Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

316 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhere, for the sake of brevity, we putf(i) = T-(P 4. ,,P 1)The term in braces is rewritten asA n, ,(P 0,P;,...,P;, t,) = [W'(n + i) - llf(n + 1)j 2 2 iW'(i) - l]f(i) + [W'(n + i) - l]f(n + 1)| (6.25)Finally, it follows from Equations (6.3), (6.4), (6.22), and (6.23) thatL„(Po,P; P 1111) = W'(n + 1) L n(P 1,,?;,...,?;,,,) (6.26)In order to compare the variances of the independent and correlated games, let usintroduce the relative variance as the ratio of the variance and the square of the expectationof the score:d 2 (P) = D 2 Js 1- S 2XP)ZM 2 Is 1- S 2](P)In the case of independent simulations MJs 1— S 2}(P) tends to zero if the perturbationvanishes, while the limit of the variance isD 2 Js 1- s 2}(P) = M 2(P) + M 2(P) - M 2 (P) - M 2 (P) — 2[M 2(P) - M 2 (P)]which is usually different from zero. Hence, the relative variance of the score differenceas estimated in independent games tends to infinity if the perturbation is vanishing. On theother hand, in a correlated gamed 2 (P) = M{(s, - s 2) 2 }(P)/M 2 {( SL- S 2)} - 1and from Equations (6.9) and (6.22) through (6.24)d 2 (P) = i jdp; JdP 1... JdP n + 1l„(p 0,p;,...,Pi +1)A n+,(P 05P;,...,P n+,)/{it JdPI JdP 1... JdP n + 1L n(P (),P;,...,P;+ 1)iW'(n + 1) - l]f(n + I)J" - 1Obviously, the kernels T and C are characterized by certain material and geometrical quantities(such as cross sections and geometrical distances) and, thus, so is the kernel L ninEquation (6.22). The perturbations in the system appear in the kernels through alteration ofsome of these quantities. If a k(k = 1, 2, . . . , K) represent these quantities in theunperturbed system and S 1are the same quantities in the perturbed system, then we canwriteL n(P 01P;,...,P n+1) -l„(p„,p;,...,P n+1|{aj)