On the stochastic theory of continuous parametric systems and its application to electron cascades

Abstract

A mathematical definition of an assembly of continuous parametric systems is given and its theory developed which makes it correspond precisely to most of the continuous parametric assemblies known in nature. Certain general theorems (formulae (19) and (22)) are deduced which hold for all such assemblies. It is shown that the usual method of treating a continuous parametric assembly by dividing up the domain of the parameter into a number of small segments, treating each segment as belonging to a discrete state of the system and then passing to the limit of making the segments infinitely small does not lead to a continuous parametric assembly of the type described above, but one of much wider generality which does not correspond to any type of physical system met with in nature. The general method and the theorems are of immediate application in calculating the fluctuations of the number of particles in chain reacting systems, and not only for systems in thermodynamic equilibrium . The general theory is applied, as an illustration, to the stochastic treatment of an electron cascade to derive the differential equations which determine the functions from which the mean number of particles in any energy interval and the mean square deviation of this number can be calculated. It is shown in the appendix how the application of the usual method to this problem leads to the same results only if particular boundary conditions are imposed on the problem.