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-sol(x) with axial periodicity length S = 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) much less than S, and small transverse momentum and axial momentum spread in comparison with the directed axial momentum p(z) = gamma(b) beta(b)c. The global rate equation is derived for the self-consistent nonlinear evolution of the statistical average [X] = N(b)(-1)integral dXdYdX'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 frame. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum [P-theta], center-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-square beam radius R-b(s)=[X-2 + Y-2](1/2). Most importantly, the present derivation of nonlinear rate equations for various statistical averages [chi] allows for general azimuthal variation (partial derivative/partial 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.