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Stochastic equations in the invariant imbedding formulation of particle transport

Journal article, 2009

Invariant imbedding theory is an alternative formulation of particle transport theory. Although stochastic
foundations of invariant imbedding have been known from the beginnings, the method itself has so far
exclusively been used for calculating first moments, i.e. expectations. The present paper attempts to
set up a probability balance equation in the invariant imbedding approach from which equations for
the first and second order densities are derived. It is shown that only the equations for the first order densities
are non-linear, while subsequent order densities obey linear equations. This is expected to considerably
simplify solution to those problems which involve second order density calculations where
invariant imbedding techniques may be profitably used. Examples of such quantities are the variance
or correlations between particles detected at two different energies or angles or the higher moments
of the emitted multiplicity distribution such as the variance from a target bombarded by incident particles.
One possible area of application of our equations is non-destructive estimation of fissile material by
the active neutron assay technique. Another area is the study of particle cascade development in sputtering
and positron backscattering from surfaces. The approach is illustrated by a simple forward–backward
scattering model for these two problems.

particle fluctuations

stochastic theory

sputtering

transport theory

Invariant imbedding