  Considering the dynamics of non-interacting particles randomly moving on a
lattice, the occurrence of a discontinuous transition in the values of the
lattice parameters (lattice spacing and hopping times) determines the uprisal
of two lattice phases. In this Letter we show that the hyperbolic hydrodynamic
model obtained by enforcing the boundedness of lattice velocities derived by
Giona (2018) correctly describes the dynamics of the system and permits to
derive easily the boundary condition at the interface, which, contrarily to the
common belief, involves the lattice velocities in the two phases and not the
phase diffusivities. The dispersion properties of independent particles moving
on an infinite lattice composed by the periodic repetition of a multiphase unit
cell are investigated. It is shown that the hyperbolic transport theory
correctly predicts the effective diffusion coefficient over all the range of
parameter values, while the corresponding continuous parabolic models deriving
from Langevin equations for particle motion fail. The failure of parabolic
transport models is shown via a simple numerical experiment.
