Chaotic crystallography: how the physics of information reveals structural order in materials

Abstract

We review recent progress in applying information-and computation-theoretic measures to describe material structure that transcends previous methods based on exact geometric symmetries. We discuss the necessary theoretical background for this new toolset and show how the new techniques detect and describe novel material properties. We discuss how the approach relates to well known crystallographic practice and examine how it provides novel interpretations of familiar structures. Throughout, we concentrate on disordered materials that, while important, have received less attention both theoretically and experimentally than those with either periodic or aperiodic order.

The ambiguity of simplicity in quantum and classical simulation

“…The Heisenberg spin chain is realized experimentally in the quasi-1D magnetic order found in antiferromagnetic KCuF 3 crystals [36][37][38]. One can then adapt the methods of 1D chaotic crystallography [39] to extract the -machine and quantum-machine descriptions of the quantum crystalline structure from scattering measurements. These and perhaps other experiments will provide an entrée to analyzing the ambiguity of simplicity in quantum systems.…”

Multivariate Dependence beyond Shannon Information

“…Due to its focus on probability distributions, it allows one to compare dissimilar systems (e.g., species abundance to ground state configurations of a spin system) and has found many successes in the physical, biological and social sciences [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] far outside its original domain of communication. Often, the issue on which it is brought to bear is discovering and quantifying dependencies [20][21][22][23][24][25].…”

Pairwise correlations in layered close-packed structures

“…It is our contention that an "-machine describing a specimen's stacking includes all of the structural information necessary to calculate physical quantities that depend on the stacking statistics (Varn & Crutchfield, 2015). In the following, we demonstrate how pairwise correlation functions can be calculated either analytically or to a high degree of numerical certainty for an arbitrary HMM and, thus, for an arbitrary "-machine.…”

“…While crystallography has historically struggled to settle on a formalism to describe disordered structures, we propose that such a framework has been identified, at least for layered materials. Based in computational mechanics, chaotic crystallography (Varn & Crutchfield, 2015) employs information theory as a key component to characterize disordered materials. Although the use of information theory in crystallography has been previously proposed by Mackay and coworkers (Mackay, 1986(Mackay, , 2002Cartwright & Mackay, 2012), chaotic crystallography realizes this goal.…”

“…For disordered materials, instead of using exact group symmetries, it has recently been suggested that the partial symmetries observed in these materials can be represented using semigroups. This new formalism, called chaotic crystallography (Varn & Crutchfield, 2015), employs information-and computation-theoretic measures to characterize and classify material structure. Chaotic crystallography provides a unified platform not only to calculate physical quantities of interest but also to give insight into their physical structure.…”