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For a class of Vlasov-Fokker-Planck equations that have frequently been used to examine beam dynamic instabilities in accelerators and storage rings, it is shown that the stability of stationary distributions can be determined by studying reduced models defined by Smoluchowski equations. This is illustrated explicitly for longitudinal particle bunches in beams subjected to a particular class of wake fields, described by a Haïssinski distribution. For this class we find that continuous Haïssinski distributions are stable because they correspond to minima of appropriately defined free energy functionals. For parameters for which continuous distributions no longer exist, discontinuous distributions may still exist but correspond to free energy maxima and saddle points and are unstable

Smoluchowski approach to nonlinear Vlasov-Fokker-Planck equations: Stability analysis of beam dynamics and Haïssinski theory