Further studies on relic neutrino asymmetry generation. I. The adiabatic Boltzmann limit, nonadiabatic evolution, and the classical harmonic oscillator analogue of the quantum kinetic equations

Abstract

We demonstrate that the relic neutrino asymmetry evolution equation derived from the quantum kinetic equations (QKE's) reduces to the Boltzmann limit that is dependent only on the instantaneous neutrino distribution functions, in the adiabatic limit in conjunction with sufficient damping. An original physical and/or geometrical interpretation of the adiabatic approximation is given, which serves as a convenient visual aid for understanding the sharply contrasting resonance behaviors exhibited by the neutrino ensemble in opposing collision regimes. We also present a classical analogue for the evolution of the difference in the ${\ensuremath{\nu}}_{\ensuremath{\alpha}}$ and ${\ensuremath{\nu}}_{s}$ distribution functions which, in the Boltzmann limit, is akin to the behavior of the generic reaction $A\ensuremath{\rightleftharpoons}B$ with equal forward and reverse reaction rate constants. A new characteristic quantity, the matter and collision-affected mixing angle of the neutrino ensemble, is identified here for the first time. The role of collisions is revealed to be twofold: (i) to wipe out the inherent oscillations, and (ii) to equilibrate the ${\ensuremath{\nu}}_{\ensuremath{\alpha}}$ and ${\ensuremath{\nu}}_{s}$ distribution functions in the long run. Studies on non-adiabatic evolution and its possible relation to rapid oscillations in lepton number generation are also featured, with the introduction of an adiabaticity parameter for collision-affected oscillations.