One-dimensional lattices topologically equivalent to three-dimensional lattices within the context of the lattice gas model

Continuum partial differential equations are obtained from a set of discrete stochastic evolution equations of both non-Markovian and Markovian processes and applied to the diffusion on a cubic lattice within the context of the lattice gas model. A procedure allowing to construct one-dimensional lattices that are topologically equivalent to a cubic three-dimensional lattice is described in detail using a successive “unfolding” process. This example shows some new features that possess the procedure and extensions are also suggested in order to provide some another uses of the present approach.