  We devise reduced-dimension metrics for effectively measuring the distance
between two points (i.e., microstructures) in the microstructure space and
quantifying the pathway associated with microstructural evolution, based on a
recently introduced set of hierarchical $n$-point polytope functions $P_n$. The
$P_n$ functions provide the probability of finding particular $n$-point
configurations associated with regular $n$-polytopes in the material system,
and a special sub-set of the standard $n$-point correlation functions $S_n$
that effectively decomposes the structural features in the systems into regular
polyhedral basis with different symmetry. The $n$-th order metric $\Omega_n$ is
defined as the $\mathbb{L}_1$ norm associated with the $P_n$ functions of two
distinct microstructures. By choosing a reference initial state (i.e., a
microstructure associated with $t_0 = 0$), the $\Omega_n(t)$ set quantifies the
evolution of distinct polyhedral symmetries and can in principle capture
emerging polyhedral symmetries that are not apparent in the initial state. To
demonstrate their utility, we apply the $\Omega_n$ metrics to a 2D binary
system undergoing spinodal decomposition to extract the phase separation
dynamics via the temporal scaling behavior of the corresponding $\Omega_n(t)$,
which reveals mechanisms governing the evolution. Moreover, we employ
$\Omega_n(t)$ to analyze pattern evolution during vapor-deposition of
phase-separating alloy films with different surface contact angles, which
exhibit rich evolution dynamics including both unstable and oscillating
patterns. The $\Omega_n$ metrics have potential applications in establishing
quantitative processing-structure-property relationships, as well as real-time
processing control and optimization of complex heterogeneous material systems.
