  We develop in detail a new formalism [as a sequel to the work of T. Champel
and S. Florens, Phys. Rev. B 75, 245326 (2007)] that is well-suited for
treating quantum problems involving slowly-varying potentials at high magnetic
fields in two-dimensional electron gases. For an arbitrary smooth potential we
show that electronic Green's function is fully determined by closed recursive
expressions that take the form of a high magnetic field expansion in powers of
the magnetic length l_B. For illustration we determine entirely Green's
function at order l_B^3, which is then used to obtain quantum expressions for
the local charge and current electronic densities at equilibrium. Such results
are valid at high but finite magnetic fields and for arbitrary temperatures, as
they take into account Landau level mixing processes and wave function
broadening. We also check the accuracy of our general functionals against the
exact solution of a one-dimensional parabolic confining potential,
demonstrating the controlled character of the theory to get equilibrium
properties. Finally, we show that transport in high magnetic fields can be
described hydrodynamically by a local equilibrium regime and that dissipation
mechanisms and quantum tunneling processes are intrinsically included at the
microscopic level in our high magnetic field theory. We calculate microscopic
expressions for the local conductivity tensor, which possesses both transverse
and longitudinal components, providing a microscopic basis for the
understanding of dissipative features in quantum Hall systems.
