Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

189Inserting the equations above into Equation (5. ill), it is seen that the expected scores inthe two games (with and without splitting) are equal ifJ dP, UP 1P 1)J dP'T(P 1PO[WL(P 5P 1) + WT(P 11P') - Wf(P 1P')+ (W - W)N 1(P')] = 0for the arbitrary density function t(P,P,). This equation ensures an unbiased estimation ofthe expected score M 1(P) in a game where, at most, one splitting is played in the first flightLet us now consider a game where two splittings are allowed in the first flight. Then accordingto the arguments above, the particles that emerge from the first splitting will yield the sameexpected score independently of whether the second splitting is played if the equality aboveholds; that is, under this condition, the games with one and two splittings are equivalentfrom the point of view of the expected score and since the game with a single splitting isequivalent to the game without splitting, so is the game with two splittings. Recursiveapplication of the arguments above shows that introduction of an arbitrary number of splittingprocedures into the first flight leaves the expected score unchanged. Furthermore, since thesplit fragments are simulated independently and splitting has an effect on the future contributionsonly, any flight where a particle is split can be considered the "first flight". Theconclusions are summarized in the following theorem.Theorem 5.6 — A game with geometrical splitting results in the same expected scoreas the game which is played with identical kernels and contribution functions, but withoutsplitting, if the following conditions hold:1. The weights of the split fragments are such thatW = Wwhere W is the weight of a particle at P' that starts a flight at P with a weight Wand enters its next collision at P' without suffering a splitting.2. The contribution assigned to the flight from the starting point P to the splitting site P 1isWf g(P,P,) = J dP'T(P,P')W'[f(P,P')- f(P,,P')]/j dP'T(P,P')= IdP 1 T(P 1 5POW[IfP 1P') - f(P,,P')] (5.1145nCondition 1 formally means that if a particle starts from P and is split into a number offragments and the fragments all collide at P', then the total expected weight of the fragmentsat P' must be equal to the weight of the original particle when it enters a collision at P''without splitting. In order to make condition 1 less abstract, let us recall the introduction,of statistical weights in Chapter 5.I.D. It was stated that in the majority of practical cases,the bias due to the selection of a free flight from a nonanaiog transition kernel is compensatedfor by multiplying the statistical weight of the particle by some weight factor w(P,P') in .;