Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

314 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsin the two systems, i.e., y = 1, c\ = c„, and f m= 00. Accordingly, conditons (6.19) and(6.20) becomea < 1/2andsup c(P) < (1 - 2a)/(l - a) 2PIt is easy to see that if in a monenergetic transport problem these conditons are violated,then the variance of the score difference is indeed unbounded. 8 - 6 7 , 8 3 8 4If in some region the survival probability c s(P) is altered so that the total cross sectionremains unchanged, then 7 = 1 and a = 0. If the maximum optical extension of the regionis T 1n, then f mT 1nand condition (6.20) givessup[6 2 (P)/c s(P)j(l e V < 1pFor an infinite homogeneous medium in momoenergetic approximation, this condition reducestoCs < Vc swhich again is not only sufficient, but also necessary for a finite variance. 8In contrast to the example above, it may also happen that condition (6.19) or (6.20)fails to hold for a problem which seems feasible. In such cases, either the conditions aretoo restrictive for the specific problem or the game is not defined properly. In practice, onehas to examine the effects of the approximations applied in the derivation of the conditionsand one also has to consider the possibiltiy of using a perturbed analog game (where thegame is played analog in the perturbed system) instead of an unperturbed analog game, itwill be seen in Section D that in certain cases this change in the simulation does indeedreduce the variance.C. CORRELATED DIFFERENCE ESTIMATORSIn the derivation of the moment Equations (6.1) and (6.9), we assumed that both theunperturbed and perturbed scores result from the same contribution function f(P,P'). Thus,if f(P,P') is a partially unbiased estimator of the unperturbed reaction rate, the change inthe reaction rate due to the perturbation is estimated via the contribution functionAf(P 1P') = (W - l)f(P,P') (6.21)where W is the weight of the nonanalog particle after the flight from P to P'. If P' = P',,the point where the particle enters its i-th collision, then W = W'(i), as given in Equation(6.8). The partially unbiased estimator of the unperturbed reaction rate, f(P,P'), may be anyof those introduced in Chapter 5. VI. The most commonly used are the collision and track -length estimators.It is, however, not necessary that the perturbed and unperturbed scores be estimated bythe same estimator. It can be easily seen 48that the derivations in Sections A and B remainvalid if the score vectorWf(P,P') =(Wf(P 5P'),f(P,P'))