Statistics and Nos\'e formalism for Ehrenfest dynamics

TL;DR

This paper formulates Ehrenfest quantum-classical dynamics within a geometric Poisson bracket framework, derives a Liouville equation, and extends Nosé formalism to enable constant temperature simulations of such systems.

Contribution

It introduces a geometric Poisson bracket formulation for Ehrenfest dynamics and extends Nosé formalism to quantum-classical systems for temperature control.

Findings

Ehrenfest dynamics can be formulated with a Poisson bracket.

The evolution is ergodic with only Hamiltonian functions as constants of motion.

The Nosé formalism is extended to Ehrenfest systems for temperature regulation.

Abstract

Quantum dynamics (i.e., the Schr\"odinger equation) and classical dynamics (i.e., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper we first show that the hybrid 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 distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of a Ehrenfest system, it is straightforward to extend the formalism of Nos\'e (invented to do constant temperature Molecular Dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.