Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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144 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhere, as before, v)i(P) is the collision density of particles at the phase-space point P andf(P) is some given weight function. R is called a reaction rate since if a r(P) is the crosssection of the reaction r andf(P) = o r(P)/ar(P) Xv(P)Xv(P) being the characteristic function of some phase-space domain V, thenR = jdPiKP)a r(P)/o-(P) = |dP (p(P) (j,.(P)is the rate of the reaction r in the domain V.Note that the integrations, unless explicitly noted otherwise, are always assumed toextend over the entire phase space. For the integrals of functions that are defined only overa part of the phase space, the definition of the functions is tacitly extended to the entirespace, with the function taken as zero outside the original domain of definition.We have seen in Chapter 4 that the integral, Equation (5.2), can be estimated in a MonteCarlo game by simulating the particles' collision density in a number, N, of historiesaccording to the transport equation and, at the i-th collision point of the a-th history, P a,,the quantity f(P„ ,) is added to the estimate. Then the estimateS ? ? f(P ° Jis unbiased with respect to R in Equation (5.2), i.e.,JdPvkp) f(P) = (^2 2 f(P,„)>Now, denotingM-(P,,,,) = Ef(Pa.i)p,(P„ „) is the score due to the history that was started from the point P a„. If M 1(P) denotesthe expectation of |x(P), then u.(P„ 0) is an unbiased estimate of M 1(P) and, if the startingpoint P a„ is selected from the source density Q(P 11„), the score~ 2M-(P 1.,) is an unbiased estimate of the integral, i.e.,R = (jjf E M-(Pa...)) = JdPQ(PjM 1(P) (5.3)It is heuristically obvious that an equation that describes the pointwise expected score,M 1(P) 1is more naturally related to the actual simulation procedure than the particle-transportequation, which does not account for the weighting function, f(P) — a quantity very characteristicof the goal and technique of the estimation. It will be seen later in this chapterthat the equations governing the various moments of the score are, indeed, very useful inthe analysis of the quality of various Monte Carlo techniques. In the next section, we showhow the equation that describes the expected score, M 1(P) 1is related to the equation adjointto the integral transport equation.
MSA. RELATION OF THE EXPECTED SCORE TO THE ADJOINT COLLISIONDENSJTYAs was seen in Chapter 4.IV, the collision density, fi/(P), satisfies the integral transportequationiMP) - ik,(P) + JdP"»|;(P")K(P",P)where >K.(P) < sthe first-flight collision density.K,(P) = J dP'Q(P')T(P',P)(SA.and Q(P) is the probability density of the source particles. Recall that the transport kerrse.iK(P",P) is related to the transition kernel T(P',P) and collision kernel C(P",P') asK(P",P) = |dP'C(P",P')T(P',P) (5.6;whereT(P',P) = T(r'->r|E)o(E' - E) 0.7;andC(P".P') = C(E"->E'jr")5(r" - r') (5.¾)while T(r'-*r|E) and C(E"-->E'[r") are given in Chapter 4.1V. In Chapter 4.VTL it wasproven that an equation adjoint to the integral transport Equation (5.4) has the formi|/*(P) = f(P) + dP"K(P,P")i|
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Library of Congress Cataloging-in-P
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III. Statistical Considerations •
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IV. Further Generalizations — 183
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Appendix 6A:Unbiased Estimation of
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2 Monte Carlo Particle Transport Me
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Chapter 2INTRODUCTIONWhen we starte
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thus,As it is proved" — and is so
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9B. THE PROBABILITY MIXING METHODTh
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11It is easy to prove 15by summing
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13but the most probable values:x, =
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ISand give a random sign (by the us
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18 Monte Carlo Particle Transport M
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25!) Monte Carlo Particle Transport
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305Chapter 6SPECIAL GAMESIn the maj
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3©7A. CORRELATED MOMENT EQUATIONSI
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309Equation (6.1) will be referred
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311The second moment of the score d
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313andC S(P',P")/C S(P',P") = y(P',
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315introduced in Equation (6.1) for
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317andL„(P 0,P;,...,Pi + 1) = L n
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319whereB„(P„,P; P:, ,) = A*(P
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E. EXAMPLES AND SPECIAL TECHNIQUESL
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323The weight factors associated wi
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325It is seen from Equation (6.37)
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327G. PARAMETRIC PERTURBATIONS: INT
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129Hall 31 gave a constructive deri
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331kernels, which remain unchanged
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333and from Equation (6.51)w';,,(i)
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335According to Equations (6,55) th
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337Minimization is performed by set
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339and letdsdaLet us consider a cor
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341whereis the length of the flight
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and from Equation (6.79)I / ds \ 2
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345Let us assume that we are intere
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347Introduction of this factor also
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3492. Let i|/ n (P) denote the solu
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3SJNow, if ( "> > 0, then k = i an
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353andk„ = S l "VS'" •" (6. if;
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3SSLet k„ denote the normalizatio
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3572. It is then proved that with t
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and start new histories until N new
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361The application of source iterat
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363we haveD 2 [k] = < 2 (k, - k,) )
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365neutron density, S 0 for every
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367if the same relation holds for t
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369Then the multiplication factors
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371in the unperturbed system, the w
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373The perturbation of the effectiv
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375where P" = (r',E'). Multiplying
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377iterate of the derivative of the
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579where P* = (r*,w*,E) = (r*,E*).
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381This estimator, unlike f, in Equ
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383tends to some stable attracting
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3§SIo the case of the next-event e
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387LDFIGURE 6.1.Geometry in scoring
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389Let us denoteThen Equation (6.18
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391is to be increased in order to o
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393Now, since bl(P) = g(P) is bound
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395is scored at every collision wit
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397and its expectation isR — R 1n
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3994. If O > 0 m, then the particle
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401plays a distinguished role), and
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403Theorem 6.7 — If x and s denot
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this a priori information, the best
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40?Thus, the bias introduced into t
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409In view of Equation (6.226) and
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41.1This means that if we use the k
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413Then the whole-sample and rare-s
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41.5According to the results of Sec
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41?(Note that vD 2 [|i|v] is indepe
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419withkOn the other hand, § is an
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421Thus, the expected ratio reads
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423St follows from Equations (6.265
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425Wc start from the observation th
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427Nowi'6.77K)kj„,and therefore(V
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42*3Similarly, for the fourth momen
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43 sOn the other hand, multiplying
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43.¾This will be proven in the lem
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435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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6. Brown. F. B. and Martin, W. R.,
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67. Rief, H. and Fioretti, A., Mont
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Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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48© Monte Carlo Particle Transport
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484 Monte Carlo Panicle Transport M
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486 Monte Carlo Panicle Transport M
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a ( X506 Monte Carlo Particle Trans
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513INDEXAAbsorptiondefined, 25photo
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515score probability in general tim
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517path stretching, 447—455splitt
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