  Disordered systems are ubiquitous in physical, biological and material
sciences. Examples include liquid and glassy states of condensed matter,
colloids, granular materials, porous media, composites, alloys, packings of
cells in avian retina and tumor spheroids, to name but a few. A comprehensive
understanding of such disordered systems requires, as the first step,
systematic quantification, modeling and representation of the underlying
complex configurations and microstructure, which is generally very challenging
to achieve. Recently, we introduce a set of hierarchical statistical
microstructural descriptors, i.e., the n-point polytope functions Pn, which are
derived from the standard n-point correlation functions Sn, and successively
include higher-order n-point statistics of the morphological features of
interest in a concise, explainable, and expressive manner. Here we investigate
the information content of the Pn functions via optimization-based realization
rendering. This is achieved by successively incorporating higher order Pn
functions up to n = 8 and quantitatively assessing the accuracy of the
reconstructed systems via un-constrained statistical morphological descriptors
(e.g., the lineal-path function). We examine a wide spectrum of representative
random systems with distinct geometrical and topological features. We find that
generally, successively incorporating higher order Pn functions, and thus, the
higher-order morphological information encoded in these descriptors, leads to
superior accuracy of the reconstructions. However, incorporating more Pn
functions into the reconstruction also significantly increases the complexity
and roughness of the associated energy landscape for the underlying stochastic
optimization, making it difficult to convergence numerically.
