Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

16!the score probability Equation (5.49) takes on the formTr(X 1U.,s) = T^ 1dx'e!x'x )^8(s - i)M Jc f , , I f 1 ,+ T - T dx'e""'-' 1 - d(x''rr(x',jx',s-- 1)J 2 J-Iwhere the integration is extended from x to X if JX > 0 and from 0 to x if JX < 0.Now, let F(s) be an arbitrary function of the score s such that it can be expanded intoa Taylor series at any value of s. The expectation of F(s) is given in Equation (5.30). Snthis special case, it readsM{F}(x. p.) = JdsF(s)i T(x,|x,s)[If F(s) = s, its expectation is just the expected score due to a particle started from x indirection jx; if F(s) = s 2 , the expectation of F(s) is the second moment of the score producedby the particle.] An equation that describes the expectations of the score function F(s) followsfrom the score probability equation after multiplying it by F(s) and integrating over s. It is,however, disturbing that the argument of the score probability TT on the RHS of the equationcontains s — 1 rather than s; therefore, its integral with F(s) will not yield a moment ofF(s), but, rather, the moment of F(s + 1). In order to establish an equation containingmoments of score functions at identical arguments (this will be important in the general casewhen more complicated functions appear in place of s — 1), let us expand F(s) into a Taylorseries around s — 1:^1 d nF(s) = 2 -TTTF(S ~ D„=o n! ds"With this expansion, the expectation of F(s) follows from the score probability equation asM{F}(x,|x) = •r- dx'e- 0 1 '""H dsF(s)8(s - 1)JjX] JJ-=°C2|'dx'e~ < x '- x , , Hi fdx'e^'-^Jdjx'j d(s - 1)1 ^,, = 0 n!• 3"— F(s - 1)ds"Tr(x',p/,s- I.)= — dx' e •