p(x1 , w) p(x2 , w) dx1 dx2

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(107)

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From this (noting that S1 = 1 2

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p(x, w)dx = 1 because p is a probability density) 1 2 x1 p(x1 , w) dx1 = xp(x, w) dx = M1 (108)

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x2 p(x2 , w) dx2 +

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This shows that the mean is the same with or without the splitting Similarly, S2 = 1 4

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2 2 (x1 + 2x1 x2 + x2 )p(x1 , w) p(x2 , w) dx1 dx2 =

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1 2 M2 + M1 2 (109)

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The knowledge of the statistical properties (Mr ) of one Monte Carlo calculation thus allows one to predict the statistical properties (Sr ) of a different Monte Carlo calculation Thus, the statistical properties using one set of Monte Carlo variance reduction parameters can be empirically derived via simulation and then the statistical properties of calculations with other parameters can be inferred Note that this analysis cannot be done without using the weight independence relationship of (103)

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Analog Monte Carlo neutron transport steps

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An analog Monte Carlo transport calculation consists of the following basic steps: 1 Begin the nth source particle history 2 Sample a source particle That is, sample a particle s phase-space coordinates (r, , E, t) = (position, direction, energy, time) from Q(r, , E, t) 3 Sample the distance to the particle s next event Let b be the distance to the boundary in direction The distance to the next event, , is sampled using the exponential distribution e (Here, is the physical interaction probability per unit length derived from the nuclear data and the exponential distribution arises naturally from the physics) If < b, then the particle collides before reaching the boundary, otherwise the particle is put on the boundary Move the particle a distance in the particle direction = v/v, where v is the particle velocity and v is the particle s speed That is, increment the position by r + r Increment the time by t + /v t (Note that the energy, E, is constant between collisions) If an interior boundary is reached (eg, an interface between concrete and iron regions), then go to step 3 If an exterior boundary is reached (eg, the particle escapes the solar system), go to 5 If no boundary is reached, the particle has collided Go to step 4

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4 Sample the collision Sample the collision nuclide i (eg, carbon-12, carbon-13, nitrogen-14, iron-56) Sample for capture (termination of the particle when the collision nuclide removes the particle) vs survival (ie, non-capture) on nuclide i If capture occurs, go to step 5 If the particle survives the collision, then sample for the j th type of interaction on nuclide i (eg, elastic scatter, inelastic scatter, ssion) If necessary, sample the number of output particles K for interaction j on nuclide i (For example, an elastic scatter always has exactly has one particle coming out of a collision, but the number of particles coming out of a ssion event is random) For interaction j (eg, ssion) on nuclide i (eg, uranium-235), sample the output phase-space coordinates for the K output particles from a probability law Ci,j ((r, , E, t) (r1 ,

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1 , E1 , t), , (rK , K , EK , t))

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(Note that the collision is instantaneous and the time does not change) If K > 1, save K 1 particles in a bank to process later Go to step 3 5 Check for banked particles If there are particles waiting in the bank to be processed, take one from the bank and go to step 3 If there are no particles in the bank, go to step 6 6 End the nth source particle history Process the statistical results for the nth source particle For example, if the nth source particle history contributes sn , increment the m moment sums (for transport, usually m = 1, 2, 3, and 4)

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m Sm + sn Sm

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so that error estimates can be made Go to step 1 and sample a new source particle

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Intuitive ideas of variance reduction

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In the analog simulation described in the previous section, every simulated particle represents one physical particle and the simulated particles are subject to the same probability laws as the physical particles The number of particles penetrating a shield, for example, can be estimated by starting N simulated particles from

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