Statistically averaged rate equations for intense non-neutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov–Maxwell equations

In this paper we present a detailed formulation and analysis of the rate equations for statistically averaged quantities for an intense non-neutral beam propagating through a periodic solenoidal focusing field ${{B}}^{\mathrm{sol}}({x})$ with axial periodicity length $\mathrm{S=}\mathrm{const}.$ The analysis is based on the nonlinear Vlasov–Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius $r_b\mathrm{\ll S,}$ and small transverse momentum and axial momentum spread in comparison with the directed axial momentum $p_z{\mathrm{=\gamma }}_b{\mathrm{m\beta }}_b\mathrm{c.}$ The global rate equation is derived for the self-consistent nonlinear evolution of the statistical average ${\mathrm{〈\chi 〉=N}}_b^{\text{−1}}{\mathrm{\int dXdYdX}}^{\prime }{\mathrm{dY}}^{\prime }{\mathrm{\chi F}}_b,$ where ${\mathrm{\chi (X,Y,X}}^{\prime }{\mathrm{,Y}}^{\prime }\mathrm{,s)}$ is a general phase function, and $F_b{\mathrm{(X,Y,X}}^{\prime }{\mathrm{,Y}}^{\prime }\mathrm{,s)}$ is the distribution function of the beam particles in the transverse phase space ${\mathrm{(X,Y,X}}^{\prime }{\mathrm{,Y}}^{\prime })$ appropriate to the Larmor frame. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular...