Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

202 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsSubstituting the explicit forms of the kernels given in Equations (5.131) and (5.128). weobtain Equation (5.150). Similarly, in the case of the equivalent nonmultiplying game definedin Section 5.III.C, the rules above do not apply; nevertheless, Equation (5.140) holds underthe alternative conditions of Equations (5.84) and (5.87).Condition 2 of the theorem restricts the class of functions that can be used as nonanalogcontribution functions (estimators). The simplest way of satisfying this condition is to choosethe nonanalog estimators identical to the analog ones, i.e., to putIl•) - f r(..)This, however, is not the only possibility. In fact, the condition states that nonanalogcontribution functions should also be partially unbiased. Indeed, in view of Theorem 5.7and Equations (5.134) and (5.141), condition 2 is equivalent to the conditionI 1(P) = KP)where I (P) is the expected score due to a flight from P followed by a collision in an analoggame with the simplest contribution f(P), as given in Equation (5.133). Correspondingly,the analog and nonanalog estimators are interchangeable and they all belong to the class ofpartially unbiased estimators. Therefore, in nonanalog games that satisfy the conditions ofTheorem 5.8, there is no need to distinguish the estimators from the analog ones. Suchgames will be called partially unbiased nonanalog games.Theorem 5.8 establishes the conditions under which the expected scores due to a unitweight starter in the analog and nonanalog games are equal. These conditions, however,ensure an unbiased nonanalog estimation only if the source densities (from which the particlesstart) are identical. In this case, the weights of the starters may be chosen equal and theequality of the expected scores per history(ies) calls forth the equality of the final scores inthe two games. In other words, the theorem gives the generation rules of the weights duringthe simulation, but it does not fix the weight of a nonanalog starter. If the nonanalog sourcedensity differs from the analog source, the statistical weight of a starter should depend onthe difference of the actual source density from the analog one. The generation rule of thestarting weight is established in the following.Theorem 5.9 — A nonanalog game will yield the same final expected score as theanalog game if the weight generation rules of Theorem 5.8 are satisfied and the weight ofa starter at P in the nonanalog game is chosen asW = w g(P) = Q(P)/Q(P)where Q(P) and Q(P) are the source densities in the analog and nonanalog games, respectively.Proof. The final expected score in a nonanalog game isR -JdPQ(P)WM 1(P)If W is chosen according to the theorem and the conditions of Theorem 5.8 are met, thenM 1(P) -M 1(P)