Application of the Green's function method to some nonlinear problems of an electron storage ring: Part 1, The Green's function for the Fokker-Planck equation

For an electron storage ring the beam size evaluation including beam-beam interaction gives an example of such a problem. Another good example is finding the beam size for a nonlinear machine. The present work gives a way to solve some of these problems, at least in principle. The approach described here is an application of the well known Green's function method, which in this case is applied to the Fokker-Planck equation governing the distribution function in the phase space of particle motion. The new step made in this paper is to consider the particle motion in two degrees of freedom rather than in one dimension, a characteristic of all the previous work. This step seems to be necessary for an adequate description of the problem, at least for the class of problems which are considered below. This work consists of the formal solution of the Fokker-Planck equation in terms of its Green's function and describing the Green's function itself. The Green's function and the description of some of its properties can be found in the Appendices. I discuss the distribution function in the transverse phase space of a particle and it's Fokker-Planck equation for a simple case of a weak focusing machine. Part of this paper is devoted to describing the Green's function and solution of this equation. Then this technique is applied to a strong focusing machine and finally there is a discussion of applicability of this method, its limitations and relation to other methods. 13 refs