Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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148 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsEquation (5.16) is rewritten assup dP'T(P,P')c(P') « B < xNow, in realistic Monte Carlo games, this condition is always satisfied sincet = sup dP'T(P.P') ss 1 (5.21)pand alsoC = SLip C(P') < +=c (5.22)p'which means that the condition holds with B =fc:sup dP"L(P,P") < t • c
149and C(P',P"). Then the first two conditions of Theorem 5,i can be interpreted as follows:The iterated kernel L n(P,P")dP" is the expected number of particles present in the simulationafter n collisions when the initial particle starts at P. Thus, condition 1 excludes the possibilityof an infinite number of progeny due to the collision of a single particle. On the other hand,condition 2 amounts to saying that after no more than N collisions, the total expected numberof particles in the system will be less than unity. Repeated use of condition 2 shows thatafter kN collisions, the expected number of particles is less than b\ i.e., it tends to zero asthe number of collisions increases. In other words, the probability of a history with aninfinite number of collisions is zero. Thus, the conditions of Theorem 5.1 ensure the feasibilityof the Monte Carlo simulation which is performed when using the transition kernelT(P.P') and collision kernel C(P',P").Similarly, the Monte Carlo interpretation of the Neumann series solution, liquation(5.23), is evident: the n-th term of the sum is the expected contribution due to the n-thcollision and (n + |)-st flight. Condition 3 makes these scores bounded and condition 2ensures the boundedness of the expected final score as well.C. ANALOG AND NONANALOG SIMULATIONIn the Monte Carlo interpretation of the mathematical results concerning the momentEquation (5.15), we have repeatedly assumed that the simulation is performed according tothe real physical source and kernels, i.e., the kernels in the game have the same forms asthose in the transport equation. Simulations of this kind are called analog, since theyreproduce the real physical phenomena (as realistically as the transport equation describesthe particles' collision density). In most applications, however, the actual Monte Carlosimulation deviates from the analog case. Examples of such deviations have been presentedin Chapter 3.II.Monte Carlo games that do not follow the real phenomena are called nonanalog As alsodiscussed in Chapter 3.II, in a nonanalog game the particles do not represent real physicalentities and in order to ensure an unbiased final estimation, statistical weights are assignedto the particles. These weights may change from event to event and, roughly speaking, theycompensate for the deviation of the particles' histories from a corresponding analog history.Nonanalog games are usually introduced in order to increase the efficiency of the simulation.In certain cases, use of nonanalog simulation is vital since, in practice, there would be nohope of obtaining reliable estimates by any analog method. (This is the case, for example,when penetration of radiation is modeled very deep in matter.)When introducing a nonanalog Monte Carlo simulation, three basic questions are to beanswered:1. How should the statistical weights of the particles be generated so that the expectedfinal score is the same as in the analog game, i.e., what are the conditions of anunbiased nonanalog simulation!2. Under what conditions is a nonanalog game feasible, in the sense of feasibility definedin the previous section?3. What gain in efficiency is expected from the nonanalog simulation, compared withthe analog and other possible nonanalog games?These questions will be reconsidered in Chapter 5.V and 5.VIiI and the considerations willbe based on the moment equations to be derived in the subsequent chapters.In what follows, the nonanalog source density will be denoted by Q(H 1 .,, s .kernel by f(P,P'), and the collision kernel by C(P',P"). With this m VD i> < j
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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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198 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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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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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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