Statistics and Nosé formalism for Ehrenfest dynamics

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2011-09, Vol.44 (39), p.395004-23
• Main Authors: Alonso, J L, Castro, A, Clemente-Gallardo, J, Cuchí, J C, Echenique, P, Falceto, F
• Format: Article
• Language: English
• Subjects: Constants; Dynamical systems; Dynamics; Exact sciences and technology; Formalism; Mathematical analysis; Mathematical models; Nose; Physics; Schroedinger equation
• ISSN: 1751-8121; 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/44/39/395004

Quantum dynamics (i.e. the Schrodinger 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 us 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 an Ehrenfest system, it is straightforward to extend the formalism of Nose (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.