Kinetic Effects in Multiple Intra-Beam Scattering

Abstract

The analysis of the evolution in phase space induced by multiple intra‐beam scattering (IBS) requires the solution of the Fokker‐Planck equation (FPE) or of similar kinetic equations. The FPE is formulated in coordinate‐momentum space (6 variables). Using the “semi‐Gaussian model” this equation is reduced to the longitudinal FPE that depends on longitudinal momentum and coordinate; drift and diffusion coefficients in this equation are presented as integrals on distribution function with kernels expressed in analytical form. The number of variables in the FPE can be reduced to three by reformulation in the space of invariants. The invariant‐vector has the following components: a longitudinal energy (for the longitudinal degree of freedom) and the Courant‐Snyder invariants (for the transverse motion). The coefficients of the FPE in invariant space are in the form of integrals over the distribution function and the invariants with the kernel in the form of many‐dimensional integrals over the longitudinal variable and over the oscillation phases. The three‐dimensional FPE can be solved numerically by application of macro‐particle codes using the different methods: 1) Langevin method; 2) binary collision map. The last method is used in the code “MOCAC” (MOnte CArlo Code) for IBS simulation. Examples of code validation and application are discussed.