Geometry and Statistics of Ehrenfest dynamics

TL;DR

This paper formulates Ehrenfest quantum-classical dynamics within a geometric framework using Poisson brackets, revealing ergodic behavior and the emergence of canonical ensembles, thus enabling stochastic methods for Ehrenfest systems.

Contribution

It extends the geometric formulation of classical and quantum dynamics to Ehrenfest systems, establishing a Liouville equation and analyzing their ergodic properties.

Findings

Ehrenfest dynamics can be formulated with a Poisson bracket.

Ehrenfest systems are ergodic with Hamiltonian as the only conserved quantity.

Canonical ensembles emerge naturally from Ehrenfest dynamics under equal a priori probabilities.

Abstract

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.