585 research outputs found
A measure of statistical complexity based on predictive information with application to finite spin systems
NOTICE: this is the author’s version of a work that was accepted for publication in 'Physical Letters A'. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in PHYSICAL LETTERS A, 376 (4): 275-281, JAN 2012. DOI:10.1016/j.physleta.2011.10.066
Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets
We introduce a family of Maxwellian Demons for which correlations among
information bearing degrees of freedom can be calculated exactly and in compact
analytical form. This allows one to precisely determine Demon functional
thermodynamic operating regimes, when previous methods either misclassify or
simply fail due to approximations they invoke. This reveals that these Demons
are more functional than previous candidates. They too behave either as
engines, lifting a mass against gravity by extracting energy from a single heat
reservoir, or as Landauer erasers, consuming external work to remove
information from a sequence of binary symbols by decreasing their individual
uncertainty. Going beyond these, our Demon exhibits a new functionality that
erases bits not by simply decreasing individual-symbol uncertainty, but by
increasing inter-bit correlations (that is, by adding temporal order) while
increasing single-symbol uncertainty. In all cases, but especially in the new
erasure regime, exactly accounting for informational correlations leads to
tight bounds on Demon performance, expressed as a refined Second Law of
Thermodynamics that relies on the Kolmogorov-Sinai entropy for dynamical
processes and not on changes purely in system configurational entropy, as
previously employed. We rigorously derive the refined Second Law under minimal
assumptions and so it applies quite broadly---for Demons with and without
memory and input sequences that are correlated or not. We note that general
Maxwellian Demons readily violate previously proposed, alternative such bounds,
while the current bound still holds.Comment: 13 pages, 9 figures,
http://csc.ucdavis.edu/~cmg/compmech/pubs/mrd.ht
Reductions of Hidden Information Sources
In all but special circumstances, measurements of time-dependent processes
reflect internal structures and correlations only indirectly. Building
predictive models of such hidden information sources requires discovering, in
some way, the internal states and mechanisms. Unfortunately, there are often
many possible models that are observationally equivalent. Here we show that the
situation is not as arbitrary as one would think. We show that generators of
hidden stochastic processes can be reduced to a minimal form and compare this
reduced representation to that provided by computational mechanics--the
epsilon-machine. On the way to developing deeper, measure-theoretic foundations
for the latter, we introduce a new two-step reduction process. The first step
(internal-event reduction) produces the smallest observationally equivalent
sigma-algebra and the second (internal-state reduction) removes sigma-algebra
components that are redundant for optimal prediction. For several classes of
stochastic dynamical systems these reductions produce representations that are
equivalent to epsilon-machines.Comment: 12 pages, 4 figures; 30 citations; Updates at
http://www.santafe.edu/~cm
Structural Information in Two-Dimensional Patterns: Entropy Convergence and Excess Entropy
We develop information-theoretic measures of spatial structure and pattern in
more than one dimension. As is well known, the entropy density of a
two-dimensional configuration can be efficiently and accurately estimated via a
converging sequence of conditional entropies. We show that the manner in which
these conditional entropies converge to their asymptotic value serves as a
measure of global correlation and structure for spatial systems in any
dimension. We compare and contrast entropy-convergence with mutual-information
and structure-factor techniques for quantifying and detecting spatial
structure.Comment: 11 pages, 5 figures,
http://www.santafe.edu/projects/CompMech/papers/2dnnn.htm
Spiral Waves in Chaotic Systems
Spiral waves are investigated in chemical systems whose underlying
spatially-homogeneous dynamics is governed by a deterministic chaotic
attractor. We show how the local periodic behavior in the vicinity of a spiral
defect is transformed to chaotic dynamics far from the defect. The
transformation occurs by a type of period doubling as the distance from the
defect increases. The change in character of the dynamics is described in terms
of the phase space flow on closed curves surrounding the defect.Comment: latex file with three postscript figures to appear in Physical review
Letter
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
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Wave chaotic behaviour generated by linear systems
It is shown that regimes with dynamical chaos are inherent not only to nonlinear system but they can be generated by initially linear systems and the requirements for chaotic dynamics and characteristics need further elaboration. Three simplest physical models are considered as examples. In the first, dynamic chaos in the interaction of three linear oscillators is investigated. Analogous process is shown in the second model of electromagnetic wave scattering in a double periodical inhomogeneous medium occupying half-space. The third model is a linear parametric problem for the electromagnetic field in homogeneous dielectric medium which permittivity is modulated in time
Many Roads to Synchrony: Natural Time Scales and Their Algorithms
We consider two important time scales---the Markov and cryptic orders---that
monitor how an observer synchronizes to a finitary stochastic process. We show
how to compute these orders exactly and that they are most efficiently
calculated from the epsilon-machine, a process's minimal unifilar model.
Surprisingly, though the Markov order is a basic concept from stochastic
process theory, it is not a probabilistic property of a process. Rather, it is
a topological property and, moreover, it is not computable from any
finite-state model other than the epsilon-machine. Via an exhaustive survey, we
close by demonstrating that infinite Markov and infinite cryptic orders are a
dominant feature in the space of finite-memory processes. We draw out the roles
played in statistical mechanical spin systems by these two complementary length
scales.Comment: 17 pages, 16 figures:
http://cse.ucdavis.edu/~cmg/compmech/pubs/kro.htm. Santa Fe Institute Working
Paper 10-11-02
Minimum memory for generating rare events
We classify the rare events of structured, memoryful stochastic processes and use this to analyze sequential and parallel generators for these events. Given a stochastic process, we introduce a method to construct a process whose typical realizations are a given process' rare events. This leads to an expression for the minimum memory required to generate rare events. We then show that the recently discovered classical-quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations
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