The Liouville equation for flavour evolution of neutrinos and neutrino   wave packets
  We consider several aspects related to the form, derivation and applications of the Liouville equation (LE) for flavour evolution of neutrinos. To take into account the quantum nature of neutrinos we derive the evolution equation for the matrix of densities using wave packets instead of Wigner functions. The obtained equation differs from the standard LE by an additional term which is proportional to the difference of group velocities. We show that this term describes loss of the propagation coherence in the system. In absence of inelastic collisions, the LE can be reduced to a single derivative equation over a trajectory coordinate. Additional time and spacial dependence may steam from initial (production) conditions. The transition from single neutrino evolution to the evolution of a neutrino gas is considered. 
Rasmus Sloth Lundkvist Hansen and Alexei Yu. Smirnov
hep-ph astro-ph.SR