Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

191It is easy to see that the procedure above grants the fulfilment of the conditions of Theorem5.6.Geometrical splitting, like most of the tricks in Monte Carlo methods, ireducing the variance of the score. This can be investigated on the basismoment equation. However, it is not easy to establish, in general, a second-ir«». > i.» >>similar in form to that concerning a game without splitting because thedepend on the weight of the particles. By arguments like those leading to Etit is easy to show that the score probability TT(P,W,S) for a game with geonsatisfies the equation-rr(P,W,s) ==|dP'T(P,P')dP (t(P,P 1)]p(P,P',W',s) * d(P'.W',s)IdP 1I(P 1P 1)I dQT(P,Q) PiP 1P 1 1W 1S)* Jg 11(P 11W)S(S) + E g k(P,,W) JpTT(P 1 1W^ 1S) (5.115)Let us consider a game in which the splitting probabilities do not depend on the particle''sweight, i.e.,g k(P„W) =g k(P,)Inserting these probabilities into Equation (5.115), multiplying the equation by s 2 , andintegrating over s, the equation for the unit weight second moment M 2(P) follows as below(details of the derivation are given in Appendix 5B):W 2 M 2(P) = dP'T(P 1P') 1 dP, 1(P 1P 1) (W') 2 [P(P 1P')2f(P,P')N,(P') +N 2(P')]dP, 1(P 1P 1) 'dQT(P-Q)] {W 2 P 2 XP 1P 1(5.11.6)+ 2WWP(P 1P 1)M 1(P 1) + W 1MKP 1) + W 2[M 2(P 1) - Mf(P 1)]}where W is the expected total weight of the split fragments as defined in Equation (.5.108),whileandW, = E &(P.) E Wf 111We recall that in the procedure discussed in this section, the functions 1(P 1P 1) and g k(P,W)and the split weights W (i)kare independent of both the kernels and the contribution functionsof the simulation. This means that even if the nonanalog kernels and contributions are fixed,for some reason the functions and weights above can be chosen almost arbitrarily and their