Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

174 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationscollision density (or flux) of the particles, while the Monte Carlo methods are able to simulatethe physical stochastic process which results in the given average value (expectation). Inthis sense, Monte Carlo methods give an essentially true description of the physical phenomenawhile the transport equation contains only reduced information about the processes.On the other hand, the quantity to be determined by Monte Carlo is related to the averagecollision density, i.e., to the solution of the transport equation (and not to the stochasticprocess that has this average). Therefore, one can forget about the physical background ofthe transport phenomena when modeling weighted integrals of the collision density. In fact,this is the case when using nonanalog Monte Carlo games and also will be done in thissection, where we propose a nonmultiplying game which, from the point of view of theresulting collision density, is equivalent to the multiplying game described in the previoussections.In the derivations above, every particle that may appear in the physical processes isindividually accounted for by using a multiplicative scattering kernel of the form in Equation(5.39). In contrast to that, in the integral transport equation formalism, Equation (5.4), thecollision densities at two successive collision points are connected by the transport kernelK(P",P) in Equation (5.6). Ignoring the physical reality, the collision density described bythe transport equation can be interpreted as some characteristic quantity (let us call it aweight) carried by some hypothetical migrating particle. The particle makes a free flightbetween the points P and P' with a probability density T(P,P') and suffers a scattering fromP' to P" according to the densityC(P',P") - C(P',P")/ dP"C(P',P")At every collision, the weight carried by the particle is changed by a factor equal to theexpected number of secondaries per collision in the physical process:c(P') =dP"C(P',P")Obviously, this interpretation of the transport equation yields a nonmultiplying analog simulationof the transport of the hypothetical particle, and the Monte Carlo game will give thesame expected score as the corresponding physical (multiplying) analog simulation (cf.Chapter 3.II).The equivalent analog game may have the advantage that no branching histories are tobe simulated, and a history consists very likely of fewer events (collisions) than the correspondingmultiplying history together with all its progenies. Therefore, the game offersthe possibility of reducing the computing time per history. (In fact, it is shown in Section5.V.F that it certainly reduces the computing time if the mean number of secondaries percollision in the multiplying game is greater than unity.) It will be seen in Section 5.VIII.Dthat in certain cases it also reduces the variance of the estimate in a history, thus having anet efficiency increasing effect. In most cases, the change of the variance and the computingtime per history are of opposite directions, and the efficiency of the game must usually beestimated by simplified models or numerical experiments (cf. Section 5. VIII.E).The heuristic arguments above are formulated more rigorously in what follows. Let usassume that the collision kernel in the multiplying ("physical") simulation is of the formin Equation (5.74). Let(5.82)