  This article analyzes several different homogenization approaches to the
long-term properties of multiphase lattice random walks, recently introduced by
Giona and Cocco, and characterized by different values of the hopping times and
of the distance between neighboring sites in each lattice phase. Both parabolic
and hyperbolic models are considered. While all the parabolic models deriving
from microscopic Langevin equations driven by Wiener processes fail to predict
the long-term hydrodynamic behavior observed in lattice models, the
discontinuous parabolic model, in which the phase partition coefficient is
a-priori imposed, provides the correct answer. The implications of this result
as regards the connection between equilibrium constraints and non-equilibrium
transport properties is thoroughly addressed.
