Showing papers by "James P. Crutchfield published in 2013"
TL;DR: The theoretical development behind a new technique for discovering and describing planar disorder in close-packed structures directly from their diffraction patterns is provided, adapting computational mechanics to describe one-dimensional structure in materials.
Abstract: In previous publications [Varn et al. (2002). Phys. Rev. B, 66, 174110; Varn et al. (2007). Acta Cryst. B63, 169–182] we introduced and applied a new technique for discovering and describing planar disorder in close-packed structures directly from their diffraction patterns. Here, we provide the theoretical development behind those results, adapting computational mechanics to describe one-dimensional structure in materials. We show that the resulting statistical model of the stacking structure – called the ∊-machine – allows the calculation of measures of memory, structural complexity and configurational entropy. The methods developed here can be adapted to a wide range of experimental systems in which power spectra data are available.
35 citations
TL;DR: Here εMSR is applied to simulated diffraction patterns from four close-packed crystals and it is found that, for stacking structures with a memory length of three or less, ε MSR reproduces the statistics of the stacking structure in the form of a directed graph called an ε-machine.
Abstract: A previous paper detailed a novel algorithm, ∊-machine spectral reconstruction theory (∊MSR), that infers pattern and disorder in planar-faulted, close-packed structures directly from X-ray diffraction patterns [Varn et al. (2013). Acta Cryst. A69, 197–206]. Here ∊MSR is applied to simulated diffraction patterns from four close-packed crystals. It is found that, for stacking structures with a memory length of three or less, ∊MSR reproduces the statistics of the stacking structure; the result being in the form of a directed graph called an ∊-machine. For stacking structures with a memory length larger than three, ∊MSR returns a model that captures many important features of the original stacking structure. These include multiple stacking faults and multiple crystal structures. Further, it is found that ∊MSR is able to discover stacking structure in even highly disordered crystals. In order to address issues concerning the long-range order observed in many classes of layered materials, several length parameters are defined, calculable from the ∊-machine, and their relevance is discussed.
15 citations
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TL;DR: The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's epsilon-machine causal-state dynamic as mentioned in this paper.
Abstract: We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's epsilon-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organization of finitary hidden Markov models and nonlinear dynamical systems with generating partitions and for the spatial organization in one-dimensional systems, including spin systems, cellular automata, and complex materials via chaotic crystallography.
7 citations
TL;DR: In this paper, the authors introduce an intersection information measure based on the G\'acs-Korner common random variable that is the first to satisfy the coveted target monotonicity property.
Abstract: The introduction of the partial information decomposition generated a flurry of proposals for defining an intersection information that quantifies how much of "the same information" two or more random variables specify about a target random variable. As of yet, none is wholly satisfactory. A palatable measure of intersection information would provide a principled way to quantify slippery concepts, such as synergy. Here, we introduce an intersection information measure based on the G\'acs-K\"orner common random variable that is the first to satisfy the coveted target monotonicity property. Our measure is imperfect, too, and we suggest directions for improvement.
2 citations
TL;DR: In the version of this review article originally published online there were several errors as mentioned in this paper, including the following: in the section entitled "Complicated yes, but is it complex?", seventh paragraph, the subscripts should appear as follows: past X:t; future Xt:; and blocks Xt :t'.
Abstract: Nature Phys. 8, 17–24 (2012); published online 22 December 2011; corrected online 17 May 2013. In the version of this Review Article originally published online there were several errors. In the section entitled 'Complicated yes, but is it complex?', seventh paragraph, the subscripts should appear as follows: past X:t; future Xt:; and blocks Xt:t'.
1 citations