Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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188 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsthe expected total weight of the fragments coming out of the splitting, the first-momentequation reduces toM,(P,W) = J dP'f(P,P')dP'T(P,P') 1 -- J dP^P.P,) W'[f(P,P') + N 1(P')dP, 1(P 1P 1) dQT(P,Q)[Wf (P,P,) + WM 1(P 1)] (5.109)where we have made use of the normalization in Equation (5.105). Any game with splittingmay only be useful if it results in the same expected score as the corresponding game withoutsplitting. According to Equation (5.75), the expected score in the game with no splittingsatisfies the equationWM 1(P) = J dP'T(P,P')W'|f(P,P') + N 1(P')] (5.110)Comparison of Equations (5.109) and (5.110) shows that the two expectations are equal ifdP.tffX,)) dQT(P 1Q)[Wf 11(P 1P 1) + WM 1(P 1)]dP'f(P,P')J dP.UPXJW'ItTP.P') + N 1(P')] = 0 (5.111)Now let us consider a particle that starts a flight from P 1with the weight W. Let W' denoteits weight after a free flight from P 1to P'. The expected score due to this particle satisfiesEquation (5.110) in the formWM(P 1) - dP'T(P,,P')W'[f(P,,P') + N 1(P')]Before substituting this equation into Equation (5.111), a number of identities are established.Notice that by interchanging the order of integrations with respect to P 1and P'and making use of Equation (5.104), the following relations hold:| dP^PX.J dQT(P,Q)[...] = J dP'T(P,P')| dP,t(P,P,)[...] (5.112)andClP 1I(P 7P 1) f dQT(P,Q) [dPT(P,,P')[...]dP'J dP,t(P,P,) I dQT(P 5Q)T(P 11POf...!dP'T(P,P')pdP,t(P,P,)[...]dP.KP.P,) dPT(P,P')[...] (5.113)
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 .;
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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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100 Monte Carlo Panicle Transport M
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102 Monte Carlo Particle Transport
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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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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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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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