  Collective neutrino oscillations can potentially play an important role in
transporting lepton flavor in astrophysical scenarios where the neutrino
density is large, typical examples are the early universe and supernova
explosions. It has been argued in the past that simple models of the neutrino
Hamiltonian designed to describe forward scattering can support substantial
flavor evolution on very short time scales $t\approx\log(N)/(G_F\rho_\nu)$,
with $N$ the number of neutrinos, $G_F$ the Fermi constant and $\rho_\nu$ the
neutrino density. This finding is in tension with results for similar but
exactly solvable models for which $t\approx\sqrt{N}/(G_F\rho_\nu)$ instead. In
this work we provide a coherent explanation of this tension in terms of
Dynamical Phase Transitions (DPT) and study the possible impact that a DPT
could have in more realistic models of neutrino oscillations and their
mean-field approximation.
