Geometrical structure of the Wigner flow information quantifiers and hyperbolic stability in the phase-space framework

TL;DR

This paper explores the geometrical structures of Wigner flow in phase space, linking information quantifiers to hyperbolic stability, with analytical results for quantum Gaussian ensembles and an application to a Harper-like system.

Contribution

It introduces a geometrical framework connecting Wigner flow quantifiers with hyperbolic stability in phase space, including explicit analytical expressions for quantum Gaussian ensembles.

Findings

Wigner flow quantifiers relate to hyperbolic stability boundaries.

Analytical expressions for equilibrium stability in quantum Gaussian ensembles.

Application demonstrates influence of quantum fluctuations on phase-space vorticity.

Abstract

Quantifiers of stationarity, classicality, purity and vorticity are derived from phase-space differential geometrical structures within the Weyl-Wigner framework, after which they are related to the hyperbolic stability of classical and quantum-modified Hamiltonian (non-linear) equations of motion. By examining the equilibrium regime produced by such an autonomous system of ordinary differential equations, a correspondence between Wigner flow properties and hyperbolic stability boundaries in the phase-space is identified. Explicit analytical expressions for equilibrium-stability parameters are obtained for quantum Gaussian ensembles, wherein information quantifiers driven by Wigner currents are identified. Illustrated by an application to a Harper-like system, the results provide a self-contained analysis for identifying the influence of quantum fluctuations associated to the emergence of phase-space vorticity in order to quantify equilibrium and stability properties of Hamiltonian non-linear dynamics.