The Chapman–Kolmogorov and Bothe–Landau Equations

Abstract

The Gaussian and asymmetric collision energy loss pdfs that are derived in the following chapter arise from solutions to the two equations derived here. The form of the Chapman–Kolmogorov equation derived here is an integro-differential equation in the collision energy loss pdf $f\left( {x,\Delta E} \right)$ where $f\left( {x,\Delta E} \right)d\left( {\Delta E} \right)$ is the probability that a charged particle has lost kinetic energy between $\Delta E$ and $\Delta E+d\left( {\Delta E} \right)$ after penetrating to a depth x in a medium. The Bothe–Landau equation, also derived in this chapter, is a solution of the Chapman–Kolmogorov equation, and we will see in the following chapter how it can yield a Gaussian pdf of collision energy loss, provided a number of important assumptions are held. The asymmetric collision energy loss pdfs of Vavilov and Landau can be derived directly from the Chapman–Kolmogorov equation.