  We investigate the rates of production and thermalization of $\nu_\mu$ and
$\nu_\tau$ neutrinos at temperatures and densities relevant to core-collapse
supernovae and protoneutron stars. Included are contributions from electron
scattering, electron-positron annihilation, nucleon-nucleon bremsstrahlung, and
nucleon scattering. For the scattering processes, in order to incorporate the
full scattering kinematics at arbitrary degeneracy, the structure function
formalism developed by Reddy et al. (1998) and Burrows and Sawyer (1998) is
employed. Furthermore, we derive formulae for the total and differential rates
of nucleon-nucleon bremsstrahlung for arbitrary nucleon degeneracy in
asymmetric matter. We find that electron scattering dominates nucleon
scattering as a thermalization process at low neutrino energies
($\epsilon_\nu\lesssim 10$ MeV), but that nucleon scattering is always faster
than or comparable to electron scattering above $\epsilon_\nu\simeq10$ MeV. In
addition, for $\rho\gtrsim 10^{13}$ g cm$^{-3}$, $T\lesssim14$ MeV, and
neutrino energies $\lesssim60$ MeV, nucleon-nucleon bremsstrahlung always
dominates electron-positron annihilation as a production mechanism for
$\nu_\mu$ and $\nu_\tau$ neutrinos.
