Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

185Proof. Let 9~(P,P') be the probability density of the point P' selected in the procedure aboveand let1, if D 13s DX(P-Q) -0, if D 1< DIt is to be shown that ST(P,P') = T(P,P'). According to steps 1 through 4, the density ofP' satisfies the equation2T(P,P') = dQ UP 1T(P 1Q)I(P 1P 1)Ix(P 11Q)S(P' - Q) + x(Q,P,)^(P,,P')ldQ 4P 1U(P 5P 1) T(P,Q)8(P' - Q) dP, dQT(P.Q) t(P, P 1)SXPi, P')or, since the densities are normalized to unityJ(P 1P') dP, KP 1,PP 11 ))] T(P t( 1P')dP, L(P 1P 1)rdQ T(P 1Q) ST(P 1,P') (5.103)First we show that T(P 1P') satisfies Equation (5.103). Inserting T into the RHS of theequation and taking into account that T(P 11P') = 0 for such P' that do riot belong to thehalf-line from P 1along to,, the second term on the RHS of Equation (5.103) becomesdP, KP 1P 1)T(P 1P')where we have made use of Equation (5.102). The sum of the two terms givesJ(P 1P') =T(P 1P')i.e., T(P 1P') does indeed satisfy Equation (5.103). To conclude the proof, it remains toshow that Equation (5.103) has a unique solution. This, however, follows from the fact thatthe integral kerne!KP 1P 1)dQT(P,Q)in Equation (5.103) has a norm definitely less than unity [unless t(P,P,)Dirac delta function], since both t and T are everywhere positive andunity. Thus, the uniqueness of the solution is ensured by Theorem 5.1.We have thus established the identityT(P 1P') - J dP,t(P,P,) T(P 1P') dP, KP 1P 1) dQT(P.Q) T(P 11P')