  We study real time classical matrix mechanics of a simplified $2\times 2$
matrix model inspired by the black hole evaporation problem. This is a step
towards making a quantitative model of real time evaporation of a black hole,
which is realized as a bound state of D0-branes in string theory. The model we
study is the reduction of Yang Mills in $2+1$ dimension to $0+1$ dimensions,
which has been corrected with an additional potential that can be interpreted
as a zero point energy for fermions. Our goal is to understand the lifetime of
such a classical bound state object in the classical regime. To do so, we pay
particular attention to when D-particles separate to check that the "off
diagonal modes" of the matrices become adiabatic and use that information to
improve on existing models of evaporation. It turns out that the naive
expectation value of the lifetime with the fermionic correction is infinite.
This is a logarithmic divergence that arises from very large excursions in the
separation between the branes near the threshold for classical evaporation. The
adiabatic behavior lets us get some analytic control of the dynamics in this
regime to get this estimate. This divergence is cutoff in the quantum theory
due to quantization of the adiabatic parameter, resulting in a long lifetime of
the bound state, with a parametric dependence of order $\log(1/\hbar)$.
