Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

139empirical variance. It can be used in any Monte Carlo game resulting in |x,, JX 2, . . . jx,,individual scores from n simulations. Thus, the introduction of the integral equation formalismdoes not influence the simple straightforward a posteriori estimation of the statisticaluncertainty.There is, however, an important benefit of the use of integral equations in the field ofvariance analysis. Equations can be derived, by the use of which efficiencies of techniquescan be estimated a priori. Though the solution of these equations is at least as complicateas that of the collision density equations, even an approximate estimation of the variancesmay clearly indicate whether a certain nonanalog procedure increases the efficiency of thegame, or not.Variance analysis is based on the moment equations. Since these equations are investigatedin detail in Chapter 5, only a short introduction is given below.A. VARIANCE ESTIMATES BY THE MOMENT EQUATIONSThe variance of a certain random variable is defined as the difference between theexpected value of the square of the random sample and the square of the expected value.If the random variable in our Monte Carlo game is the score (x, then the variance is:D 2 O) - ill 2 ) - (li) 2In particle transport, if Q(r,E)dr dE particles start from the phase-space element dr dEabout (r,E) and the score depending on the starting point is denoted by |x(r,E) then the totalscore, i.e., the reaction rate, is(jji) = R =clrdEOir.EKp.(r.E»where parentheses denote the expectation over all possible histories started from (r,E). IfM 1(r.E) denotes the first moment of the score, i.e., the expected score due to a particlestarted from P thenM 1(F 5E) = {(x(r,E))and the reaction rate reads:R = j JdFdEQ(F.Ei M 1(F 1E) (4.128)It is obvious from the comparison of either the definitions, or, formally, Equations(4.128) and (4.100) that the first moment is equivalent to the value of the particles leavingcollisions:M,(r.E)--X*(r,E)Consequently, the integral equation of the first moment is (see Equation [4.1.00]):M,(r,E) == f x(r,E)JJdr'dE'T(r-> r'|E)C(E-»E'|r')M,(r\E') (4.129)The reader should be reminded, here, of our comment at the end of Chapter 4.III.C.that in most papers the collision density of particles entering a collision is used solelySimilarly, in most treatments the value of the ingoing particles (»j;*) is the only adjoint