The Ehrenfest approximation for electrons coupled to a phonon system

We investigate the accuracy of the Ehrenfest approximation for a solid modelled as an independent electron system coupled to a many-mode phonon system. We construct a quantum to classical correspondence which we use to set the initial conditions of the Ehrenfest approximation given an initial density matrix. We find that the Ehrenfest approximation is accurate when the temperature of the phonon system is much higher than the temperature of the electron system. It is therefore reasonable to use the Ehrenfest approximation in the simulation of the later stages of radiation damage processes. We trace the failure of the Ehrenfest approximation to the absence of a term accounting for spontaneous phonon emission. The analytical work is validated numerically for the model system of a single oscillator coupled to electrons on a one-dimensional chain.