  We study the early evolution of the electron fraction (or, alternatively, the
neutron-to-proton ratio) in the region above the hot proto-neutron star formed
after a supernova explosion. We study the way in which the electron fraction in
this environment is set by a competition between lepton (electron, positron,
neutrino, and antineutrino) capture processes on free neutrons and protons and
nuclei. Our calculations take explicit account of the effect of nuclear
composition changes, such as formation of alpha particles (the alpha effect)
and the shifting of nuclear abundances in nuclear statistical equilibrium
associated with cooling in near-adiabatic outflow. We take detailed account of
the process of weak interaction freeze-out in conjunction with these nuclear
composition changes. Our detailed treatment shows that the alpha effect can
cause significant increases in the electron fraction, while neutrino and
antineutrino capture on heavy nuclei tends to have a buffering effect on this
quantity. We also examine the effect on weak rates and the electron fraction of
fluctuations in time in the neutrino and antineutrino energy spectra arising
from hydrodynamic waves. Our analysis is guided by the Mayle & Wilson supernova
code numerical results for the neutrino energy spectra and density and velocity
profiles.
