493 research outputs found
Diffraction Patterns of Layered Close-packed Structures from Hidden Markov Models
We recently derived analytical expressions for the pairwise (auto)correlation
functions (CFs) between modular layers (MLs) in close-packed structures (CPSs)
for the wide class of stacking processes describable as hidden Markov models
(HMMs) [Riechers \etal, (2014), Acta Crystallogr.~A, XX 000-000]. We now use
these results to calculate diffraction patterns (DPs) directly from HMMs,
discovering that the relationship between the HMMs and DPs is both simple and
fundamental in nature. We show that in the limit of large crystals, the DP is a
function of parameters that specify the HMM. We give three elementary but
important examples that demonstrate this result, deriving expressions for the
DP of CPSs stacked (i) independently, (ii) as infinite-Markov-order randomly
faulted 2H and 3C stacking structures over the entire range of growth and
deformation faulting probabilities, and (iii) as a HMM that models
Shockley-Frank stacking faults in 6H-SiC. While applied here to planar faulting
in CPSs, extending the methods and results to planar disorder in other layered
materials is straightforward. In this way, we effectively solve the broad
problem of calculating a DP---either analytically or numerically---for any
stacking structure---ordered or disordered---where the stacking process can be
expressed as a HMM.Comment: 18 pages, 6 figures, 3 tables;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dplcps.ht
Chaotic Crystallography: How the physics of information reveals structural order in materials
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.Comment: 9 pages, two figures, 1 table;
http://csc.ucdavis.edu/~cmg/compmech/pubs/ChemOpinion.ht
The Origins of Computational Mechanics: A Brief Intellectual History and Several Clarifications
The principle goal of computational mechanics is to define pattern and
structure so that the organization of complex systems can be detected and
quantified. Computational mechanics developed from efforts in the 1970s and
early 1980s to identify strange attractors as the mechanism driving weak fluid
turbulence via the method of reconstructing attractor geometry from measurement
time series and in the mid-1980s to estimate equations of motion directly from
complex time series. In providing a mathematical and operational definition of
structure it addressed weaknesses of these early approaches to discovering
patterns in natural systems.
Since then, computational mechanics has led to a range of results from
theoretical physics and nonlinear mathematics to diverse applications---from
closed-form analysis of Markov and non-Markov stochastic processes that are
ergodic or nonergodic and their measures of information and intrinsic
computation to complex materials and deterministic chaos and intelligence in
Maxwellian demons to quantum compression of classical processes and the
evolution of computation and language.
This brief review clarifies several misunderstandings and addresses concerns
recently raised regarding early works in the field (1980s). We show that
misguided evaluations of the contributions of computational mechanics are
groundless and stem from a lack of familiarity with its basic goals and from a
failure to consider its historical context. For all practical purposes, its
modern methods and results largely supersede the early works. This not only
renders recent criticism moot and shows the solid ground on which computational
mechanics stands but, most importantly, shows the significant progress achieved
over three decades and points to the many intriguing and outstanding challenges
in understanding the computational nature of complex dynamic systems.Comment: 11 pages, 123 citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/cmr.ht
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
Language extraction from ZnS
Perhaps the most fundamental questions we can ask about a solid are What is it made of? and How are the constituent parts assembled? This is so elementary, and yet so basic to any detailed understanding of the thermal, electrical, magnetic, optical, and elastic properties of materials. At the beginning of the twenty-first century, concern over the placement of the atoms in a solid seems quaint and anachronistic, more suited to the dawn of the twentieth century. X-ray diffraction, electron diffraction, optical microscopy, x-ray diffraction tomography, to name a few, are powerful techniques to uncover structure in solids. With this arsenal of tools, and the efforts of many researchers, surely we can have nothing novel to say about the discovery and description of structure in solids, save perhaps the refinement of well-worn techniques or the analysis of particularly obstinate cases. But careful examination of present technology reveals that while we are quite good at finding and describing periodic order in nature, cases that lack such order are much more difficult. Certainly in the complete absence of structural order, as in a gas, statistical methods exist that permit a satisfying understanding of the properties of the system without knowing ( or even wanting to know) the details of the microscopic placement of the constituents. But it is the in-between cases, where order and disorder coexist, that has proven so elusive to both analyze and describe. In this thesis, we will tackle these in-between cases for a special type of layered material, called polytypes. They exhibit disorder in one dimension only, making the analysis more tractable. We will give a method for determining the structure of these solids from experimental data and demonstrate how this structure, both the random and the non-random part, can be compactly expressed. From our solution, we will be able to calculate the effective range of the inter-layer interactions, as well as the configurational energies of the disordered stacking sequences
Fraudulent white noise: Flat power spectra belie arbitrarily complex processes
Power spectral densities are a common, convenient, and powerful way to analyze signals, so much so that they are now broadly deployed across the sciences and engineering - from quantum physics to cosmology and from crystallography to neuroscience to speech recognition. The features they reveal not only identify prominent signal frequencies but also hint at mechanisms that generate correlation and lead to resonance. Despite their near-centuries-long run of successes in signal analysis, here we show that flat power spectra can be generated by highly complex processes, effectively hiding all inherent structure in complex signals. Historically, this circumstance has been widely misinterpreted, being taken as the renowned signature of "structureless"white noise - the benchmark of randomness. We argue, in contrast, to the extent that most real-world complex systems exhibit correlations beyond pairwise statistics their structures evade power spectra and other pairwise statistical measures. As concrete physical examples, we demonstrate that fraudulent white noise hides the predictable structure of both entangled quantum systems and chaotic crystals. To make these words of warning operational, we present constructive results that explore how this situation comes about and the high toll it takes in understanding complex mechanisms. First, we give the closed-form solution for the power spectrum of a very broad class of structurally complex signal generators. Second, we demonstrate the close relationship between eigenspectra of evolution operators and power spectra. Third, we characterize the minimal generative structure implied by any power spectrum. Fourth, we show how to construct arbitrarily complex processes with flat power spectra. Finally, leveraging this diagnosis of the problem, we point the way to developing more incisive tools for discovering structure in complex signals
How soft materials control harder ones: routes to bioorganization
Ordered structures are remarkably common, even without direct human guidance or direction. The ordering can be at the atomic scale or on the macroscopic scale or at the mesoscale. The term 'self-organization' is often used, but this description is facile, giving no hint as to the range or variety of mechanisms. Ordering can occur in circumstances commonly associated with disorder, as in the irradiation of metals to high doses; it can also occur when soft, flexible materials organize structures of harder, rigid structures. My review attempts to analyse some of these widely varying behaviours, both to seek evidence of common underlying principles and to assess how organization might be controlled, and with what level of accuracy
Spectral Simplicity of Apparent Complexity, Part II: Exact Complexities and Complexity Spectra
The meromorphic functional calculus developed in Part I overcomes the
nondiagonalizability of linear operators that arises often in the temporal
evolution of complex systems and is generic to the metadynamics of predicting
their behavior. Using the resulting spectral decomposition, we derive
closed-form expressions for correlation functions, finite-length Shannon
entropy-rate approximates, asymptotic entropy rate, excess entropy, transient
information, transient and asymptotic state uncertainty, and synchronization
information of stochastic processes generated by finite-state hidden Markov
models. This introduces analytical tractability to investigating information
processing in discrete-event stochastic processes, symbolic dynamics, and
chaotic dynamical systems. Comparisons reveal mathematical similarities between
complexity measures originally thought to capture distinct informational and
computational properties. We also introduce a new kind of spectral analysis via
coronal spectrograms and the frequency-dependent spectra of past-future mutual
information. We analyze a number of examples to illustrate the methods,
emphasizing processes with multivariate dependencies beyond pairwise
correlation. An appendix presents spectral decomposition calculations for one
example in full detail.Comment: 27 pages, 12 figures, 2 tables; most recent version at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt2.ht
- …