  We compare the decay rates of excited populations directly calculated within
a Keldysh formalism to the equation of motion of the population itself for a
Hubbard-Holstein model in two dimensions. While it is true that these two
approaches must give the same answer, it is common to make a number of
simplifying assumptions within the differential equation for the populations
that allows one to interpret the decay in terms of hot electrons interacting
with a phonon bath. Here we show how care must be taken to ensure an accurate
treatment of the equation of motion for the populations due to the fact that
there are identities that require cancellations of terms that naively look like
they contribute to the decay rates. In particular, the average time dependence
of the Green's functions and self-energies plays a pivotal role in determining
these decay rates.
