Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

255We have thus completed the construction of the nonanalog kernels that result in a gamewith pointwise (or, better, starterwise) zero variance. The transition kernel follows fromEquations (5.281) and (5.283) asT(P,P') = T(P 1P') [f(P,P') + M 1(PO]ZiVI 1(P) (5.288)The multiplication probability is obtained from Equations (5.285) and (5.286) asC n(P') = Hc n(POrO n(POZM 1(P') (1289)while the postcollision densities are defined by Equations (5.282) and (5.285) asC„(P',P") = C 11(P', F)M 1(FVm n(PO (5.290)whereTr) 11(P') =JdFC 11(POPOM 1(P")We note in passing that according to Equations (5.9) and (5.12), the transition kernel (5.288)can be expressed by the adjoint collision density iJf(P') asT(P 1P') = T(P 1POJf(P 1P') - f(P) +t|/*(P'Vi/M,(P)where f(P) is the weighting function in the reaction rate [Equation (5.2)] to be estimated.It is apparent that the absoiption probability in the nonanalog game is zero since1 - c a(P') = S C n(PO = S HC n(POfTl n(POZM 1(P') - 1.and the nonanalog kernels are normalized to unity. We have thus proved the following.Theorem 5.19 — Given an analog unbiased game with a nonnegative contributionfunction assigned to the intercollision flights. If one chooses the kernels of a nonanaloggame according to Equations (5.288) through (5.290), then the game with the analog contributionfunction will be partially unbiased and any starter will produce an estimate withzero variance.LJTwo comments are to be made here. First, since the kernels are defined through theexpected score which is unknown at the time of the simulation, such a game, of course,cannot be realized. Nevertheless, approximations to the kernels above may substantiallyreduce the resulting variance, as will be seen in Section 7.III. Second, the game so definedgives a zero-variance estimate of the reaction rate due to a starter from any source pointHowever, it does not garantee zero variance of the total score R unless the nonanalog sourcedensity is specifically chosen. Indeed, the variance of the total score readsD 2 IR] - JdPQ(P)W 2 Mf(P) JIdPQ(P)M 11(P)