Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

251therefore result in finite variance. In this case, the form of the estimator will, in turn,influence the resulting variance.In the first part of this section, we define the kernels of a general multiplying, partiallyunbiased nonanalog game that results in pointwise zero variance, i.e., that define a gamein which the scores in a history depend only on the starting point of the history, but areidentical for every starter from a given point. Then we introduce a nonanalog source densitythat, together with the pointwise error-free game, yields a zero-variance estimate of thereaction rate in question. The schemes proposed in the works referred to above are specialcases of the game to be derived here.In the derivation below, the estimators will be fixed and nonanalog kernels leading tozero variance will be determined. The opposite way, i.e., derivation of suitable contributionfunctions with fixed kernels, seems equally reasonable and. indeed, it will be seen in SectionF that for any analog game, partially unbiased estimators exist that yield zero variance.For the sake of simplicity, let us assume that scores result from intercollision flightsonly, i.e., that the estimator used in the simulation is of the form S{f,{()}}. Without loss orgenerality, we can write the analog collision kernel in the formC(P',F') - c a(P')&(P"-P) + V nc„(P')C 11(P',P")n= )where C 11(P') is the probability that n secondaries are emitted in a collision at P' and thedensity function of the postcollision coordinates in an n-foid multiplication is C n(P' ,P") itis easy to see that by choosingC 1(P') -- c s(P') +CXPOq 1(P')C n(P') - c,(P')q n(P')and replacing C 1(P',P") in Equation (5.74) by[c s(P')C 5(P',P') 4- C 1(POq 1(POC 1(P',P")]/ Cj(P')the collision kernel in Equation (5.74) (for an analog game) takes on the form proposedhere. Similarly, let us write the nonanalog collision kernel (to be determined below) asC(P',P") - c a(P')8(P"-~P) + E "C n(POC 11(P',P")O = 1Consider an analog game with the estimator f(P,P') S= 0 that results in the required expectedscore. The first moment of the analog game is governed by the equationM 1(P) = fdP'T(P,P')f(P,P') + jdP'T(P,P0 E HC N(POJDFC n(P^)M 1(P"). (5.273)Now, let H(P,P') and H n(P') be (for the moment) arbitrary functions. Simple manipulationsyield a first-moment equation equivalent to Equation (5.273):M 1(P) - JdP'T(P,P')|f(P,PO + E nc„(P')[l - H(P 1POH n(POIm 11(PO}+ jdP'T(P,P')H(P,P') E Rc n(POH n(POJdFC 1 1(POF)M 1(P") (5.274)