  Quantum dynamics (e.g., the Schr\"odinger equation) and classical dynamics
(e.g., Hamilton equations) can both be formulated in equal geometric terms: a
Poisson bracket defined on a manifold. The difference between both worlds is
due to the presence of extra structure in the quantum case, that leads to the
appearance of the probabilistic nature of the measurements and the
indetermination and superposition principles. In this paper we first show that
the quantum-classical dynamics prescribed by the Ehrenfest equations can also
be formulated within this general framework, what has been used in the
literature to construct propagation schemes for Ehrenfest dynamics. Then, the
existence of a well defined Poisson bracket allows to arrive to a Liouville
equation for a statistical ensemble of Ehrenfest systems. The study of a
generic toy model shows that the evolution produced by Ehrenfest dynamics is
ergodic and therefore the only constants of motion are functions of the
Hamiltonian. The emergence of the canonical ensemble characterized by the
Boltzmann's distribution follows after an appropriate application of the
principle of equal a priori probabilities to this case. This work provides the
basis for extending stochastic methods to Ehrenfest dynamics.
