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

C. J. Gommes1,2,*, Y. Jiao2,†, and S. Torquato2,3,4,5,6,‡

  • 1Department of Chemical Engineering, University of Liège, Liège 4000, Belgium
  • 2Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
  • 3Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
  • 4Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
  • 5Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
  • 6Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA

  • *cedric.gommes@ulg.ac.be
  • †yjiao@princeton.edu
  • ‡torquato@electron.princeton.edu
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Phys. Rev. E 85, 051140 – Published 29 May, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.051140

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 [Gommes, Jiao, and Torquato, Phys. Rev. Lett. 108, 080601 (2012)], 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.

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Phys. Rev. E 85, 051140– Published 29 May, 2012
  • Received 16 March 2012

©2012 American Physical Society

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