Detailed balance in Ehrenfest mixed quantum-classical dynamics

We examine the equilibrium limits of self-consistent field (Ehrenfest) mixed quantum-classical dynamics. We derive an analytical expression for the equilibrium mean energy of a multistate quantum oscillator coupled to a classical bath. We show that, at long times, for an ergodic system, the mean energy of the quantum subsystem always exceeds the temperature of the classical bath that drives it. Furthermore, the energy becomes larger as the number of states increases and diverges as the number of quantum levels approaches infinity. We verify these results by simulations.