Entropy and canonical ensemble of hybrid quantum classical systems

Abstract

We generalize von Neumann entropy function to hybrid quantum-classical systems by considering the principle of exclusivity of hybrid events. For non-interacting quantum and classical subsystems, this entropy function separates into the sum of the usual classical (Gibbs) and quantum (von Neumann) entropies, whereas if the two parts do interact, it can be properly separated into the classical entropy for the marginal classical probability, and the conditional quantum entropy.We also deduce the hybrid canonical ensemble (HCE) as the one that maximizes this entropy function, for a fixed ensemble energy average. We prove that the HCE is additive for non-interacting systems for all thermodynamic magnitudes, and reproduces the appropriate classical-and quantumlimit ensembles. Furthermore, we discuss how and why Ehrenfest dynamics does not preserve the HCE and does not yield the correct ensemble averages when time-averages of simulations are considered -even if it can still be used to obtain correct averages by modifying the averaging procedure.