  Phase-space features of the Wigner flow for generic one-dimensional systems
with a Hamiltonian, $H^{W}(q,\,p)$, constrained by the $\partial ^2 H^{W} /
\partial q \partial p = 0$ condition are analytically obtained in terms of
Wigner functions and Wigner currents. Liouvillian and stationary profiles are
identified for thermodynamic (TD) and Gaussian quantum ensembles to account for
exact corrections due to quantum modifications over a classical phase-space
pattern. General results are then specialized to the Harper Hamiltonian system
which, besides working as a feasible test platform for the framework here
introduced, admits a statistical description in terms of TD and Gaussian
ensembles, where the Wigner flow properties are all obtained through analytical
tools. Quantum fluctuations over the classical regime are therefore quantified
through probability and information fluxes whenever the classical Hamiltonian
background is provided. Besides allowing for a broad range of theoretical
applications, our results suggest that such a generalized Wigner approach works
as a probe for quantumness and classicality of Harper-like systems, in a
framework which can be extended to any quantum system described by Hamiltonians
in the form of $H^{W}(q,\,p) = K(p) + V(q)$.
