Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

267An Immediate consequence of this equation is that the quantities SF(P') and J 1(P') in Equations(5.320) and (5.323), respectively, are equal. [Note that they represent the expectedfinal score due to a particle entering a collision at P'. This quantity was denoted by N 1(P)in Chapter 5.111.] Introducing the notationA(P) = M 2(P) -Ji 2(P)the difference of the variances in the analog and equivalent games, it satisfies the equationA(P) = J dP'T(P.P')[Ss(P') -JS 2(P')]+ JdP'T(P 5P')[I - c(P')]JdP"C(P',P")[M 2(P") - Mf(P")]+ dP'T(P,P')c(P') dP" C(P',F)A(P")Now, A(P) > 0 if c(P') < 1 and J 2(P') > J 2(P'). In order to keep the derivation as shortas possible, we do not proceed further in full generality, but we assume thatf a(P') = O 5f s(P',P") = f n(P',P") - f c(P',P")in the analog game, i.e., we consider a game in which no score is assigned to an absorptionand the scores in a collision do not depend on the type of reaction in the collision. Thissimplification is also justified by the fact that the great majority of the commonly usedestimators are such indeed. Thenf*(P',P") =f t(P',P")satisfies Equation (5.326) and^ 2(P') - J 2(P') -- [1 - c(P')]JdP"C(P',P")M 2 (P")+ c f(P') E n(n - l)q n(P')|JdP"C„(P' 5P")[f c(P',P") + M 1(P")])^ (5.327)is positive If c(P') < 1. Thus, we have the following theorem.Theorem 5.22 — The variance of an analog game, with the contribution functions f(P,P')and f c(P',P") is not lower than that of the equivalent nonanalog game with the same contributionfunctions ifc(P') ^ 1A number of comments are proper here.1. Since SF — J 2in Equation (5.327) usually is definitely positive for c(P') < I, it isexpected that the variance of the equivalent game is lower than that of the analoggame even if c(P') is slightly greater than unity.