  We investigate the spatial coarse-graining of interactions in host-guest
systems within the framework of the recently proposed Interacting Pair
Approximation (IPA). Basically, the IPA method derives local effective
interactions from the knowledge of the bivariate histograms of the number of
sorbate molecules (occupancy) in a pair of neighboring subvolumes, taken at
different values of the chemical potential. Here we extend the IPA approach to
the case in which every subvolume is surrounded by more than one class of
neighbors, and we apply it on two systems: methane on a single graphene layer
and methane between two graphene layers, at several temperatures and sorbate
densities. We obtain coarse-grained (CG) adsorption isotherms and reduced
variances of the occupancy in a quantitative agreement with reference atomistic
simulations. A quantitative matching is also obtained for the occupancy
correlations between neighboring subvolumes, apart from the case of high
sorbate densities at low temperature, where the matching is refined by
pre-processing the histograms through a quantized bivariate Gaussian
distribution model.
