Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

163This is the general moment equation concerning nonmultiplying nonanalog ;forms of the equation with selected functions F(s) will be detailed in the next icturning to the discussion of the moment equations, however, a number of idestablished by clarifying the role of the statistical weights in the contributiorin the expectation.I _1. It is reasonable to assume that the contribution functions f are linear homogeneousfunctions of the particles' weights, i.e., if a particle undergoing an event with a weightof unity contributes to the total score a value f, then the contribution of a particleWITHa weight W in the same event is Wf. In short,f(P,P',W) =Wf(P,P')(5.5 i)f,(P',W a ) =Wf 11(P')(5.52)andf s(P',P",W") = W"f s(P',P")(5,53)2. Similarly, the expectation of any linear combination of some functions is the linearcombination of the expectations of the functions:MJS a,F,(s)J(P,W) = V 3iMjF 1(S)KP,W) (5.54)3. Since the score in a history is proportional to the starting weight of the particle, thescore probability satisfies the relation*ir(P,l,s)ds = 7r(P,W,Ws)d(Ws)Consequently, for the expectation of any functionM{F(s)}(P,W) = M{F(Ws)}(P, 1) (5-55)C SPECIAL CASES: EXPECTATION AND SE(OM) MOMENT OF THESCOREIn the majority of the applications, the score function F(s) takes on very simple forms.The most important among them areF(s) = s rwhich give the r-th moment of the score. Trivially, for r = O (F = 1), the constant M„ ==M{1} = i satisfies Equation (5.50), which is not surprising since, TT being normalized tounity:M 0= ds TT(P,1,s) = 1This relation and therefore also Equation (5.55) is only true if the simulation is independent of the statistics!weight of the particle. In the majority of the cases discussed in this chapter, it is indeed so. An example ofthe opposite case will be given in Section 5.III.D in connection with the splitting procedure.