  Two-point correlation functions provide crucial yet incomplete
characterization of microstructures because different microstructures may have
the same correlation function. In an earlier Letter [Phys. Rev. Lett. 108,
080601 (2012)], we addressed the degeneracy question: What is the number of
microstructures compatible with a specified correlation function? We computed
this degeneracy, i.e., configurational entropy, in the framework of
reconstruction methods, which enabled us to map the problem to the
determination of ground-state degeneracies. Here, we provide a more
comprehensive presentation and additional results. Since the configuration
space of a reconstruction problem is a hypercube on which a Hamming distance is
defined, we can calculate analytically an energy profile corresponding to the
average energy of all microstructures at a given Hamming distance from a ground
state. The steepness of this profile is a measure of the roughness of the
energy landscape, which can be used as a proxy for ground-state degeneracy. The
relationship between roughness metric and ground-state degeneracy is calibrated
using a Monte Carlo algorithm for determining the degeneracy of a variety of
microstructures, including hard disks and Poisson point processes as well as
those with known degeneracies (single disks of various sizes and a particular
crystalline microstructure). We show that our results can be expressed in terms
of the information content of the two-point correlation functions. From this
perspective, the a priori condition for a reconstruction to be accurate is that
the information content, expressed in bits, should be comparable to the number
of pixels in the unknown microstructure. We provide a formula to calculate the
information content of any two-point correlation function, which makes our
results broadly applicable to any field in which correlation functions are
employed.
