642 research outputs found
Anatomy of a Spin: The Information-Theoretic Structure of Classical Spin Systems
Collective organization in matter plays a significant role in its expressed
physical properties. Typically, it is detected via an order parameter,
appropriately defined for each given system's observed emergent patterns.
Recent developments in information theory, however, suggest quantifying
collective organization in a system- and phenomenon-agnostic way: decompose the
system's thermodynamic entropy density into a localized entropy, that solely
contained in the dynamics at a single location, and a bound entropy, that
stored in space as domains, clusters, excitations, or other emergent
structures. We compute this decomposition and related quantities explicitly for
the nearest-neighbor Ising model on the 1D chain, the Bethe lattice with
coordination number k=3, and the 2D square lattice, illustrating its generality
and the functional insights it gives near and away from phase transitions. In
particular, we consider the roles that different spin motifs play (in cluster
bulk, cluster edges, and the like) and how these affect the dependencies
between spins.Comment: 12 pages, 8 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/ising_bmu.ht
Optimizing Quantum Models of Classical Channels: The reverse Holevo problem
Given a classical channel---a stochastic map from inputs to outputs---the
input can often be transformed to an intermediate variable that is
informationally smaller than the input. The new channel accurately simulates
the original but at a smaller transmission rate. Here, we examine this
procedure when the intermediate variable is a quantum state. We determine when
and how well quantum simulations of classical channels may improve upon the
minimal rates of classical simulation. This inverts Holevo's original question
of quantifying the capacity of quantum channels with classical resources. We
also show that this problem is equivalent to another, involving the local
generation of a distribution from common entanglement.Comment: 13 pages, 6 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/qfact.htm; substantially updated
from v
Extreme Quantum Advantage for Rare-Event Sampling
We introduce a quantum algorithm for efficient biased sampling of the rare
events generated by classical memoryful stochastic processes. We show that this
quantum algorithm gives an extreme advantage over known classical biased
sampling algorithms in terms of the memory resources required. The quantum
memory advantage ranges from polynomial to exponential and when sampling the
rare equilibrium configurations of spin systems the quantum advantage diverges.Comment: 11 pages, 9 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/eqafbs.ht
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
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
Synchronization and Control in Intrinsic and Designed Computation: An Information-Theoretic Analysis of Competing Models of Stochastic Computation
We adapt tools from information theory to analyze how an observer comes to
synchronize with the hidden states of a finitary, stationary stochastic
process. We show that synchronization is determined by both the process's
internal organization and by an observer's model of it. We analyze these
components using the convergence of state-block and block-state entropies,
comparing them to the previously known convergence properties of the Shannon
block entropy. Along the way, we introduce a hierarchy of information
quantifiers as derivatives and integrals of these entropies, which parallels a
similar hierarchy introduced for block entropy. We also draw out the duality
between synchronization properties and a process's controllability. The tools
lead to a new classification of a process's alternative representations in
terms of minimality, synchronizability, and unifilarity.Comment: 25 pages, 13 figures, 1 tabl
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
Existence and Stability of Steady Fronts in Bistable CML
We prove the existence and we study the stability of the kink-like fixed
points in a simple Coupled Map Lattice for which the local dynamics has two
stable fixed points. The condition for the existence allows us to define a
critical value of the coupling parameter where a (multi) generalized
saddle-node bifurcation occurs and destroys these solutions. An extension of
the results to other CML's in the same class is also displayed. Finally, we
emphasize the property of spatial chaos for small coupling.Comment: 18 pages, uuencoded PostScript file, J. Stat. Phys. (In press
Forecasting in the light of Big Data
Predicting the future state of a system has always been a natural motivation
for science and practical applications. Such a topic, beyond its obvious
technical and societal relevance, is also interesting from a conceptual point
of view. This owes to the fact that forecasting lends itself to two equally
radical, yet opposite methodologies. A reductionist one, based on the first
principles, and the naive inductivist one, based only on data. This latter view
has recently gained some attention in response to the availability of
unprecedented amounts of data and increasingly sophisticated algorithmic
analytic techniques. The purpose of this note is to assess critically the role
of big data in reshaping the key aspects of forecasting and in particular the
claim that bigger data leads to better predictions. Drawing on the
representative example of weather forecasts we argue that this is not generally
the case. We conclude by suggesting that a clever and context-dependent
compromise between modelling and quantitative analysis stands out as the best
forecasting strategy, as anticipated nearly a century ago by Richardson and von
Neumann
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