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Microstructural degeneracy associated with a two-point correlation function and its information content

Author(s): Gommes, Cedric J.; Jiao, Yang; Torquato, Salvatore

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Abstract: A two-point correlation function provides a crucial yet an incomplete characterization of a microstructure because distinctly different microstructures may have the same correlation function. In an earlier Letter, we addressed the microstructural 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 of the methodology and analyses, as well as additional results. Since the configuration space of a reconstruction problem is a hypercube on which a Hamming distance is defined, we can calculate analytically the energy profile of any reconstruction problem, corresponding to the average energy of all microstructures at a given Hamming distance from a ground state. The steepness of the energy profile is a measure of the roughness of the energy landscape associated with the reconstruction problem, which can be used as a proxy for the ground-state degeneracy. The relationship between this roughness metric and the ground-state degeneracy is calibrated using a Monte Carlo algorithm for determining the ground-state degeneracy of a variety of microstructures, including realizations of hard disks and Poisson point processes at various densities as well as those with known degeneracies (e.g., 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. © 2012 American Physical Society.
Publication Date: May-2012
Electronic Publication Date: 29-May-2012
Citation: Gommes, C.J., Jiao, Y., Torquato, S.. (2012). Microstructural degeneracy associated with a two-point correlation function and its information content. Physical Review E, 85 (5), 10.1103/PhysRevE.85.051140
DOI: doi:10.1103/PhysRevE.85.051140
ISSN: 1539-3755
EISSN: 1550-2376
Pages: 85, 051140-1 - 051140-16
Type of Material: Journal Article
Journal/Proceeding Title: Physical Review E
Version: Final published version. Article is made available in OAR by the publisher's permission or policy.
Notes: Volume 85, Issue 5, 2012, Article number 051140

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