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Beam dynamics calculations that are based on the Vlasov equation do notpermit the the treatment of stochastic phenomena such as intra-beam scattering.If the nature of the stochastic process can be regarded as a Markov process, weare allowed to use the Fokker-Planck equation to describe the change of thephase space volume the beam occupies. From the Fokker-Planck equation we deriveequations of motion for the beam envelopes and for the rms-emittances. Comparedto previous approaches based on Liouville's theorem, these equations containadditional terms that describe the temperature balancing within the beam. Ourformalism is applied to the effect of intra-beam scattering relevant for beamscirculating in storage rings near thermodynamical equilibrium. In this case,the Fokker-Planck coefficients can be treated as adiabatic constants of motion.Due to the simplified analysis based on `beam moments', we obtain fairly simpleequations that allow us to estimate the growth rate of the beam emittance.

Improved envelope and emittance description of particle beams using the Fokker-Planck approach