The violation of detailed balance poses a serious problem for the majority of
current quasiclassical methods for simulating nonadiabatic dynamics. In order
to analyze the severity of the problem, we predict the long-time limits of the
electronic populations according to various quasiclassical mapping approaches,
by applying arguments from classical ergodic theory. Our analysis confirms that
regions of the mapping space that correspond to negative populations, which
most mapping approaches introduce in order to go beyond the Ehrenfest
approximation, pose the most serious issue for reproducing the correct
thermalization behaviour. This is because inverted potentials, which arise from
negative electronic populations entering into the nuclear force, can result in
trajectories unphysically accelerating off to infinity. The recently developed
mapping approach to surface hopping (MASH) provides a simple way of avoiding
inverted potentials, while retaining an accurate description of the dynamics.
We prove that MASH, unlike any other quasiclassical approach, is guaranteed to
describe the exact thermalization behaviour of all
quantum\unicode{x2013}classical systems, confirming it as one of the most
promising methods for simulating nonadiabatic dynamics in real condensed-phase
systems