Annals of Nuclear Energy 94 (2016) 856–868

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Annals of Nuclear Energy

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Monte Carlo power iteration: Entropy and spatial correlations

http://dx.doi.org/10.1016/j.anucene.2016.05.0020306-4549/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: jlmiao@mit.edu (J. Miao), andrea.zoia@cea.fr (A. Zoia).

Michel Nowak a, Jilang Miao b, Eric Dumonteil a,c, Benoit Forget b, Anthony Onillon c, Kord S. Smith b,Andrea Zoia a,⇑aDen-Service d’études des réacteurs et de mathématiques appliquées (SERMA), CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, FrancebDepartment of Nuclear Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 24-107, Cambridge, MA 02139, United Statesc IRSN, 31 Avenue de la Division Leclerc, 92260 Fontenay aux Roses, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 March 2016Received in revised form 29 April 2016Accepted 1 May 2016Available online 8 May 2016

Keywords:EntropyClusteringPower iterationOpenMCTRIPOLI-4�

The behavior of Monte Carlo criticality simulations is often assessed by examining the convergence of theso-called entropy function. In this work, we shall show that the entropy function may lead to a mislead-ing interpretation, and that potential issues occur when spatial correlations induced by fission events areimportant. We will support our analysis by examining the higher-order moments of the entropy functionand the center of mass of the neutron population. Within the framework of a simplified model based onbranching processes, we will relate the behavior of the spatial fluctuations of the fission chains to the keyparameters of the simulated system, namely, the number of particles per generation, the reactor size andthe migration area. Numerical simulations of a fuel rod and of a whole core suggest that the obtainedresults are quite general and hold true also for real-world applications.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Monte-Carlo simulation is often used in criticality calculationsto assess the asymptotic distribution of the neutron populationwithin a system, which corresponds to the fundamental eigen-mode of the Boltzmann critical equation (Lux and Koblinger,1991). The most widely used and simplest numerical methodallowing the neutron population to converge to the fundamentaleigenmode is the power iteration (Lux and Koblinger, 1991; Riefand Kschwendt, 1967; Brown, 2005): in Monte Carlo methods, aninitial arbitrary source particle distribution is transported untilall neutrons have been either absorbed or leaked (forming aso-called generation). The secondary neutrons coming from thefission events within a generation g are banked and provide thesource for the following generation g þ 1. The algorithm is theniterated over many generations, until the fission sources for asufficiently large g statistically attain a spatial and energeticequilibrium, as ensured by the power iteration method. The effectsof higher eigenmodes on the neutron population are expected tofade away, and eventually the neutron population will bedistributed according to the fundamental eigenmode. The ratiobetween the population size at generation g þ 1 and the popula-tion size at generation g converges to the fundamental eigenvaluekeff for large g (Lux and Koblinger, 1991).

In this context, two key issues are known to affect the neutronpopulation during power iteration and have therefore attractedintensive research efforts: fission source convergence andcorrelations.

Concerning the former, a slow exploration of the viable phasespace by the population implies a poor source convergence. Inparticular, it has been shown that the convergence of keff mightbe faster than that of the associated fundamental eigenmode,which is expected on physical grounds, the former being an inte-gral property of the system and the latter being a local property(Lux and Koblinger, 1991). The rate of convergence depends onthe separation between the first and the second eigenvalue ofthe Boltzmann equation, the so-called dominance ratio: the closerto one the dominance ratio becomes, the poorer the convergence(Lux and Koblinger, 1991; Ueki et al., 2004; Dumonteil andMalvagi, 2012). If Monte-Carlo tallies are scored before attainingequilibrium, biases on the estimation of the variance may appear,and monitoring the convergence of keff might be insufficient soas to determine the convergence of the whole population (Uekiand Brown, 2003; Dumonteil et al., 2006; L’Abbate et al., 2007;Ueki, 2005).

Several tools have been proposed to assess the spatial conver-gence of fission sources, among which, one of the most popular,is the entropy of the fission sources (Ueki et al., 2003; Ueki,2012; Ueki and Brown, 2003; Ueki, 2005). The idea behind theentropy function is to superimpose a regular Cartesian mesh tothe viable space and to record the number of fission sites for each

M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868 857

cell of the mesh, at each generation. This allows computing the so-called Shannon entropy S (Li and Vitanyi, 1997), which is definedas

SðgÞ ¼ �Xi;j;k

pi;j;kðgÞlog2½pi;j;kðgÞ�; ð1Þ

where pi;j;kðgÞ is the (statistically weighted) number of fission sourceparticles in the cell of index i; j; k at generation g divided by the totalnumber of source particles in all cells at generation g. The entropyfunction is expected to provide a measure of the phase space explo-ration as a function of the number of generations (Li and Vitanyi,1997; Cover and Thomas, 1991): when the neutron distributionattains its stationary shape, the entropy S converges. A prominentadvantage of the entropy is that S is a single scalar value whoseevolution condenses the required information on the spatial repar-tition. Moreover, as apparent from Eq. (1), the entropy S of thesource distribution at generation g is bounded, namely,

0 6 SðgÞ 6 log2B; ð2Þ

where B is the number of cells of the spatial mesh. This propertyensures in particular that the variations of S will be bounded, andthat the highest value of the entropy will be reached in the caseof a perfect equipartition. The entropy function is nowadays a stan-dard tool for most production Monte Carlo codes, although someconcerns have been raised about possible issues related to its usein convergence diagnostics for loosely coupled multiplying systems(see, e.g., the analysis in Shi and Petrovic, 2010a,b; Ueki, 2005).

The latter issue with power iteration in Monte Carlo simulationconcerns the impact of correlations induced by fission events:physically speaking, a neutron can only be generated in the pres-ence of a parent particle, which induces generation-to-generationcorrelations (Lux and Koblinger, 1991; Sjenitzer andHoogenboom, 2011). This is a widely recognized problem, whichis expected to affect the convergence of Monte Carlo scores andin particular make the applicability of Central Limit Theorem ques-tionable (Ueki, 2012; Brown, 2009; Ueki, 2005). Correlationsbetween generations have been often studied within the mathe-matical framework provided by the eigenvalue analysis of theBoltzmann critical equation (Brown, 2005; Sutton, 2014a,b). Fur-ther work on correlations has concerned techniques aimed atimproving the standard deviation estimates of Monte Carlo scores(Gelbard and Prael, 1990; Ueki et al., 2003, 2004; Dumonteil andMalvagi, 2012; Ueki, 2005). More recently, it has been pointedout that, due to the asymmetry between correlated births by fis-sion and uncorrelated deaths by capture and leakage,1 neutrons ini-tially prepared at equilibrium will be preferentially found clusteredclose to each other after a few generations (Dumonteil et al.,2014). This peculiar phenomenon, named neutron clustering, mightinduce a strongly heterogeneous spatial repartition of the neutronpopulation, which randomly evolves between generations(Dumonteil et al., 2014; Zoia et al., 2014; de Mulatier et al., 2015).The impact of neutron clustering has been determined to be inver-sely proportional to the number of neutrons per generation(Dumonteil et al., 2014).

In this paper, we will show that neutron clustering, not surpris-ingly, also affects the convergence of the fission sources: because offission-induced correlations, the entropy function might in turn beineffective at detecting potential deviations of the neutron popula-tion with respect to the expected equilibrium.

1 This phenomenon has been first investigated in the context of theoretical ecology,especially in relation to the evolution of biological communities: see, e.g., Young et al.(2001) and Houchmandzadeh (2008), and can be better understood in the frameworkof branching random walks (Athreya and Ney, 1972; Williams, 1974; Pázsit and Pál,2008; Zoia et al., 2012).

This manuscript is organized as follows. In Section 2, we willinitially consider a simplified reactor model where exact analyticalresults can be established, and show that in some cases the conver-gence of the entropy is achieved, although the neutron populationis still affected by strong spatial fluctuations. Then, in Section 3, wewill refine our analysis based on the spatial moments of theentropy function and on the center of mass of the neutron popula-tion: these statistical tools can be used together with regularentropy so as to extract information concerning the simulated sys-tem, and thus improve the diagnosis of fission source equilibrium.In Section 4 we will then relate the behavior of such spatial fluctu-ations to the key system parameters, namely, the reactor size, thenumber of neutrons per generation, and the migration area, basedon the theory of branching processes. In Section 5 we will numer-ically explore the behavior of a fuel rod and a full core withdetailed geometry and compositions and continuous-energy treat-ment, and show that the theoretical findings for the simple reactormodel actually apply more generally to realistic configurations.Conclusions will be drawn in Section 6.

2. Neutrons in a box and the behavior of the entropy

In order to assess the behavior of the entropy function in thepresence of fission-induced correlations, we will work out anexample that is simple enough for exact results to be analyticallyderived and compared to the Monte Carlo simulations, and yetretains the key physical features of a real system (Miao et al.,2016).

Let us therefore consider a prototype model of a reactor coreconsisting of a collection of N neutrons undergoing scattering, cap-ture and fission within a box of volume V ¼ L3. To simplify thematter, we will assume that neutrons can only be reflected at theboundaries. The random displacements of the neutrons will bemodeled by branching exponential flights with constant speed v;scattering and fission will be taken to be isotropic in the centerof mass frame. The physical parameters of this prototype reactorwill be the following:

Rs ¼ 0:27; Rc ¼ 0:02; Rf ¼ Rc

�m� 1:0; ð3Þ

where Rs is the scattering cross section, Rc is the capture cross sec-tion, and Rf is the fission cross section (in cm�1). The parameter �mdenotes the average number of secondary neutrons per fission.Observe that on the basis of the cross sections defined above thesystem is exactly critical, i.e.,

keff ¼�mRf

Rc þ Rf¼ 1; ð4Þ

for any choice of �m. For our simulations, we have set �m ¼ 2:5. Thefundamental eigenmode associated to keff ¼ 1 corresponds to anequilibrium distribution that is spatially homogeneous over thebox, as expected on physical grounds.

This prototype reactor model can be easily implemented andsolved by power iteration within a Monte Carlo code. In order toprobe the effects of neutron clustering on the convergence ofpower iteration, we have performed several Monte Carlo criticalsimulations of such system by varying the number N of particlesper generation and the size L of the box, the other physical param-eters being unchanged. For all configurations, we have assumedthat the power iteration is started with a point source consistingof N neutrons located at the center of the box. As the number ofgeneration increases, the neutron population spreads over thewhole box, and is forced to converge towards the fundamentaleigenmode by the power iteration. However, this spread is

858 M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868

counter-reacted by the spatial correlations induced by fissionchains, and clustering might then come into play.

The distribution of the fission sites as obtained during thepower iteration for a fixed number N ¼ 104 of initial particlesand different box sizes L is displayed in Fig. 1. When the neutrondensity is high (i.e., L is small for a given N), the fission sites con-verge to an equilibrium configuration where neutrons are homoge-neously spread over the whole volume, with mild fluctuationsmostly due to scattering. As L increases, spatial fluctuations dueto the competing mechanisms of fission, absorption and scatteringbecome more apparent. For even larger L, the neutron populationdisplays patchiness, with neutrons randomly moving around thebox grouped into a large cluster. Previous investigations based onthe diffusion theory approximation have indeed shown that spatialclustering is quenched when L2 � NM2, where M2 is the migra-tion area (de Mulatier et al., 2015). Fluctuations due to correlationsare therefore expected to decrease for decreasing box size L whenkeeping N constant, which is coherent with our numerical findings.

It is instructive to compute the Shannon entropy SðgÞ for thepower iteration simulations examined here. To fix the ideas, wewill assume that each box side L is partitioned into 8 spatialmeshes, which implies B ¼ 83 ¼ 512. For the simple model consid-ered in this Section, the theoretical ideal entropy associated to thefundamental eigenmode, i.e., the expectation of �log2ðpÞ corre-sponding to a uniform spatial distribution, can be exactly com-puted, and reads

S1 ¼ log2ðBÞ ¼ 9: ð5ÞActually, for a finite number N of particles per generation, the the-oretical expected Shannon entropy is always lower than the idealvalue S1. For the homogeneous reactor, the expected Shannonentropy at finite N can be explicitly computed (derivation is pro-vided in Appendix A) and reads

SN ¼ log2ðNÞ �B1�N

N

XNk¼0

N

k

� �ðB� 1ÞN�kklog2ðkÞ; ð6Þ

Fig. 1. Distribution of fission sites during Monte Carlo power iteration as a functionof generations g, for three different reactor sizes L. The guess source at g ¼ 0consists of N ¼ 104 neutrons located at the center of the cube. Power iteration is runfor 1000 generations. Top: L ¼ 100 cm; center: L ¼ 200 cm; bottom: L ¼ 400 cm.

which depends on the number of spatial meshes B and of the num-ber of particles N. It can be shown that SN < S1; when the numberof neutrons N is very large, the expected entropy converges to theideal reference value, namely, SN ! S1. In the following, we willcompare the measured entropy SðgÞ as a function of generationsto the expected value SN .

The behavior of the measured entropy SðgÞ corresponding tothe reactor configurations presented in Fig. 1 is displayed inFig. 2. The number m of generations taken by the neutron popula-tion to achieve spatial convergence (i.e., to explore the whole reac-tor) starting from a point source can be roughly estimated bym ’ L2=‘2; ‘2 being the mean square displacement of a particleper generation. The quantity ‘2 can be estimated during the MonteCarlo simulation, and for the example discussed here we have‘2 ’ 175 cm2 for L ¼ 100 cm, ‘2 ’ 185 cm2 for L ¼ 200 cm, and‘2 ’ 190 cm2 for L ¼ 400 cm, which yields m ’ 57 for L ¼ 100 cm,m ’ 215 for L ¼ 200 cm, and m ’ 830 for L ¼ 400 cm, respectively.This is consistent with the number of generations taken by themeasured entropy SðgÞ to attain convergence, as shown in Fig. 2.

When L ¼ 100 cm, the computed SðgÞ asymptotically convergesto the expected value SN for large g: in this case, the entropy func-tion correctly mirrors the equilibrium attained by the neutron pop-ulation. As L increases by keeping N fixed, spatial clusteringstrongly affects the neutron population during the power iteration:the neutron population still attains a stationary equilibrium distri-bution, which is nonetheless quite different from the flat funda-mental eigenmode. In particular, a larger fraction of empty cellsis observed. The measured entropy SðgÞ consequently convergesto an asymptotic value for large gwhich is lower than the expectedSN and decreases with increasing L. Numerical analysis shows thatthe ratio between the measured and the asymptotic value of theShannon entropy scales as ðSN � SðgÞÞ=SN / 1=L for fixed N.

Fig. 2. Homogeneous cube reactor. The behavior of the measured Shannon entropySðgÞ during Monte Carlo power iteration as a function of generations g, for threedifferent reactor sizes L and fixed number N of neutrons per generation. The guesssource at g ¼ 0 consists of N ¼ 104 neutrons located at the center of the cube. Poweriteration is run for 1000 generations. Upper red curve: L ¼ 100 cm; central greencurve: L ¼ 200 cm; lower blue curve: L ¼ 400 cm. The dashed red line representsthe expected entropy value SN as in Eq. (6), and the solid black line is the idealexpected entropy S1 for an infinite number of particles per generation as in Eq. (5).(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 4. Homogeneous cube reactor. The behavior of the measured Shannon entropySðgÞ during Monte Carlo power iteration as a function of generations g, for differentinitial population sizes N and fixed L ¼ 400 cm. The guess source at g ¼ 0 consists ofN neutrons located at the center of the cube. Power iteration is run for 1000generations. Upper red curve: N ¼ 105; central green curve: N ¼ 104; lower bluecurve: N ¼ 103. The dashed lines represent the expected entropy value SN as in Eq.(6) (red: N ¼ 105, green: N ¼ 104 and blue: N ¼ 103, respectively), and the solidblack line is the ideal expected entropy S1 for an infinite number of particles pergeneration as in Eq. (5). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868 859

For the homogeneous reactor examined here, where SN can beexactly determined, measuring the discrepancy between SðgÞ andSN at convergence allows in principle the anomalous behavior ofthe fission source convergence to be detected. In general, however,it is not possible to compute the expected entropy value SN , whichmeans that in the presence of strong spatial clustering assessingthe convergence of the Shannon entropy may turn out to be insuf-ficient to ensure a proper spatial convergence of the fissionsources.

We have carried a similar analysis for the power iteration byvarying the number N of initial neutrons per generation at fixedreactor size L. The results are displayed in Fig. 3. The pattern fol-lowed by the neutron population during convergence is similarto that discussed in the analysis carried out above. When the neu-tron density is high (i.e., N is large for a given L), the fission sitesconverge to a spatial equilibrium with neutrons homogeneouslydistributed over the whole volume, with mild fluctuations mostlydue to scattering. As N decreases, spatial fluctuations become moreapparent, and for even smaller populations neutron clusteringeventually sets in. On the basis of the argument discussed above,fluctuations due to correlations are expected to increase fordecreasing population size N when keeping L constant, which iscoherent with our numerical findings.

The behavior of the measured Shannon entropy SðgÞ for theconfigurations presented in Fig. 3 is displayed in Fig. 4. Theexpected entropy value SN depends on the number of particlesper generation, whereas the ideal asymptotic value S1 isunchanged and reads S1 ¼ 9. The number m of generations takenby the neutron population to achieve spatial convergence startingfrom the point source is again m ’ L2=‘2, with ‘2 ’ 190 cm2 andL ¼ 400 cm, which yields m ’ 830, independently of the numberof simulated neutrons. This is consistent with the number of gen-erations taken by the measured entropy SðgÞ to attain conver-gence, as shown in Fig. 4.

When the number of initial neutrons N is sufficiently large, theShannon entropy converges to an asymptotic value that is veryclose to SN . As the relevance of the spatial clustering increasesfor decreasing N, the fraction of empty cells at equilibrium

Fig. 3. Homogeneous cube reactor. Distribution of fission sites during Monte Carlopower iteration as a function of generations g, for three different initial populationsizes N and fixed L ¼ 400 cm. The guess source at g ¼ 0 consists of N neutronslocated at the center of the cube. Power iteration is run for 1000 generations. Top:N ¼ 105; center: N ¼ 104; bottom: N ¼ 103.

increases, and the asymptotic value of SðgÞ attained at convergencebecomes progressively lower than SN . Numerical analysis showsthat the ratio between the measured and expected entropy scalesas ðSN � SðgÞÞ=SN / 1=N. Similarly as observed above, the Shannonentropy may thus become ineffective in diagnosing fission sourceconvergence in the presence of spatial clustering.

One might wonder whether the results discussed in this Sec-tion are specific to exactly critical configurations (i.e., to havingchosen keff ¼ 1). Actually, this is not the case. We have performedseveral other Monte Carlo power iteration simulations for super-or sub-critical reactors by varying the system parameters, andthe outcomes are qualitatively similar to those presented here.Finally, the behavior of the entropy function with respect to themesh number B is presented in Fig. 5. The convergence to theasymptotic value SN for any fixed N decreases for increasing B,which is clearly understood on physical grounds. Indeed, whenthe number of mesh B is larger, the number of particles N requiredto smooth out the effects of correlations in each spatial bin must bealso larger. In particular, we have numerically observed that wehave the scaling ðSN � SðgÞÞ=SN / B1=3 for fixed N and L.

3. Beyond entropy: extracting information from spatialmoments

The main drawback of the entropy function as a tool for the sta-tistical analysis of Monte Carlo power iteration is that the spatialfluctuations of the neutron population are somehow averagedout by the sum over all cells in Eq. (1): an asymptotic convergencecan be ultimately attained even if neutrons are subject to strong(but statistically stationary in space) patchiness. Clustering effectswill then go undetected when using the standard definition of SðgÞduring power iteration, unless the theoretical value SN is known in

Fig. 5. Homogeneous cube reactor. The behavior of the measured Shannon entropySðgÞ during Monte Carlo power iteration as a function of generations g, for differentmesh number B, and fixed L ¼ 400 cm. The guess source at g ¼ 0 consists of Nneutrons located at the center of the cube. Power iteration is run for 2� 104

generations. Upper red curves: B ¼ 16� 16� 16; central green curves:B ¼ 8� 8� 8; lower blue curve: B ¼ 4� 4� 4. For any fixed B;N is progressivelyincreased, i.e., N ¼ 5� 103; N ¼ 104; N ¼ 5� 104; N ¼ 105. For reference, thecorresponding theoretical entropy values S1 are displayed as solid lines. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Fig. 6. Homogeneous cube reactor. Monte Carlo power iteration as a function ofgenerations g, for three different reactor sizes L and fixed number N of neutrons pergeneration. The guess source at g ¼ 0 consists of N ¼ 104 neutrons located at thecenter of the cube. Power iteration is run for 1000 generations. Top. The behavior ofthe generalized Shannon entropy S�

1;0;0ðgÞ. Red curve: L ¼ 100 cm; green curve:L ¼ 200 cm; blue curve: L ¼ 400 cm. Bottom. The behavior of the center of massxcomðgÞ. Red curve: L ¼ 100 cm; green curve: L ¼ 200 cm; blue curve: L ¼ 400 cm.(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

860 M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868

advance, so that the ratio SðgÞ=SN can be computed. Generallyspeaking, this is however not possible but for the simplest reactorconfigurations, such as the homogeneous cube considered above.

In order to explicitly include the effects of spatial correlations inour statistical analysis, a convenient choice consists in generalizingthe entropy function defined in Eq. (1) as follows:

S�u;v ;w ¼ �

Xi;j;k

LuðniÞLvðnjÞLwðnkÞpi;j;klog2½pi;j;k�; ð7Þ

where LqðfÞ are the Legendre polynomials (other basis sets couldalso be used) of order q and argument f, and ni is the x-coordinate(nj is the y-coordinate, and nk is the z-coordinate, respectively) ofthe center of the cell i; j; k, normalized to the interval ½�1;1�. Whenu ¼ v ¼ w ¼ 0, we recover the regular Shannon entropy function,namely, S�

0;0;0ðgÞ ¼ SðgÞ. For q ¼ 1, we have L1ðfÞ ¼ f, so that thequantity S�

1;0;0ðgÞ can be interpreted as the first spatial moment ofthe entropy function along the x direction (S�

0;1;0ðgÞ in the y directionand S�

0;0;1ðgÞ in the z direction, respectively).By construction, the generalized entropy function S�

u;v ;wðgÞ isbetter suited than SðgÞ in detecting spatial fluctuations duringthe Monte Carlo power iteration. In order to illustrate our argu-ment, let us revisit the homogeneous cubic reactor of the previousSection, with a fixed number of meshes B ¼ 512. The behavior ofS�1;0;0ðgÞ for the reactor configurations obtained by keeping the

number of neutrons N fixed and varying the box size L is illustratedin Fig. 6 (top). The first spatial moment of the entropy along the xdirection is expected to be zero due to the symmetry of neutrondistribution. Since the source for the power iteration is initiallyplaced at the center of the box, there is no appreciable convergencephase for S�

1;0;0ðgÞ, as opposed to SðgÞ. Fluctuations of S�1;0;0ðgÞ

around zero clearly mirror the spatial correlations: as L increases,

correlations become stronger and the fluctuations of S�1;0;0ðgÞ

become wilder. A similar behavior is found when decreasing thenumber N of simulated neutrons per generations at fixed reactorsize L, as shown in Fig. 7 (top). When N is large, the S�

1;0;0ðgÞ is againclose to zero, and the fluctuations increase by decreasing N. Forsymmetry reasons, the results for S�

0;1;0ðgÞ and S�0;0;1ðgÞ are (statis-

tically) identical to those obtained for S�1;0;0ðgÞ and will thus not

be shown here.A second, and perhaps more intuitive, approach to the analysis

of spatial fluctuations during power iteration consists in comput-ing the center of mass rcomðgÞ of the neutrons as a function of gen-erations g, as recently suggested for instance by Wenner andHaghighat (2007, 2008). To be more precise, for a collection of Nparticles having coordinates x1; y1; z1; . . . xi; yi; zi; . . . xN; yN; zNf g atthe fission sites, the center of mass is defined as

rcomðgÞ ¼P

iwiriPiwi

; ð8Þ

where ri is the vector of components ri ¼ xi; yi; zif g, and wi are thestatistical weights of the neutrons. The components of the centerof mass along each direction are defined as

xcomðgÞ ¼P

iwixiPiwi

; ycomðgÞ ¼P

iwiyiPiwi

; zcomðgÞ ¼P

iwiziPiwi

: ð9Þ

Observe that by construction the center of mass rcomðgÞ does notdepend on the number of spatial meshes.

The analysis of the center of mass estimator for the reactor con-figurations obtained by keeping the number of neutrons N fixedand varying the box size L is illustrated in Fig. 6 (bottom), wherewe display xcomðgÞ. Since the source for the power iteration is ini-tially placed at the center of the box, there is no appreciable con-vergence phase for xcomðgÞ. The evolution of xcomðgÞ clearlymirrors the spatial fluctuations of the neutron population: whenL is small, and the population is uniformly distributed within thebox, xcomðgÞ fluctuates around the symmetry center (here,x ¼ y ¼ z ¼ 0), and the fluctuations are rather mild. As L increases,

Fig. 7. Homogeneous cube reactor. Monte Carlo power iteration as a function ofgenerations g, for three different initial population sizes N and fixed L ¼ 400 cm. Theguess source at g ¼ 0 consists of N neutrons located at the center of the cube. Poweriteration is run for 1000 generations. Top. The behavior of the generalized Shannonentropy S�

1;0;0ðgÞ. Red curve: N ¼ 105; green curve: N ¼ 104; blue curve: N ¼ 103.Bottom. The behavior of the center of mass xcomðgÞ. Red curve: N ¼ 105; green curve:N ¼ 104; blue curve: N ¼ 103. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868 861

the effects of spatial correlations becomes stronger, and the evolu-tion of xcomðgÞ becomes increasingly erratic. The qualitative behav-ior of xcomðgÞ is closely related to that of S�

1;0;0ðgÞ, apart from thenormalizing factor that is imposed by construction in the Legendrepolynomials for S�

u;v ;wðgÞ.A similar behavior is again found for xcomðgÞ when decreasing

the number N of simulated neutrons per generations at fixed reac-tor size L, as shown in Fig. 7 (bottom). The fluctuations of xcomðgÞincrease by decreasing N, and the qualitative evolution of xcomðgÞis strikingly similar to that of S�

1;0;0ðgÞ. For symmetry reasons, theresults for ycomðgÞ and zcomðgÞ are (statistically) identical to thoseobtained for xcomðgÞ and will thus not be shown here.

2 This mechanism has been first introduced in the theoretical ecology (Zhang et al.,1990; Meyer et al., 1996), where similar large-scale constraints have been shown toquench the wild fluctuations in the number of individuals that are expected for anunconstrained community.

4. The impact of clustering on the spatial moments

A deeper understanding of the qualitative behavior of the gen-eralized entropy and by the center of mass that we have observedin the previous Section can be achieved by relating the features ofthese estimators to the key physical parameters that govern theevolution of the Monte Carlo power iteration. This is actually pos-sible by resorting to the theory of branching stochastic processes.Since the features of the generalized entropy are almost identicalto those of the center of mass, for our analysis in the followingwe will focus on this latter.

The nuclear reactor model described above can be conceptuallyrepresented as a collection of N particles undergoing scattering,reproduction and absorption within a homogeneous box of finitevolume V, with reflecting (mass-preserving) boundaries. In orderto keep notation simple, and yet retain the key ingredients of themodel, we will approximate the exponential paths of the neutronsby regular Brownian motion with a constant diffusion coefficient D(in other words, we are assuming that the diffusion approximationholds). For the same reason, instead of working with discrete gen-erations we will introduce a continuous time t. The diffusingwalker undergoes a birth–death event at rate b ¼ vRf : the neutrondisappears and is replaced by a random number k of descendants,

distributed according to a law qk with average �mf ¼P

kkqk. We willassume that exactly two neutrons are emitted at fission.

In the Monte Carlo power iteration, some population controlmechanisms are typically applied (such as Russian roulette andsplitting), and the neutron population is typically normalized atthe end of each generation in order to prevent the number of indi-viduals from either exploding or shrinking to zero. For our aims,the effect of such population control mechanisms on our modelcan be mimicked by imposing that the total number N of neutronsin V is preserved. The simplest way to ensure a constant N is to cor-relate fission and capture events (Zhang et al., 1990; Meyer et al.,1996): at each fission, a neutron disappears and is replaced by 2descendants, and 1 neutron is simultaneously captured (i.e.,removed from the collection) in order to ensure the conservationof total population.2

Analysis of this model shows that the evolution of the neutronpopulation is governed by two distinct time scales: a mixing timesD / V2=3=D and a renewal time sR / N=b. The quantity sD physi-cally represents the time over which a particle has explored thefinite viable volume V by diffusion. The emergence of the timescale sD is a distinct feature of confined geometries having a finitespatial size: for unbounded domains, sD ! 1. The quantity sR rep-resents the time over which the system has undergone a popula-tion renewal, and all the individuals descend from a singlecommon ancestor. When the concentration N=V of individuals inthe population is large (and the system is spatially bounded), itis reasonable to assume that sR > sD.

Consider then a collection of N such particles moving aroundrandomly, whose vector positions riðtÞ; i ¼ 1; . . . ;N, are recordedat time t. The spatial behavior of the individuals can be character-ized in terms of several moments, namely, the square center ofmass

hr2comiðtÞ �1N

Xi

riðtÞ�����

�����2* +

; ð10Þ

the mean square displacement

hr2iðtÞ � 1N

Xi

h riðtÞj j2i; ð11Þ

and the mean square distance between pairs of particles

hr2piðtÞ �1

NðN � 1ÞXi;j

jriðtÞ � rjðtÞj2D E

: ð12Þ

Brackets denote the expectation with respect to the ensemble ofpossible realizations.

By construction, these three quantities are related to each other.By developing the series in the definitions above, we can in partic-ular express the center of mass as a function of the mean squaredisplacement and of the particle pair distance (Meyer et al.,1996), namely,

hr2comiðtÞ ¼ hr2iðtÞ � 12N � 1N

hr2piðtÞ: ð13Þ

In the following, we will explicitly compute the center of mass forthe homogeneous reactor and relate its behavior to the modelparameters.

862 M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868

4.1. Neutron density and correlations

Let us denote by nðx; tÞ the instantaneous density of neutronslocated at x at time t. For a critical reactor, the average neutrondensity at a point x reads

nðx; tÞh i ¼ Nqðx; tÞ; ð14Þwhere we have set

qðx; tÞ ¼Z

dx0Qðx0ÞGðx; x0; tÞ: ð15Þ

Here Q is the spatial probability distribution function of the neu-trons at time t ¼ 0, and the Green’s function Gðx; x0; tÞ satisfies thebackward diffusion equation

@

@tGðx;x0; tÞ ¼ Dr2

x0Gðx;x0; tÞ; ð16Þ

with the appropriate boundary conditions (de Mulatier et al., 2015).Assuming that the initial neutron population has a uniform spatialdistribution, we have Q ¼ 1=V , and the average neutron density atany time will be spatially uniform, namely, hnðx; tÞi ¼ N=V . For arbi-trarily distributed sources at t ¼ 0, the neutron density will asymp-totically converge towards hnðx; t ! 1Þi ¼ N=V for long times.

By construction, the average density does not convey any infor-mation concerning the spatial fluctuations of the neutron popula-tion. In order to go beyond the average behavior, we introducethe pair correlation function hðx; y; tÞ between positions x and y,which is proportional to the probability density of finding a pairof particles with the former at x and the latter at y (de Mulatieret al., 2015). It is important to stress that the correlation lengthcan be extracted from the shape of the function h: if h is almost flatin space, then the correlations will have the same relevance at anyspatial site; on the contrary, the presence of a peak in h at shortdistances might reveal clustering phenomena (Zhang et al., 1990;Meyer et al., 1996). The overall intensity of the correlations is sim-ply provided by the amplitude of h. When the particles can onlydiffuse (i.e., if the probability of fission and capture are artificiallysuppressed), the neutron population behave as an ideal gas of Nindependent random walkers, and the corresponding correlationfunction yields

hidðx; y; tÞ ¼ NðN � 1Þqðx; tÞqðy; tÞ: ð17ÞFor the particular configuration where the particles are uniformlydistributed at the initial time, hidðx; y; tÞ ¼ NðN � 1Þ=V2.

For the homogeneous reactor model defined above, the pair cor-relation function h can be exactly computed by resorting to theapproach originally proposed in Meyer et al. (1996) for infinitedomains and later refined by de Mulatier et al. (2015) for boundeddomains. The calculations are developed in Appendix B, and yield

hðx; y; tÞ ¼ N N � 1ð ÞV2 e�bpt þ b

NV

Z t

0dt0e�bpt

0 Gðx; y;2t0Þ ð18Þ

when imposing the initial uniform source Q ¼ 1=V . The quantity bp

is a shorthand for bp ¼ b=ðN � 1Þ. When branching events areabsent (b ¼ 0), hðx; y; tÞ ! hidðx; y; tÞ, since spatial correlations aresuppressed. The integral of the Green’s function appearing in Eq.(18) is bounded thanks to the exponential term, and at long timesthe correlation function converges to an asymptotic shape. Thesame asymptotic behavior is expected for arbitrary initial sources.

4.2. Relating the spatial moments to q and h

The spatial moments of the neutron population defined abovecan be formally expressed in terms of the particle density qðx; tÞ

and of the correlation function hðx; y; tÞ. In particular, for the meansquare displacement we have

hr2iðtÞ ¼Z

jxj2qðx; tÞdx: ð19Þ

As for the mean square distance between pairs of particles, we have

hr2piðtÞ ¼Rdx

Rdyjx� yj2hðx; y; tÞRdx

Rdyhðx; y; tÞ ; ð20Þ

which is to be compared to the ideal average square distance of anuncorrelated population uniformly distributed in the viable volume,namely,

hr2piid ¼1V2

Zdx

Zdyjx� yj2 ¼ 1

2V

23: ð21Þ

Deviations of hr2piðtÞ from the ideal behavior hr2piid allow quantifyingthe impact of spatial clustering (Meyer et al., 1996; de Mulatieret al., 2015).

4.3. Analysis of the homogeneous reactor

Let us now consider the case of the homogeneous cube reactorintroduced above, with size L and V ¼ L3. At the boundaries, weimpose reflecting (Neumann) conditions. The Green’s function forthis system reads (Grebenkov and Nguyen, 2013)

Gðx; x0; tÞ ¼X1i;j;k

uiðuÞuyi ðu0ÞujðvÞuy

j ðv0ÞukðwÞuykðw0Þe�ai;j;kt;

where

uqðfÞ ¼ cosqpfL

� �ð22Þ

are the eigen-modes of the Laplace operator and

ai;j;k ¼ DpL

� �2i2 þ j2 þ k2

� �ð23Þ

are the associated eigenvalues. The vector x is defined by its compo-nents, namely, x ¼ u;v;wf g. By inspection, the mixing time of theneutron population is identified with sD / L2=ðp2DÞ. The functionsuy

qðfÞ are found by imposing ortho-normalization of the eigen-modes, which yields

uyqðfÞ ¼

1Lcos

qpfL

� �ð24Þ

for q ¼ 0, and

uyqðfÞ ¼

2Lcos

qpfL

� �ð25Þ

for q P 1.Assuming a uniform spatial distribution Q ¼ 1=L3 at time t ¼ 0,

the average density simply reads

hnðx; tÞi ¼ N

L3: ð26Þ

The mean square displacement can be easily computed, and yields

hr2iðtÞ ¼Z

jxj2qðx; tÞdx ¼ L2

4: ð27Þ

From Eq. (18) we get the pair correlation function

hðx; y; tÞ ¼ NðN � 1ÞL6

þ bN

L3X1i;j;k

uiðuÞuyi ðqÞujðvÞuy

j ðrÞukðwÞuykðsÞ

� 1� e�ð2ai;j;kþbpÞt

2ai;j;k þ bp;

Table 1Measured and predicted values for the fluctuations of the x component of the centerof mass for the homogeneous cube reactor. Here the number of neutrons pergeneration is kept fixed at N ¼ 104, and the reactor size L varies.

L [cm] ‘2 [cm2] r xcom [cm] r̂ x

com [cm]

100 174 1.2 1.5200 185 4.7 5.9400 192 18.3 19.9

Table 2Measured and predicted values for the fluctuations of the x component of the centerof mass for the homogeneous cube reactor. Here the reactor size L is kept fixed atL ¼ 400 cm, and the number N of neutrons per generation varies.

N r xcom [cm] r̂ x

com [cm]

105 5.8 7.3

104 18.3 19.9

103 57.8 64.4

M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868 863

where the sum is extended to all indexes i P 0; j P 0; k P 0, excepti ¼ j ¼ k ¼ 0, and we have set x ¼ u;v;wf g and y ¼ p; r; sf g. The ser-ies appearing at the right-hand side is bounded, and for timest � sD we obtain the asymptotic shape of the pair correlationfunction

h1ðx; yÞ ¼ NðN � 1ÞL6

þ bN

L3X1i;j;k

uiðuÞuyi ðqÞujðvÞuy

j ðrÞukðwÞuykðsÞ

2ai;j;k þ bp:

ð28ÞAs for the average square distance, at time t ¼ 0 we have

hr2pið0Þ ¼ hr2piid, as expected. The asymptotic behavior of hr2piðtÞ attimes t � sD can be computed exactly based on Eqs. (28) and(20), and reads

hr2pi1 ¼ limt!1

hr2piðtÞ ¼ 12Dbp

1�ffiffiffiffiffiffiffiffiffiffi8DbpL

2

stanh

ffiffiffiffiffiffiffiffiffiffibpL

2

8D

s0@ 1A24 35: ð29Þ

By recalling the definitions of the mixing time sD and the renewaltime sR, we can rewrite Eq. (29) as

hr2pi1 ¼ 12L2

p2

sRsD

1�ffiffiffiffiffiffiffiffiffiffiffiffiffi8p2

sRsD

stanh

ffiffiffiffiffiffiffiffiffiffiffiffiffip2

8sDsR

s0@ 1A24 35; ð30Þ

which shows that the impact of the spatial correlations is ruled bythe dimensionless ratio between the renewal time and the mixingtime. Intuitively, we expect the effects of the correlations to bestronger when the typical time scale of fission renewal is in compe-tition with diffusive mixing (i.e., sR ’ sD), and to be weaker whendiffusive mixing is faster than renewal (i.e., sD � sR).

Let us first consider the case of a system where the effects of thespatial correlations induced by clustering are very weak (i.e.,sR ! 1), which is obtained for a very large number of particlesor a vanishing fission rate. By taking the limit of b ! 0 or equiva-lently N ! 1, we have

hr2pi !L2

2¼ hr2piid ð31Þ

and we recover the ideal case corresponding to uncorrelated trajec-tories. In this case, the center of mass of the population obeys

hr2comiid ¼ hr2i � 12N � 1N

hr2piid ¼L2

4N¼ 1

Nhr2iid; ð32Þ

which basically means that for a collection of independent particlesthe mean square displacement of the center of mass is equal to themean square displacement of a single particle of the collection,divided by the number of particles.

Consider now a finite reproduction rate b and a large but finitenumber of particles N � 1. In this case, we can expand Eq. (30) forsD � sR, which yields

hr2pi1 ’ L2

21� p2

20sDsR

þ �

¼ hr2piid 1� 120

L2

NM2 þ " #

; ð33Þ

where we have used the definition of the migration area M2 ¼ D=b.This result relates the typical inter-particle distance to the physicalparameters of the reactor model, namely, N; L, and M2, and impliesin particular that hr2pi1 will be smaller than in the uncorrelated casebecause of the effects of spatial clustering.

As for the center of mass, we finally get

hr2comi1 ¼ hr2i � 12N � 1N

hr2pi1 ’ L2

4N1þ p2

20NsDsR

þ �

¼ hr2comiid 1þ 120

L2

M2 þ " #

; ð34Þ

which again relates the mean square displacement of the center ofmass to the physical parameters N; L, and M2. In particular, hr2comi1will be larger than that of an uncorrelated system. The correctionfactor increases for increasing system size L, and decreases forincreasing migration area M2, as expected on physical grounds.

4.4. Numerical findings for the homogeneous reactor

It is interesting to compare the numerical values of hr2comi1 pre-dicted by Eq. (34) to the behavior of the center of mass displayed inFig. 6 (bottom) and Fig. 7 (bottom). In particular, rcom ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihr2comi1

pintuitively represents the typical amplitude of the fluctuations ofthe center of mass around the average. The value ofM2 can be esti-mated by computing ‘2 in each Monte Carlo simulation and thensetting M2 ¼ ‘2=6 according to diffusion theory. Because of thegeometrical symmetries of the cube configuration, along a givenaxis, say x, we get r x

com ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihr2comi1=3p

.

When the number of particles is kept fixed at N ¼ 104, and thesize L varies, from Eq. (34) we get for r x

com the values reported inTable 1. These findings are entirely consistent with the typical sizeof the fluctuations of the x component of the center of massobserved in Fig. 6 (bottom): for comparison, the standard deviationr̂ x

com of the recorded statistical series is also reported in Table 1.When the reactor size is kept fixed at L ¼ 400 cm (‘2 ’ 192 cm2)

and N varies, from Eq. (34) we get the r xcom values reported in

Table 2. Comparison with Fig. 7 (bottom) shows that these predic-tions are again entirely consistent with the typical size of the fluc-tuations of the x component of the center of mass observed in ourMonte Carlo simulations, whose standard deviation r̂ x

com is alsoreported in Table 2.

5. Application to realistic reactor configurations

In the previous Sections, we have applied our analysis to a sim-plified reactor model. In the following, we would like to ascertainwhether the conclusions that were drawn in the case of space-and energy-independent neutron transport actually carry over tomore realistic configurations. To this aim, we will examine thebehavior of Monte Carlo power iteration for a fuel rod and for a fullreactor core. Monte Carlo simulations have been performed byresorting to the reference code TRIPOLI-4�, developed at CEA (Brunet al., 2015; Tripoli-4 Project Team, 2008).

Fig. 9. The fuel rod. The behavior of the measured Shannon entropy SðgÞ duringMonte Carlo power iteration as a function of generations g, for a fixed fuel celllength L ¼ 400 cm and different numbers N of neutrons per generation. The guesssource at g ¼ 0 consists of N neutrons uniformly distributed along the fuel pin.Power iteration is run for 8� 103 generations. Upper red curve: N ¼ 5� 105 ;central green curve: N ¼ 5� 104 ; lower blue curve: N ¼ 104. The mesh chosen forthe entropy computation is fixed to 8 identical meshes along the total length of thefuel rod. The theoretical value of the entropy given by Eq. (5) reads S1 ¼ 3:0. (Forinterpretation of the references to colour in this figure caption, the reader isreferred to the web version of this article.)

864 M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868

5.1. A fuel rod

Let us begin by considering a fuel rod. This configuration iscomposed of UO2 fuel at 3:25% enrichment, with radius0:407 cm. The fuel pellets are enclosed in a Zircaloy cladding ofouter radius 0:477 cm, and a water moderator surrounds the clad-ding. All materials are kept at 300 K. Reflective boundary condi-tions have been applied on the system, so that the expectedequilibrium distribution for the flux is axially flat in space. The cho-sen nuclear data library is ENDF/B-VII.0.

The measured entropy function SðgÞ for the fuel rod is shown inFig. 8 for fixed number N of simulated particles per generation andvarying fuel rod sizes L, and in Fig. 9 for fixed L and varying N,respectively. When L is varied and N ¼ 104, the entropy functionconverges to a stationary value after about m ¼ L2=‘2 ’ 100 gener-ations for L ¼ 100 cm (‘2 ’ 100 cm2), m ¼ L2=‘2 ’ 370 generationsfor L ¼ 200 cm (‘2 ’ 108 cm2), and m ¼ L2=‘2 ’ 1400 generationsfor L ¼ 400 cm (‘2 ’ 112 cm2). Our predictions are consistent withthe numerical findings for SðgÞ. The entropy at convergencedepends on the number N of simulated particles per generations,and an asymptotic value SN is eventually reached in the limit oflarge N, with a / 1=N scaling.

The behavior of S�0;0;1ðgÞ for the fuel rod with fixed N fixed and

varying L is illustrated in Fig. 10 (top), z being the axis alignedalong the rod. As L increases, the fluctuations of S�

0;0;1ðgÞ becomestronger. A similar behavior is found when decreasing the numberN of simulated neutrons per generations at fixed reactor size L, asshown in Fig. 11 (top): the fluctuations increase by decreasing N.The analysis of the center of mass estimator for the fuel rod withfixed N and varying L is illustrated in Fig. 10 (bottom), where wedisplay zcomðgÞ: when L is small, zcomðgÞ fluctuates around the sym-metry center z ¼ 0, and the fluctuations are rather mild. As Lincreases, the evolution of zcomðgÞ becomes increasingly erratic.

Fig. 8. The fuel rod. The behavior of the measured Shannon entropy SðgÞ duringMonte Carlo power iteration as a function of generations g, for three different fuelrod lengths L and fixed number N of neutrons per generation. The guess source atg ¼ 0 consists of N ¼ 104 neutrons uniformly distributed along the fuel rod. Poweriteration is run for 2� 104 generations. Upper red curve: L ¼ 100 cm; central greencurve: L ¼ 200 cm; lower blue curve: L ¼ 400 cm. The mesh chosen for the entropycomputation is fixed to 8 identical meshes along the total length of the fuel rod. Thetheoretical value of the entropy given by Eq. (5) reads S1 ¼ 3:0. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

Fig. 10. The fuel rod. Monte Carlo power iteration as a function of generations g, forthree different reactor sizes L and fixed number N of neutrons per generation. Theguess source at g ¼ 0 consists of N ¼ 104 neutrons uniformly distributed alongthe fuel rod. Power iteration is run for 2� 104 generations. Top. The behavior of thegeneralized Shannon entropy S�

0;0;1ðgÞ. Red curve: L ¼ 100 cm; green curve:L ¼ 200 cm; blue curve: L ¼ 400 cm. Bottom. The behavior of the center of masszcomðgÞ. Red curve: L ¼ 100 cm; green curve: L ¼ 200 cm; blue curve: L ¼ 400 cm.(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 11. The fuel rod. Monte Carlo power iteration as a function of generations g, fora fixed fuel length L ¼ 400 cm and different numbers N of neutrons per generation.The guess source at g ¼ 0 consists of N neutrons uniformly distributed along thefuel rod. Power iteration is run for 8� 103 generations. Top. The behavior of thegeneralized Shannon entropy S�

0;0;1ðgÞ. Red curve: N ¼ 5� 105; green curve:N ¼ 5� 104; blue curve: N ¼ 104. Bottom. The behavior of the center of masszcomðgÞ. Red curve: N ¼ 5� 105; green curve: N ¼ 5� 104; blue curve: N ¼ 104. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Table 4Measured and predicted values for the fluctuations of the z component of the centerof mass for the fuel rod. Here the fuel rod size size L is kept fixed at L ¼ 400 cm, andthe number N of neutrons per generation varies.

N r zcom [cm] r̂ z

com [cm]

105 2 1.7

5� 104 6.2 4.9

104 13.8 15.4

M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868 865

The qualitative behavior of zcomðgÞ is similar to that of S�0;0;1ðgÞ. A

similar behavior is found for fixed L and varying N, as shown inFig. 11 (bottom).

In order to relate the numerical simulation results for zcomðgÞ tothe theory developed above, we have solved Eq. (18) for a one-dimensional homogeneous rod geometry, where neutrons areallowed to only move back and forth along a line of size L. Neu-mann boundary conditions are applied to the ends of the segment.The rod model equations yield

hz2comi1 ¼ hz2comiid 1þ 120

L2

M2 þ " #

; ð35Þ

with hz2comiid ¼ L2=ð12NÞ and M2 ¼ ‘2=2. For the case of fixed

N ¼ 104 and varying L, the rod model formula yields the r zcom values

reported in Table 3. Comparison with Fig. 10 (bottom) shows thatthese predictions are in good agreement with the standard devia-tion r̂ z

com of the recorded statistical series, which is also reportedin Table 3.

When L ¼ 400 cm and N is varied, Eq. (35) yields the r zcom values

reported in Table 4. These predictions are again in good agreementwith the standard deviation r̂ z

com of the recorded statistical seriesshown in Fig. 11 (bottom), which is also reported in Table 4.

5.2. The Hoogenboom–Martin benchmark

Let us now turn our attention to a full-scale reactor core model,namely, the Hoogenboom–Martin benchmark (Hoogenboom and

Table 3Measured and predicted values for the fluctuations of the z component of the centerof mass for the fuel rod. Here the number of neutrons per generation is kept fixed atN ¼ 104, and the fuel rod size L varies.

L [cm] ‘2 [cm2] r zcom [cm] r̂ z

com [cm]

100 100 1 0.9200 108 2.6 3.3400 112 13.8 15.4

Martin, 2010). This benchmark considers a simplified PWR coremade of 241 identical fuel assemblies. The fuel composition is rep-resentative of a typical depleted core configuration. Two modera-tor zones are used so as to model the decreasing bottom to topcoolant density. We will focus on the cold zero power core config-uration (CZP: all materials are at room temperature, 300 K), andthe chosen nuclear data library is ENDF/B-VII.0 (Chadwick,2006).

By analogy with the simulations that were carried out for thehomogeneous cube reactor, we will first examine the evolutionof the fission site distribution as a function of the number of gen-erations, for different values N of simulated neutrons per genera-tion. The results for an axial cut at mid-plane are displayed inFig. 12. We have chosen a uniform neutron source at g ¼ 0. WhenN ¼ 105, the population evolves toward the fundamental eigen-mode, and the fluctuations around the average density are rathermild. When N decreases to N ¼ 104, the impact of spatial correla-tions becomes stronger, and for even smaller N ¼ 5� 103 neutronsare clearly clustered. Apart from geometrical effects due to theshape of the reactor core, the behavior of the Monte Carlo poweriteration of the Hoogenboom–Martin reactor model is not entirelydissimilar to that of the homogeneous cube.

The corresponding entropy function SðgÞ is shown in Fig. 13.The mean square displacement per generation has been estimatedwithin the Monte Carlo power iteration and reads ‘2 ’ 356 cm2.

Fig. 12. The Hoogenboom–Martin benchmark. Distribution of fission sites duringMonte Carlo power iteration as a function of generations g, for three different initialpopulation sizes N. The guess source at g ¼ 0 consists of N neutrons uniformlydistributed across the reactor. Power iteration is run for 12� 104 generations. Top:N ¼ 105; center: N ¼ 104; bottom: N ¼ 5� 103.

Fig. 13. The Hoogenboom–Martin benchmark. The behavior of the measuredShannon entropy SðgÞ during Monte Carlo power iteration as a function ofgenerations g, for different initial population sizes N. The guess source at g ¼ 0consists of N neutrons uniformly distributed in the core. Power iteration is run for12� 104 generations. Upper red curve: N ¼ 105; central green curve: N ¼ 104;lower blue curve: N ¼ 5� 103. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 14. The Hoogenboom–Martin benchmark. Monte Carlo power iteration as afunction of generations g, for three different initial population sizes N. The guesssource at g ¼ 0 consists of N neutrons located at the center of the cube. Poweriteration is run for 12� 104 generations. Top. The behavior of the generalizedShannon entropy S�

0;0;1ðgÞ. Red curve: N ¼ 105; blue curve: N ¼ 5� 104; yellowcurve: N ¼ 104; green curve: N ¼ 5� 103. Bottom. The behavior of the center ofmass zcomðgÞ. Red curve: N ¼ 105; blue curve: N ¼ 5� 104; yellow curve: N ¼ 104;green curve: N ¼ 5� 103.

Table 5Measured and predicted values for the fluctuations of the z component of the centerof mass for the Hoogenboom–Martin benchmark. The number N of neutrons pergeneration varies.

N r zcom [cm] r̂ z

com [cm]

105 3.5 1.8

5� 104 5 2.2

104 11.3 5.8

5� 103 15.9 8.6

866 M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868

Similarly as in the case of the homogeneous cube reactor, theentropy function converges to a stationary value after aboutm ¼ H2=‘2 ’ 380 generations, where we have used a characteristicreactor size H ’ 366 cm corresponding to the active fuel length(and approximately to the core diameter). The entropy at conver-gence depends on the number N of simulated particles per gener-ations, and an asymptotic value SN is eventually reached in thelimit of large N, with a / 1=N scaling.

The behavior of the spatial moments of the entropy functionand of the center of mass for the Hoogenboom–Martin benchmarkare displayed in Fig. 14 (top) and (bottom), respectively. When N islarge, the S�

0;0;1ðgÞ is close to zero, and the fluctuations increase bydecreasing N. Similar results are found for the x and y axis,although geometrical effects come into play due to the cylindricalsymmetry. The fluctuations of zcomðgÞ also increase by decreasing N,and the qualitative evolution of zcomðgÞ is very close to that ofS�0;0;1ðgÞ. Eq. (18) can be developed for a homogeneous cylinder

with Neumann boundaries, which conceptually corresponds toassuming that the water reflector surrounding the reactor core actsas a perfect mirror for neutrons. In this case, for the z component ofthe center of mass fluctuations we would get

hz2comi1 ¼ hz2comiid 1þ 120

H2

M2 þ " #

; ð36Þ

with hz2comiid ¼ H2=ð12NÞ. Numerical estimates for r zcom are reported

in Table 5, where we used M2 ¼ ‘2=6 ’ 60 cm2. Comparison withFig. 14 (bottom) shows that these predictions systematically over-estimate (roughly by a factor of 2) the standard deviation r̂ z

com ofthe recorded statistical series, which is also reported in Table 5.

Deviations of the theoretical formula in Eq. (34) from theobserved behavior of the fluctuations are mostly due to theapproximation of entirely neglecting the effects of leakages inour simple model. Nonetheless, the order of magnitude of the fluc-tuation amplitude and the 1=

ffiffiffiffiN

pscaling are correctly captured by

Eq. (34).

6. Conclusions

The statistical behavior of the neutrons in Monte Carlo poweriteration has been analyzed within the framework of branchingstochastic processes. In particular, we have shown that it is possi-ble to relate the spatial distributions of neutrons to the key param-eters of the simulated configuration, namely, the number ofneutrons per generation, the system size, and the migration area.By resorting to a simple homogeneous cube reactor model, wehave illustrated some possible shortcomings of the entropy func-tion in detecting the convergence of the neutron population toequilibrium. We have supported our investigation by analyzingthe higher moments of the entropy and the center of mass of thepopulation, whose behavior is directly related to the spatial fluctu-ations of the neutron population.

A deeper understanding of the spatial behavior of the neutronpopulation during Monte Carlo power iteration has been achievedby resorting to the theory of branching processes, which allowsformally relating the spatial moments (the square pair distanceand the square center of mass) to the system parameters. Thedeveloped formalism is amenable to exact results (within the dif-fusion approximation) that are in fairly good agreement with thebehavior observed in the homogeneous cube model.

We have also tested our theoretical findings on more realisticreactor configurations, including a full reactor core and a fuelrod. For the former, we have shown that the formulas derived for

M. Nowak et al. / Annals of Nuclear Energy 94 (2016) 856–868 867

a homogeneous cylindrical reactor with reflecting boundaries cancapture the scaling of the center of mass fluctuations with respectto the number of particles per generation, although our predictionsglobally overestimate the fluctuation amplitude. We conjecturethat this discrepancy is due to the approximation of havingassumed reflecting boundaries: more sophisticated formulasallowing for arbitrary boundary conditions would be needed, andthis issue will be the subject of future research work. Finally, inthe case of the fuel rod (with reflective boundaries) our simplifiedmodel is actually capable of correctly assessing the amplitude ofthe center of mass fluctuations and their scaling with respect tothe number of simulated particles per generation and to the fuelrod length.

Acknowledgments

E.D., M.N. and A.Z. thank AREVA and Electricité de France (EdF)for partial financial support. J.M., B.F. and K.S. thank the Consor-tium for Advanced Simulation of Light Water Reactors (CASL), anEnergy Innovation Hub for Modeling and Simulation of NuclearReactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725, for providing partial financial support.

Appendix A. Expectation of the Shannon entropy

We develop the calculations for the expected Shannon EntropySN as defined in Eq. (1). Assume that there are B tally regions in thesystem, with N identical neutrons. The probability for each neutronto be found at region i is Pi. After each simulation, Pi is estimated as

bPi ¼ kiN

ðA:1Þ

where ki is the number of neutrons in region i. Therefore, for theexpected entropy we have

SN ¼ �EXBi¼1

bPi log2ð bPiÞ" #

¼ log2ðNÞ �1N

XBi¼1

E kilog2ðkiÞ½ �: ðA:2Þ

For each region i; ki represents the number of occurrences of a neu-tron fallen in this region. As such, it can be expressed as a sum of Nbinomial random variables. Therefore, we obtain

E kilog2ðkiÞ½ � ¼XNki¼0

N

ki

� �ð1� PiÞN�ki Pki

i kilog2ðkiÞ: ðA:3Þ

For the homogeneous reactor model considered in Section 2, wesimply have Pi ¼ 1=B and SN reduces to

SN ¼ log2ðNÞ �B

NBN

XNki¼0

N

ki

� �ðB� 1ÞN�ki kilog2ðkiÞ: ðA:4Þ

Appendix B. The pair correlation function

For the sake of completeness, we briefly report here the argu-ment developed in de Mulatier et al. (2015) for the derivation ofthe pair correlation function h. Suppose that a given time t wechoose a pair of (distinct) neutrons located at positions x and y,respectively. We would like to determine whether these particlecome from a common ancestor, i.e., a fission event occurred at atime 0 < t0 < t in the past, in which case the two neutrons are cor-related. In our model, we have iposed that the number of individ-uals in the population is exactly preserved. Then, the fraction ofnew paris of individuals that can be produced during a time inter-val ðt0; t0 þ dtÞ is given by bdt=ðN � 1Þ (Meyer et al., 1996).

The probability for two selected particles not to share a com-mon ancestor in the past is e�bpt . It follows thus that the probabilitydensity for the occurrence of an ancestor at time t0 is

wtðt0Þ ¼ bpe�bpðt�t0Þ; ðB:1Þ

when the observation time is t. We have denoted bp ¼ b=ðN � 1Þ. Itis convenient to split h into the contributions of uncorrelated and

correlated events, namely, h ¼ hð1Þ þ hð2Þ. For the former, by con-

struction we simply have hð1Þðx; y; tÞ ¼ e�bpt hidðx; y; tÞ, which is pre-cisely the contribution of uncorrelated neutrons. As for thecorrelated events, it can be shown that we have (de Mulatieret al., 2015)

hð2Þðx; y; tÞ ¼ NðN � 1ÞZ t

0dt0

ZVdx0Gðx; x0; t � t0Þ

� Gðy;x0; t � t0Þwtðt0Þqðx0; t0Þ: ðB:2ÞThe pair correlation function h finally yields

hðx; y; tÞ ¼ N N � 1ð ÞV2 e�bpt þ b

NV

Z t

0dt0e�bpt

0 Gðx; y;2t0Þ

when imposing Q ¼ 1=V .

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