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survey of techniques in transport (Because of this focus, the reader should note that the reference list is skewed toward publications associated with MCNP; see Lux and Koblinger [22] for a more representative reference list) The review process for this chapter indicated that transport Monte Carlo differs in signi cant ways from the Monte Carlo used in other elds, for example operations research Not only are the terminology and methods quite different in many cases, there is sometimes even a conceptual difference in the viewpoints of what a Monte Carlo calculation is This chapter rst discusses these differences in terminology and viewpoint, to the extent that the author and reviewer have identi ed them In the major portion of this chapter, some of the methods used in rare event transport simulations are described and then demonstrated in the context of a sample transport problem In addition to the methods illustrated on the sample problem, the comb is discussed because few people are aware of the comb technique, despite some interesting theoretical and practical aspects Inasmuch as the implications of using importance function information is not always well understood, some guidance is given The chapter concludes with practical comments about when users should stop variance reduction efforts, followed by some comments on the future of Monte Carlo transport methods
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Scope of particle transport problems considered
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The eld of particle transport contains deterministic (ie, non-stochastic) solution techniques as well as the Monte Carlo techniques discussed in this chapter Almost all of the deterministic techniques solve the transport problem by solving the Boltzmann transport equation [1] for the particle ux (particle density times velocity) as a function of position, energy, angle and time Monte Carlo is often viewed as an alternative approach to solving transport equations, so most of the texts implicitly assume in their discussions that the Monte Carlo codes are being used to estimate quantities such as particle densities, currents, and uxes This assumption is often invalid, as explained in the paragraph after next Those not familiar with transport can get some idea of the issue by considering the planar heat equation 2 T = 0 in a homogeneous medium with boundary conditions T = 1 on some part of the boundary and T = 0 on the rest of the boundary The solution to the heat equation at a point P can be shown to be [2] the probability that a particle starting at P = (x, y) with an isotropic random direction and randomly walking on circles reaches the T = 1 boundary rather than the T = 0 boundary Thus, the random walk is solving the heat equation Note that if two particles are started at P with particle 2 s direction opposite to particle 1 s direction, the random walk of either one solves the heat equation If instead one changes the question to estimating the probability that both particles reach the T = 1 boundary, one needs to write a different equation involving the joint density of particles 1 and 2 because the heat equation gives no information about the joint density
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There are many transport problems for which solving the standard Boltzmann transport equation is irrelevant because it describes the behavior of individual particles Any particle transport problems that depend on the collective behavior of several particles must be treated differently From a Monte Carlo standpoint, the estimates made are dependent on collections of particles and therefore the collection of particles carries a statistical weight rather than the individual particles [4 6, 25] One example is the coincident physical detection of a pair of gamma rays from an electron positron annihilation event The physical detector system responds only when both gamma rays enter the detector within a very short time interval ( in coincidence ) If two gamma rays enter the detector, but not in coincidence, the detector does not respond because the gamma rays could not be from the annihilation event The Boltzmann equation provides no information about the probability of coincidence in the detector For simplicity, this chapter considers only those types of Monte Carlo calculations for which individual particles carry weights
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