  Diffusion is the macroscopic manifestation of disordered molecular motion.
Mathematically, diffusion equations are partial differential equations
describing the fluid-like large-scale dynamics of parcels of molecules.
Spatially inhomogeneous systems affect in a position-dependent way the average
motion of molecules; thus, diffusion equations have to reflect somehow this
fact within their structure. It is known since long that in this case an
ambiguity arises: there are several ways of writing down diffusion equations
containing space dependence within their parameters. These ways are all
potentially valid but not equivalent, meaning that the different diffusion
equations yield different solutions for the same data. The ambiguity can only
be resolved at the microscopic level: a model for the stochastic dynamics of
the individual molecules must be provided, and a well-defined diffusion
equation then arises as the long-wavelength limit of this dynamics. In this
work we introduce and employ the integro-differential Master Equation (ME) as a
tool for describing the microscopic dynamics. We show that is possible to
provide a parameterized version of the ME, in terms of a single numerical
parameter (alpha), such that the different expressions for the diffusive fluxes
are recovered at different values of alpha. This work aims to fill a gap in the
literature, where the ME was shown to deliver just one diffusive limit. In the
second part of the paper some numerical computer models are introduced, both to
support analytical considerations, and to extend the scope of the ME to more
sophisticated scenarios, beyond the simplest alpha-parameterization.
