"Classical" instabilities and "quantum" speed-up in the evolution of neutrino clouds

We study some examples of collective behavior in neutrino clouds governed by the neutral-current neutrino-neutrino interaction. The standard equations for analyzing such systems are rederived in a two-step process: first, a replacement of the full interaction Hamiltonian with a "forward" Hamiltonian that contains only the momentum states that were initially occupied by a neutrino of one flavor or another; second, a factorization assumption that reduces the time evolution problem to the solution of coupled equations for the expectations of various bilinear forms in the neutrino fields. We designate the latter as the "classical" equations. We analyze some solutions of these equations in cases in which the initial momentum and flavor distributions of neutrinos are strongly anisotropic in space. In some cases we find an instability that leads to rapid evolution of the flavor-angle distribution, even when it is seeded by a very small initial flavor mixing (or alternatively by a very small neutrino mass$^2$ term). Turning to the more complete case in which we do not assume the classical factorization, but instead solve for the evolution under the influence of the full "forward" Hamiltonian, we find the possibility of rapid evolution, under our definition, even when there is no seeding from conventional neutrino mixing.