  Dense neutrino gases form in extreme astrophysical sites, and the flavor
content of the neutrinos likely has an important impact on the subsequent
dynamical evolution of their environment. Through coherent forward scattering
among neutrinos, the flavor content of the gas evolves under a time-dependent
potential which can be modeled in a quantum many-body formalism as an
all-to-all coupled spin-spin interaction. This two-body potential generically
introduces entanglement and greatly complicates the study of these systems. In
this work we study the evolution of the quantum many-body problem as well as
the typically employed mean-field approximation to it for a small number of
neutrinos ($N = 16$). We consider randomly chosen one- and two-body couplings
in the Hamiltonian, and the resulting evolution of several initial product
states. We subsequently compare many-body and mean-field predictions for
one-body observables, and we consider one- and two-body entanglement to assess
under what conditions the many-body and mean-field predictions are likely to
disagree. Except for a special category of prototypical initial conditions, we
find that the typically employed mean-field approximation is insufficient to
capture the evolution of one-body operators in the systems we consider. We also
observe a loss of coherence in one- and two-body trace-reduced subsystems which
suggests that the evolution may be well approximated as a classical mixture of
separable states.
