Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

157to be carried by the particles in order to make the simulation unbiased. A typical nonanalogmoment equation will have the following form:WM 1(P) = JdP'T(P,P')W'f(P,P') + J dP'T(P,P')JdP"C(P',HWM 1(P") (5.41)where M 1(P) is the expected score due to a starter from P which has a weight of unity. Thisequation will be obtained after having shown that the first score moment defined in SectionD is a homogeneous function of the statistical weight, i.e., thatM 1(P 1W) =WM 1(P)The nonanalog first-moment equation (5.41) can be interpreted similarly to the analogequation and it reflects that if a starter at P carries a weight W, this weight becomes W'after a flight to P' and W" after the flight and a collision from P' to P". it is also seen thatif the analog partial score is f(P,P'), the same events in the nonanalog game contribute tothe score by W'f(P,P'). Thus, the moment equations reflect the role of the statistical weightsin the scoring procedure, as was detailed in steps 1 through 6 in Section CIn the same Section, three questions were posed concerning nonanalog games which weanswered with the aid of the moment equations. The first question concerns the choice ofthe weights. The answer is again based on the uniqueness of the solution of Equation (5.40):the weights W' and W" are to be chosen such that Equations (5.41) and (5.40) becomeidentical. Having introduced such weights, the resulting equation can be examined from thepoint of view of the feasibility of the nonanalog game; this will answer the second questionin Section C. The considerations outlined above are described in Section 5.V.The third question concerns the relative merits of different games. A general measureof the quality of a game is its efficiency. The inverse of the efficiency was seen to be theproduct of the variance per history and the computing time per history. In most cases, thecomputing time is roughly proportional to the number of collisions (cf. Chapter 5.V.F).The expected number of collisions per history will be seen to satisfy a first-momentequation of the type in Equation (5.40). The second moment of the score will again be seento be governed by a moment equation. Comparing the second-moment equations concerningtwo unbiased games, in certain cases we will be lead to definite statements about the variancerelations of the two games. (cf. Chapter 5.VIII). An efficiency comparison in general is notpossible, but approximate solutions to the moment equations will make it possible to investigatethe efficiency of specific games. These questions will be addressed in Chapter 7.The second-moment equation in an analog game has the typical formM 2(P) = JdP'T(PX^)P(PX') + 2 J dP' T(P,P') f(P,P') J dP"C(P' ,F') M 1(P')+ JdP' T(P,P')JdP"C(P' ,P") M 2(P")This equation again has a reasonable interpretation. Roughly speaking, it expresses the factthat the expected square of the sum of the first flight and collided scores is the sum of therespective expected squares plus twice the expectation of their product. Comparison of suchequations concerning various estimators, f(P,P'), will show the relative merits of the estimators.In what follows, moment equations concerning more and more complex games will bederived with mathematical rigor. The most common unbiased nonanalog games will beintroduced by appropriate transformations of the moment equations. The bulk of the den-