STATISTICALLY AVERAGED RATE-EQUATIONS FOR INTENSE NONNEUTRAL 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 ra te equations for statistically averaged quantities for an intense non- neutral beam propagating through a periodic solenoidal focusing field B-sol(x) with axial periodicity length S = const. The analysis is base d on the nonlinear Vlasov-Maxwell equations in the electrostatic appro ximation, assuming a thin beam with characteristic beam radius r(b) mu ch less than S, and small transverse momentum and axial momentum sprea d in comparison with the directed axial momentum p(z) = gamma(b) beta( b)c. The global rate equation is derived for the self-consistent nonli near evolution of the statistical average [X] = N(b)(-1)integral dXdYd X'dY'chi F-b, where chi(X,Y,X',Y',s) is a general phase function, and F-b(X,Y,X',Y',s) is the distribution function of the beam particles in the transverse phase space (X,Y,X',Y') appropriate to the Larmor fram e. The results are applied to investigate the nonlinear evolution of t he generalized entropy, mean canonical angular momentum [P-theta], cen ter-of-mass motion for [X] and [Y], mean kinetic energy (1/2)[X'(2)+ Y '(2)], mean-square beam radius (X-2 + Y-2), and coupled rate equations for the unnormalized transverse emittance epsilon(s) and root-mean-sq uare beam radius R-b(s)=[X-2 + Y-2](1/2). Most importantly, the presen t derivation of nonlinear rate equations for various statistical avera ges [chi] allows for general azimuthal variation (partial derivative/p artial derivative theta not equal 0) of the distribution function and self-field potential, and therefore represents a major generalization of earlier calculations carried out for the case of axisymmetric beam propagation. (C) 1998 American Institute of Physics.