Stochastic problems in transport theory

Abstract

The application of the theory of discontinuous Markov processes and stochastic population processes to problems in the field of trans­ port theory is investigated. In Chapter 1 it is shown how a scattering process (in which there may, at each collision, be a constant probability that the scattered particle is absorbed and lost to the system) can be char­acterized by transition probabilities Pk (S;0₁...,0k|w;s) of a transition w-S in distance s with exactly k scatterings, the angles of scattering 0₁,...,0k (in order of occurrence) being such that 0₁ϵ 0₁.... .0k ϵ 0k. Such a characterization is of importance when there is a change of state of the particle at each collision and the corresponding transition probabilities depend on the angle of deflection. For one-dimensional processes the problems simplify considerably owing to the fact that the angular deflections, 0i, can take only two values, 0 or π, A detailed investigation of one-dimensional problems in which the probability of a collision in a small element of path- length 6s is λ𝛅s + o (𝛅s) is made in Chapter 2. In order to apply the theory to scattering in three dimensions an approximation is made in which scattered particles are assumed to move in one of a set of 30 directions in space. This model is considered in detail in Chapters 6 and 7.

Chapter 3 deals with a particular one-dimensional problem which arises in the analysis of bubble chamber tracks and is concerned with the stochastic population process generated by the distances