Particle Moderation: Random Functional Approach

The slowing down of a particle by a homogeneous isotropic moderator is considered. It is shown that finding the collision density of the particle as a function of space and energy is equivalent to finding the probability distribution of a certain random functional. By means of this random functional, expressions for the spatial moments are obtained without imposing any restrictions on the variations of the scattering kernel or cross section with energy. These moments are then used to obtain the age equation, the derivation given here differing from others in that no a priori assumptions are made on the collision density itself. Finally, as a special case of the above, the time-energy moments are found.