Particle distribution parameters in an electron storage ring

The distribution function of particles in a plasma can often be found by solving the Fokker-Planck equation. This technique can also be used to find the average particle distribution in an electron storage ring in the presence of coupling between the particle coordinates and momenta. Using a smooth approximation, the beam distribution parameters, such as the transverse beam size, can be described in terms of the eigenvalues and eigenvectors of the coupling matrix. For some case, analytic expressions for these quantities can be obtained directly. This method offers us a straightforward means to find the functional dependence of the beam-shape parameters upon various machine parameters which are essential for storage-ring design