Entropy functionals and equilibrium states in mixed quantum-classical dynamics

The computational challenges posed by many-particle quantum systems are often overcome by mixed quantum-classical (MQC) models in which certain degrees of freedom are treated as classical while others are retained as quantum. One of the fundamental questions raised by this hybrid picture involves the characterization of the information associated to MQC systems. Based on the theory of dynamical invariants in Hamiltonian systems, here we propose a family of hybrid entropy functionals that consistently specialize to the usual R\'enyi and Shannon entropies. Upon considering the MQC Ehrenfest model for the dynamics of quantum and classical probabilities, we apply the hybrid Shannon entropy to characterize equilibrium configurations for simple Hamiltonians. The present construction also applies beyond Ehrenfest dynamics.