In high dimensional systems very few exact results are known about quantum dynamics. One of the most important exact conditions we can try to satisfy is detailed balance, ie: dynamics should equilibrate to the correct statistical distribution, a Fermi distribution for electrons, at long times.
Lots of people are familiar with Surface Hopping, Redfield theory, and Ehrenfest dynamics, but actually you can’t use any of these methods to produce a Fermi distribution exactly. Based on our work simulating relaxation, we’ve actually been able to derive an equation of motion which does obey Fermi-Dirac statistics. In obtaining the derivation, we learned useful tricks that are going to help us treat mixed-states on the same footing as pure states. We’re super jazzed about these things.
There are cool experimental consequences, for example that non-radiative relaxation rates are not constant with time. You can check out the whole story on ArXiv for the time being: