , where M bA is an orthogonal operator and DbA is a

diagonal operator containing the eigenvalues of cA. It fol-lows that the calculation of the ensemble average of (19),

hUyðxÞi ¼ M bAhe�Ny ðxÞDbA iM�1bA ;

reduces to evaluate the ensemble average of the diagonaloperator

W y;xðxÞ ¼ e�Ny;xðxÞDbA :

The treatment that follows parallels that used in Sanchez(submitted for publication) so we shall sketch only themain steps. The treatment is based on the density of prob-ability Pk(s;x), such that Pk(s;x)ds is the probability for arealization x to have s(x,x) 2 (s,s + ds) and x(x) = k. Forbinary Markov statistics these probabilities obey the Liou-ville master equation (Sanchez, submitted for publication):

oxP k ¼ �osðNkP kÞ þ P k0=kk0 � P k=kk; s > 0;

P kðs; xÞ ¼ pkðxÞdðsÞ; x ¼ 0:

�ð20Þ

Next, we observe that operator Wy,x obeys the stochasticequation

ðox þ NxDbAÞW y;x ¼ 0; y < x < R;

W y;xðxÞ ¼ I; x ¼ y:

�Passing to density distance, x! sðx;xÞ ¼

R xy NxðzÞdz, and

defining bW yðsÞ ¼ W y;xðxÞ one obtains a deterministic equa-tion for bW yðsÞ:

ðos þ DbAÞ bW y ¼ 0; s > 0;bW yðsÞ ¼ I; s ¼ 0:

(Finally, we consider the case when x is a binary Markovprocess. Then, the joint process {Wy,x,x} is also Markov.Therefore

hW yðxÞi ¼ W y;1ðxÞ þ W y;2ðxÞ;where the W y;kðxÞ ¼

RXkðxÞW y;xðxÞpðxÞdx for k = 1,2 are

diagonal operators. One can now write W y;kðxÞ ¼R1

0bW y

ðsÞP kðs; xÞds and, proceeding as in Sanchez (submittedfor publication), use dynamic Eq. (20) to obtain a coupledset of evolution equations for the Wy,k(x):

ðox þ N kDbAÞW y;k ¼ W y;k0=kk0 � W y;k=kk; y < x < R;

W y;kðxÞ ¼ pkðyÞI; x ¼ y

�with k = 1,2 and k 0 = 3 � k.

1 To put aside mathematical details, from here on we shall consider afinite-dimensional space, as it is the case for numerical applications. Thereader can then replace ‘operator’ by ‘matrix’ and rely on the familiarmatrix algebra. In the continuum case, operator cA diagonalizes when it isselfadjoint compact. For a finite space this implies that A is symmetrical.The latter condition is, in particular, true for one-group transport inrotationally invariant media.

We note that operator DbA is diagonal and that the initialcondition ensures that the off-diagonal terms of Wy,k van-ish, so we can write W y;kðxÞ ¼

PcP cw

ckðxÞ, where the sum

is over all the eigenvalues of cA (the entries of DbA) andPc is the projector over the eigenvector corresponding toeigenvalue c, that is, Pc is diagonal with all diagonal termsequal to 0 except for that corresponding to the position of cin DbA that is set to1. Moreover, the operators wc

kðxÞ obeythe equations

ðox þ cN kÞwk ¼ wk0=kk0 � wk=kk; y < x < R;

wkðxÞ ¼ pkðyÞ; x ¼ y;

�ð21Þ

where we have omitted upper index c.The final expression for the ensemble average propaga-

tor is

hUyðxÞi ¼ M bAXc

P c½wckðxÞ þ wc

k0 ðxÞ�M�1bA : ð22Þ

For the case of homogeneous statistics and uniform densi-ties Eq. (21) accept as solution a linear combination ofexponentials. Moreover, the propagator is invariant undertranslations and so are the ensemble average operators, forinstance ÆUy(x)æ = ÆU(x � y)æ. Then, the solution of (21) isreadily obtained as w!ðxÞ ¼ expð�BxÞ p!, where w! and p!are the vectors with components wk and pk for k = 1, 2,respectively, and

B ¼b1 �c2

�c1 b2

� �: ð23Þ

with bk = cNk + ck and ck = 1/kk. By diagonalizing thismatrix one has

w!ðxÞ ¼ MBe�zþx

e�z�x

� �M�1

B p!;

where M�1B is the matrix

MB ¼c2 c2

b1 � zþ zþ � b2

� �:

and z± are the two eigenvalues of B, see (Sanchez, submit-ted for publication) for more details. Such an approach hasbeen used to solve the STM equations for the rod problem(Akcasu and Williams, 2004; Akcasu, 2006).

3.2. STM equations for the material fluxes

The STM model originally proposed by Akcasu aims tothe calculation of solely the ensemble average flux for abimaterial Markov process. However, in most physicalapplications the interesting quantities are the materialfluxes wk(x,v). Here we analyze how to evaluate thesefluxes in terms of the propagators. Our approach is to writea STM equation for the material flux in terms of an appro-priate propagator, as it was done earlier for Æw(x,v)æ.

For density statistics and for x(x) Markov the joint pro-cess {Uy,x,x} is also Markov and we have

hUyðxÞi ¼ p1ðxÞhUyðxÞi1 þ p2ðxÞhU yðxÞi2; ð24Þ


where the material averaged propagators hUyðxÞik ¼RXkðxÞ U y;xðxÞpðxÞdx=pkðxÞ obey the coupled evolution

equations (Akcasu, submitted for publication)

ðlox þ AkÞhU yik ¼ l pk0pkkk0ðhU yik0 � hUyikÞ; y < x < R;

hUyðxÞik ¼ I; x ¼ y:

(ð25Þ

We shall consider general bimaterial Markov statistics withstochastic boundary conditions and sources: win

L;x ¼ winL;k for

x(L) = k, winR;x ¼ win

R;k for x(R) = k, and Sx(x,v) = Sk(x,v)for x(x) = k. By averaging Eq. (5) over Xk(x) one obtainsthe STM formulation for the material fluxes:

hwðx; vÞik � ðhU LikhwLikÞðx; vÞ þ hQðx; vÞik; ð26Þwhere hQðx; vÞik ¼

RXkðxÞ Qxðx; vÞpðxÞdx=pkðxÞ. Similarly to

the STM formulation for the ensemble average flux, theapproximation invoked in this equation is to neglect thecorrelation between the propagator and the left boundaryflux.

For the two types of sources considered earlier, theexpressions for ÆQæk are very similar to those obtainedfor ÆQæ. As was done in (13), by neglecting correlations,ÆUyl

�1Sæk � Æ Uyækl�1Sk, one writes

hQðx; vÞik �Z x

LðhUyikl�1SkÞðx; vÞdy; ð27Þ

while, for a source as in (16), we get

hQðx; vÞik ¼ ½ðI� hULikÞA�1s�ðx; vÞ

�Z x

LðhU yikA�1oysÞðx; vÞdy: ð28Þ

We have then

hQðx; vÞi ¼ p1ðxÞhQðx; vÞi1 þ p2ðxÞhQðx; vÞi2for deterministic and density-like sources.

Form this relation and from Eq. (24) we see that Eq.(26) for the Æwæk are consistent with Eq. (11) for the ensem-ble average flux only if

hwLi1 ¼ hwLi2 ¼ hwLi:This observation brings us to propose two different meth-ods to consistently compute Æwæ and the Æwæk from theSTM approach:

I A first way to obtain the material fluxes is to evaluateEq. (26) at x = R to compute the left exiting materialflux hwout

L ik,

winR;kðvÞ ¼ ðhUþL ikw

inL;kÞðR; vÞ þ ðhU�L ikw

outL;k ÞðR; vÞ þ hQðR; vÞik;

ð29Þ

and use Eq. (26) to compute the material fluxes. Finally, theensemble average flux is given by Æwæ = p1Æwæ1 + p2Æwæ2 and,even with deterministic entering fluxes, does not satisfyEq. (11) with the left boundary condition hwLi ¼

P2k¼1pk

ðLÞhwLik.

II The second way is to use standard STM model (11) todetermine Æwæ, that is, use (12) to get hwout

L i from theensemble average boundary fluxes and sources, andthen obtain the material fluxes Æwæk from Eq. (26)with ÆwLæk = ÆwLæ.

Clearly, both approaches differ. The reason is that withSTM Method I the ensemble average flux does not satisfy aSTM equation. The difference can also be illustrated byconsidering the material fluxes for a simple case with deter-ministic boundary conditions. From approach I we get thematerial fluxes satisfying Eq. (26) with hwout

L i1 6¼ hwoutL i2,

while from STM Method II the material fluxes are com-puted with the same value:

hwoutL i1 ¼ hw

outL i2 ¼ hw

outL i: ð30Þ

3.3. From the STM model to the MLP equations

The solution of the STM equations for the materialfluxes requires the numerical evaluation of the materialpropagators ÆUyæk. From Eqs. (24) and (22) one gets

hU yðxÞik ¼ M bAXc

P cwckðxÞM�1bA ;

where the wck are obtained from Eq. (21) but using now the

boundary condition wk(x) = 1.However, with the exception of the rod problem with

homogenous statistics and cross sections, the numericalevaluation of the propagators is very involved. Therefore,instead of working with the STM Eq. (26), it would bemuch easier to deal with an equivalent differential formula-tion for the material fluxes.

In this section we derive such a formulation and discussits limitations. Because the MLP equations and the STMmethod are different ways of solving the same problem,the reader should not be surprised that, when the assump-tions for both formulations are identical, the differentialform of the STM equations yields the MLP equations.

To derive a differential-like form of STM Eq. (26) forthe material fluxes we simply apply the operator lox + Ak

to these equations. The result is

ðlox þ AkÞhwik ¼ Sk þ l pk0pkkk0ðhwik0 � hwik � Ckk0 Þ; L < x < R;

hwðx; vÞik ¼ winL;kðvÞ; x ¼ L; l > 0;

hwðx; vÞik ¼ winR;kðvÞ; x ¼ R; l < 0;

8>><>>:ð31Þ

where use has been made of (25). Also, for the source in(27) we have

Ckk0 ¼ f½hULik0 ðwL;k0 � wL;kÞ�ðx; vÞ

þZ x

L½hU yik0l�1ðSk � Sk0 Þ�ðx; vÞdyg;

while for the density-like source in (28)


Ckk0 ¼ ½hULik0 ðwL;k0 � wL;kÞ�ðx; vÞ

þ pk0

pkkk0

� ��1

ðA�1oxsÞðx; vÞ: ð32Þ

This result shows that the differential form of the STMmodel contains, in general, the integral term Ckk0 . It is onlywhen this term vanishes that we recover the MLP equa-tions. For density-like sources as in (16) and for determin-istic sources this happens when the left boundary fluxes aredeterministic. The implication is that only Method II leadsto the MLP equations, while Method I must be solvedusing the propagator formulation, except for the HSSDproblem when it is equivalent to the MLP equations. Nev-ertheless, although the MLP equations can be used to eval-uate the material fluxes with Method II, this does not gowithout pain in that a new numerical difficulty is that onehas to abandon the natural boundary conditions and re-place them with the left boundary condition ÆwLæk = ÆwLæwith the value of ÆwLæ obtained from STM Eq. (11) forthe ensemble average flux.

Finally, we observe that Eq. (31) with Ckk0 ¼ 0 are noth-ing else but the MLP heuristic model that we have intro-duced and numerically tested in Sanchez (submitted forpublication). Although they both use natural boundaryconditions, the MLP model appears to differ radically fromSTM Method I in that the latter requires the extra term Ckk0

in Eq. (32).

4. Numerical examples for the STM models

In our tests we have considered the so-called rod prob-lem. In this problem, particles move on a line sufferingabsorptions and forward and backward collisions. Forhomogeneous Markov density statistics and cross sectionsthe propagators can be evaluated from analytical expres-sions (see Appendix B). We have programmed the STMmodel equations for this simple problem, including thelimit case for non absorbing materials, and will presenthere some of our results. Our numerical solutions havebeen checked against the analytical solutions of Akcasu(submitted for publication) for the simple cases of noabsorption (c = 0) and of a purely scattering medium

Fig. 2. Ensemble average fluxes for a rod of length 10 with a unit flux enteringand R1 = 1, material 2 has k2 = 5 and R2 = 2, c1 = 0.99 for both materials.

(c = 1), but we are interested here in exploring the limita-tions and possibilities of STM Methods I and II for morerealistic problems. Our aim is to compare STM MethodsI and II and to check their precision.

Our results are compared with the results provided bythe Levermore–Pomraning (LP) method (Pomraning,1991), and the accuracy is checked with reference solutionsobtained by averaging the results from 100,000 randomlygenerated realizations. We note that, as opposed to theSTM solutions, which are obtained by evaluating analyti-cal expressions, the LP and the reference calculations weredone by numerically solving the equations with a finite-dif-ference scheme with 10,000 regions and converged to 10�7

in relative precision.Our first three cases concern propagation problems. The

first two calculations are aimed to examine the symmetry ofthe STM solutions: in one case particles enter the left of therod while in the other case they enter the right. The resultsfor ÆUæ, ÆUæ1 and ÆUæ2 are compared, side-by-side for thetwo opposite entering modes, in Figs. 2–4, respectively.While the LP and the reference solutions are symmetrical(the reference only to statistical precision), neither of theSTM solutions are symmetrical. Moreover, the STM solu-tions are far from the reference solution.

For both STM methods the ensemble average and thematerial average currents satisfy the exact boundary condi-tion on the left. However, while STM I also satisfies theexact right boundary condition for all fluxes, STM II doesit only for the ensemble average flux. For the case with leftentering flux STM II gives hwin

R i ¼ 1, hwinR i1 ¼ 3:978� 10�2

and hwinR i2 ¼ �2:388� 10�2, while, for the case with right

entering flux, it gives hwinR i ¼ 1; hwin

R i1 ¼ 8:831� 10�1 andhwin

R i2 ¼ 1:070. As discussed earlier, the reason why thematerial fluxes of Method II do not fulfill the right bound-ary condition is that one uses (30) instead as determiningthe hwout

L ik from the right boundary condition, as it is donein Method I.

In the third propagation case we have calculated a prop-agation problem when one of the materials is vacuum. Thedata and the results for this problem are given in Figs. 5and 6. The results show an overall better performance forthe LP method. The differences between STM I and II

the left side (left figure) or the right side (right figure): material 1 has k1 = 3

Fig. 3. Ensemble average material fluxes for material 1 for a rod of length 10 with a unit flux entering the left side (left figure) or the right side (rightfigure): material 1 has k1 = 3 and R1 = 1, material 2 has k2 = 5 and R2 = 2, c1 = 0.99 for both materials.

Fig. 4. Ensemble average material fluxes for material 2 for a rod of length 10 with a unit flux entering the left side (left figure) or the right side (rightfigure): material 1 has k1 = 3 and R1 = 1, material 2 has k2 = 5 and R2 = 2, c1 = 0.99 for both materials.

Fig. 5. Ensemble average fluxes for a rod of length 20 with a unit fluxentering the left side: material 1 has k1 = 3, R1 = 1 and c1 = 0.99, material2 is vacuum with k2 = 6.

Fig. 6. Ensemble average material fluxes for a rod of length 20 with a unit flux eis vacuum with k2 = 6.


are small but, because in STM Method I the right incomingflux is forced to the true boundary value, this method givesbetter results than STM II.

Our last three examples are for problems with sources.We explore, first, a symmetric problem with either a deter-ministic or a density source and unit fluxes entering bothsides of the rod. The idea is to analyze the behavior ofthe models for deterministic and density sources. The inten-sity of the density sources has been adjusted so as to pro-duce on the average 1 particle per unit length, as for thedeterministic source. The results for ÆUæ, ÆUæ1 and ÆUæ2

are compared, side-by-side for the two opposite enteringmodes, in Figs. 7–9, respectively.

ntering the left side: material 1 has k1 = 3, R1 = 1 and c1 = 0.99, material 2

Fig. 7. Ensemble average fluxes for a rod of length 10 with unit fluxes entering the left and right sides and a unit deterministic (left figure) or density (rightfigure) source: material 1 has k1 = 3 and R1 = 1, material 2 has k2 = 5, R2 = 2, c = 0.8 for both materials.

Fig. 8. Material average fluxes for material 1 for a rod of length 10 with unit fluxes entering the left and right sides and a unit deterministic (left figure) ordensity (right figure) source: material 1 has k1 = 3 and R1 = 1, material 2 has k2 = 5, R2 = 2, c = 0.8 for both materials.

Fig. 9. Material average fluxes for material 2 for a rod of length 10 with unit fluxes entering the left and right sides and a unit deterministic (left figure) ordensity (right figure) source: material 1 has k1 = 3 and R1 = 1, material 2 has k2 = 5, R2 = 2, c = 0.8 for both materials.

Fig. 10. Ensemble average fluxes for a rod of length 10 with a unit fluxentering the right side and a deterministic unit source: material 1 hask1 = 3 and R1 = 2, material 2 has k2 = 3.5, R2 = 3, c = 0.8 for bothmaterials.


The results show that STM fails for the deterministicsource, yielding even negative values for material 1. Thetwo STM solutions are much better with the densitysource, although they are not totally symmetric, withSTM Method I giving the best results of the two. Webelieve that the reason for this is that the STM (MLP) for-mulation is asymptotically exact for density statistics and

sources. With finite slabs the precision of the STM modelsdepends on the correlation between the propagator and theleft exiting flux. With density sources the correlation van-ishes as the thickness of the slab goes to infinity, while thisis not the case for deterministic sources. This explains whySTM I method gives better results than STM II for finiteslabs with density sources.

Our last case in an asymmetric rod with a deterministicunit source and a unit flux entering the right side, and it is

Fig. 11. Ensemble average material fluxes for a rod of length 10 with a unit flux entering the right side and a deterministic unit source: material 1 hask1 = 3 and R1 = 2, material 2 has k2 = 3.5, R2 = 3, c = 0.8 for both materials.


meant to show the failure of the STM models for this typeof problems with particles entering from the left. The dataand the results are given in Figs. 10 and 11. For this prob-lem, both STM methods give unphysical negative fluxes,while LP holds tightly to the reference solution. It shouldbe mentioned, that the negative fluxes disappear when bothmaterials have the same mean chord length as well as whenthe source is a density source but, even in this case, the pre-cision of the STM results are not acceptable.

5. Conclusions

Our numerical comparisons for the rod problem showthe shortcomings of the STM models, which were also dis-cussed in our companion paper (Sanchez, submitted forpublication) for the equivalent MLP model, and whichoriginate in the lack of positiveness and symmetry of theseequations. Our numerical explorations show that the twoSTM methods proposed here give accurate results for theensemble average flux with density statistics, a fact thatshould not come as a surprise, given that the equationsare asymptotically exact (for a half-space) with densitysources. Method I gives the best results for the materialaverage fluxes, whereas Method II is inaccurate for the esti-mation of these fluxes.

Another property that has been claimed for the STMsolution of MLP is that for a finite slab this model yieldsthe correct relaxation lengths for the spatial modes(Akcasu, 2006, submitted for publication). This claim isbased on the fact that in STM the statistics are expressedvia the ensemble average propagator and that, for bima-terial Markov density statistics, the latter is calculatedwith no approximations. However, in spite of this, theSTM equation is not exact in a finite slab because itneglects the correlation between the left outgoing fluxand the propagator. While this correlation disappearsfor a half-space, the fact is that we cannot predict theimpact of the correlation for a finite slab and, further-more, it is improbable that the exact ensemble average(ÆULwLæ)(x,v) will yield the asymptotic relaxation modesfor a finite slab, which are incorporated with the STMformulation . What is true is that, as the width of the slabincreases, the outgoing flux will tend to be deterministic

and, then, the true modes for the ensemble average fluxwill be closer to the asymptotic ones given by the STMapproach. In conclusion, for finite slabs, only an exactsolution will preserve the modes that are actually excitedby the natural boundary conditions of the problem. How-ever, this exact solution is not known.

There is also the issue of symmetry. As we have previ-ously said (Sanchez, submitted for publication), the MLPmodel does not give symmetric solutions when the problemhas symmetric boundary conditions and sources, and thisproperty is inherited by the equivalent STM formulations.An analysis of the STM equations for the rod problem withuniform statistics and cross sections shows (see AppendixB) that symmetry is preserved only for the ensemble aver-age flux as calculated with STM Method II when the med-ium is non absorbing (c = 1) and has deterministic sources(Akcasu, 2006). This surprising finding should be carefullyweighted in the light that the STM equations are asymptot-ically exact for density statistics and sources, but not withdeterministic sources.

A different approach, based on the material averagingof the integral form of the transport equations, was devel-oped earlier (Pomraning, 1991; Sanchez, 1989) and hasbeen proved to give accurate numerical results for stan-dards tests (Zuchuat et al., 1994). This approach allowsfor renewal statistics, which generalize Markov statisticsand allow for a more realistic modeling in that the mate-rial chords may remain bounded. The renewal methodleads to two sets of coupled integral equations for thematerial fluxes. The lower set of equations is exact butincorporates new statistical flux averages. These are theso-called interface fluxes and also new average fluxesappearing in the collision term. The second set of equa-tions gives the interface fluxes in terms of still higher sta-tistical flux moments. For purely absorbing media theselast flux moments equal the interface fluxes and therenewal equations are exact. A heuristic model can thenbe invoked by adopting the closure for purely absorbingmedia and an approximation for the new moments inthe collision term.

A comparison can be made between the integral STMapproach and the renewal equations. In the renewalapproach the integral operator does not account for colli-


sions but the correlation between the integral kernel andthe incoming fluxes or the sources are accounted forexactly, however, to the prize of introducing new ensembleaverage fluxes that, in particular, affect the collision term.In STM the propagator accounts for collisions but thereis an approximation regarding the correlations betweenthe propagator and the entering fluxes and the externalsources. The STM approach is much better because thepropagator incorporates collisions and the approximationis done only for the external sources, while, even with col-lisions, the lower set of renewal equations is exact butincorporates the unknown interface average fluxes. A pos-sibility to improve the renewal model could consist ofreplacing the second integral equation with a STM-likeequation and use STM to better account for the collisionterms. Although this program seems complicated, a firstviable step can consist of deriving the equations for thematerial average propagators for renewal statistics.

Acknowledgements

Many thanks are due to Ziya Akcasu for comments anddiscussions, as well as for providing unpublished material.Although there has been much controversy and our opin-ions are sometimes drastically different, the email ex-changes helped me to understand better STM. Clearly,the opinions advanced in this paper are the sole responsi-bility of the author.

Appendix A. Analysis of an inverse problem

We consider the one-group transport problem in a finiteslab with a left incoming flux win

L ðlÞ and, otherwise, zerosources and zero right incoming flux. The angular flux leav-ing the right boundary is given by

woutR ðlÞ ¼

1

l

Z R

Le�sðL;xÞ=lðcRÞðxÞðQwin

L Þðx; lÞ

þ e�sðL;RÞ=lwinL ðlÞ; l > 0; ð33Þ

where

ðQwinL Þðx; lÞ ¼

Z 1

�1

hðx; l0 ! lÞwðx; l0Þdl0

with w(x,l) solution of the problem

ðlox þ AÞw ¼ S; L < x < R;

wðx; lÞ ¼ winL ðlÞ; x ¼ L; l > 0;

wðx; lÞ ¼ 0; x ¼ R; l < 0:

8><>: ð34Þ

In operator notation we write:

f ¼ esðL;RÞ=lwoutR ¼ ð1þ AÞwin

L ;

where operator A is es(L,R)/l times the integral operator in(33). We introduce the Banach spaces X = L1(0, 1) andXw = L1[(0, 1),w] with norms kf kX ¼

R 1

0jf ðlÞjdl and

kf kX w¼R 1

0 jf ðlÞje�sðL;RÞ=ldl, so that 1 + A: X! Xw. Givenf 2 Xw, the equation accepts a unique solution win

L 2 X is

the operator 1 + A has an inverse. In particular, the inverseexists if kAkLðX ;X wÞ < 1.

We proceed to estimate a bound for the norm of A:

kAwinL kX w

¼Z 1

0

1

l

Z R

Le�sðL;xÞ=lðcRÞðxÞjðQwin

L Þðx; lÞjdxdl

6

Z 1

0

1

l

Z R

Le�sðL;xÞ=lðcRÞðxÞdxdl

�Z 1

0

Z R

LjðQwin

L Þðx; lÞjdxdl:

The first integral can be bounded asZ 1

0

1

l

Z R

Le�sðL;xÞ=lðcRÞðxÞdxdl 6 cmax

Z 1

0

ð1� e�sðL;RÞ=lÞdl

¼ cmaxð1� E2½sðL;RÞ�Þ;

where cmax = maxx2(L,R)c(x) and EnðsÞ ¼R 1

0ln�2e�s=ldl is

the well-known exponential integral function. For the sec-ond integral we haveZ 1

0

Z R

LjðQwin

L Þðx; lÞjdxdl 6 hmax wk kY ;

where hmax ¼ maxx2ðL;RÞ;l;l02ð�1;1Þhðx; l0 ! lÞ and Y = L1[(L,R) · (�1,1)] is the Banach space with norm kf kY ¼R R

L

R 1

�1jf ðx; lÞjdxdl.

Finally, we use the fact that problem (34) accepts abounded solution, that is, for cmax < 1 there exists a posi-tive constant a such that kwkY 6 akwin

L kX . Putting our find-ings together we have

kAkLðX ;X wÞ 6 cmaxð1� E2½sðL;RÞ�Þhmaxa

and we conclude that winL can be uniquely calculated from f

for

cmax < C ¼ 1

ð1� E2½sðL;RÞ�Þhmaxa:

This proves that the operator P�G�R ðLÞ accepts an inversein the interval cmax 2 [0,C). Possibly this range could be ex-tended to [0,1) by passing to complex values and using ana-lytical prolongation. Also, for strictly positive total crosssections, the above proof can be extended to the case withvariable v.

Appendix B. STM method for the rod problem

We summarize here the formulation of the STM meth-ods for constant statistics and cross sections for a rod withx 2 [0, R] and anisotropic sources and collisions. The‘transport’ equation for the rod problem reads

ðox þ RAÞw!ðxÞ ¼ Q!ðxÞ; 0 < x < R;

where

w!¼ wþ

w�

� �; Q!¼ S

qþ�q�

� �; A ¼

aþ �a�a� �aþ

� �: ð35Þ


Here w+ and w� are the angular fluxes for particles movingto the right and to the left, respectively, S is the intensity ofthe source, q+ is the probability for source emission to theright, q� = 1 � q+, a+ = 1 � cp+ and a� = cp�, where c isthe number of secondaries, p+ is the probability of forwardscattering and p� = 1 � p+. Matrix A has two real eigen-values ±c = ± {(1 � c)[1 � c(2p+ � 1)]}1/2.

In the following we adopt Akcasu’s notation R ¼ðR1 þ R2Þ=2, dR = (R1 � R2)/2, �k ¼ ð1=k1 þ 1=k2Þ=2 anddk = (1/k1 � 1/k2)/2.

The calculation of ÆU(x)æ and the Æ U(x)æk are done fol-lowing the approach outlined in Sections 3.1 and 3.3. Theresult is that ensemble and material fluxes are obtainedfrom equations of the type

w!ðxÞ ¼ uI þ du

1

cA

� �w!

0 þ wI þ dw1

cA

� �Q!; ð36Þ

where I is the unit 2 · 2 matrix and for a density sourceQ!! Q

!=c. Also, in Eq. (36) we have

uðxÞ ¼ ½ucðxÞ þ u�cðxÞ�=2; du ¼ ½ucðxÞ � u�cðxÞ�=2

and similar expressions for �w and dw in terms of the func-tions wc(x).

The function uc(x) is a combination of exponentialmodes

ucðxÞ ¼ ucþe�zþx þ uc

�e�z�x; ð37Þ

where z� ¼ cRþ �k� D, with D2 ¼ ðcdRÞ2 þ 2cdkdRþ �k2,are the two eigenvalues of matrix B in (23) with nowbk = cRk + 1/kk, and the values of the constants uc

� are

ðuc�Þ1 ¼ ½1� ð�k� cdRÞ=D�=2;

ðuc�Þ2 ¼ ½1� ð�kþ cdRÞ=D�=2

for the material average fluxes and

uc� ¼ ½1� ð1þ cdRdk=�k2Þ�k=D�=2

for the ensemble average flux.For deterministic sources the expressions for the wc(x)

are like the ones in (37) but replacing exp(�zx)![1 � exp(�zx)]/z, while for a density source �w ¼ �du anddw = 1 � �u.

B.1. Limit for c! 1

The limiting expression for w!ðxÞ can also be written

similarly as in (36):

w!ðxÞ ¼ ðuI þ duAÞw!0 þ ðwI þ dwAÞQ!;

where the Q!

is the one in (35) for both types of sources.For the material average fluxes we have

u� ¼ 1; du� ¼ �½a�xþ t�ð1� e�2�kxÞ=2�;

where we adopt the convention that + and � denote,respectively, materials 1 and 2, a� ¼ R½1� dRdk=ðR�kÞ�and t� ¼ ðdR=�kÞ½ðdk=�kÞ � 1�. For the ensemble average fluxwe have

�u ¼ 1; du ¼ �a�x:

For the deterministic source we have

�w� ¼ x; dw� ¼ �fa�x2 þ t�½x� ð1� e�2�kxÞ=ð2�kÞ�g=2

for the material fluxes and

�w ¼ x; dw ¼ �a�x2=2

for the ensemble average flux.Finally, for the density source we obtain

�w� ¼ �du�;

dw� ¼ f�ða�xÞ2 þ ½2�kb� t�ða� � aþe�2�kxÞ�xþ b�ð1� e�2�kxÞg=2

for the material fluxes, and

�w ¼ du; dw ¼ ½bð2�kxþ e�2�kx � 1Þ � a2�x2�=2

for the ensemble average flux. The new constants in theseexpressions are b ¼ ½ðdkÞ2=�k2 � 1�ðdRÞ2=ð2�k2Þ and b� ¼ð1=2ÞðdR=�kÞ2½1� 3ðdk=�kÞ2� � ðdR=�kÞ2ðdk=�kÞ.

B.2. Symmetry

We consider a symmetric problem with p+ = q+ = 1/2and win

0 ¼ winR ¼ w. Then, the ensemble or material averages

of the scalar flux and the current are given by theexpressions

hUi¼ ½ð1þqwÞ�uþð1�qwÞdu=c�winþ½2dw=cþqSð�u�du=cÞ�S=2;

hJi¼ ½ð1�qwÞ�uþð1þqwÞð1�cÞdu=c�winþf2w�qS ½�u�ðð1�cÞdu=cÞ�gS=2;

ð38Þ

where for the density source S! S/c and where

qw ¼ ½1� ðc=2ÞduðRÞ=c�=q; qS ¼ ½wðRÞ � dwðRÞ=c�=q

and q = �u(R) � (1 � c/2)du(R)/c are constants arisen fromthe calculation of wout

0 in terms of the right boundaryflux.

The solution is symmetric when wþk ðxÞ ¼ w�k ðR� xÞ or,equivalently, when ÆU(x)æ = ÆU(R � x)æ and ÆJ(x)æ =� ÆJ(R � x)æ. For the case with c < 1, an examination of(38) shows that the symmetry is, in general, not realized.The only exception is the case with density sourcesS = 2(1 � c)win for which we have ÆUæ = 2win and ÆJæ = 0.

For the case with c = 1 we have c = 1 in (38) and theonly symmetric result is for the ensemble average flux withc = 1. Therefore, only STM Method II will preserve thesymmetry for this particular case.

References

Akcasu, A.Z., Williams, M.M.R., 2004. An analytical study of particletransport in spatially random media in one dimension: Mean andvariance calculation. Nucl. Sci. Eng. 148, 403.

Akcasu, A.Z., 2006. A critique of the Levermore–Pomraning equations.Trans. Am. Nucl. Soc. 95, 545–546.

Akcasu, A.Z., submitted for publication. Modified-Levermore–Pomra-ning equation: its derivation and its limitations. Ann. Nucl.Energy.


Pomraning, G.C., 1991. Linear Kinetic Theory and Particle Transport inStochastic Mixtures. World Scientific, Singapore.

Sanchez, R., 1989. Linear kinetic theory in stochastic media. J. Math.Phys. 30, 2511–2948.

Sanchez, R., 1998. Duality, Green’s functions and all that. Trans. Theor.Stat. Phys. 27 (5-7), 445–478.

Sanchez, R., submitted for publication. A critique of the modifiedLevermore–Pomraning equations. Ann. Nucl. Energy.

Zuchuat, O., Sanchez, R., Zmijarevic, I., Malvagi, F., 1994. Transport inrenewal statistical media: Benchmarking and comparison with models.J. Quant. Spectrosc. Radiat. Transfer 51, 689–722.

A critique of the stochastic transition matrix formalism

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