127,659 research outputs found
A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics
From the output produced by a memoryless deletion channel from a uniformly
random input of known length , one obtains a posterior distribution on the
channel input. The difference between the Shannon entropy of this distribution
and that of the uniform prior measures the amount of information about the
channel input which is conveyed by the output of length , and it is natural
to ask for which outputs this is extremized. This question was posed in a
previous work, where it was conjectured on the basis of experimental data that
the entropy of the posterior is minimized and maximized by the constant strings
and and the alternating strings
and respectively. In the present
work we confirm the minimization conjecture in the asymptotic limit using
results from hidden word statistics. We show how the analytic-combinatorial
methods of Flajolet, Szpankowski and Vall\'ee for dealing with the hidden
pattern matching problem can be applied to resolve the case of fixed output
length and , by obtaining estimates for the entropy in
terms of the moments of the posterior distribution and establishing its
minimization via a measure of autocorrelation.Comment: 11 pages, 2 figure
Polydisperse Adsorption: Pattern Formation Kinetics, Fractal Properties, and Transition to Order
We investigate the process of random sequential adsorption of polydisperse
particles whose size distribution exhibits a power-law dependence in the small
size limit, . We reveal a relation between pattern
formation kinetics and structural properties of arising patterns. We propose a
mean-field theory which provides a fair description for sufficiently small
. When , highly ordered structures locally identical
to the Apollonian packing are formed. We introduce a quantitative criterion of
the regularity of the pattern formation process. When , a sharp
transition from irregular to regular pattern formation regime is found to occur
near the jamming coverage of standard random sequential adsorption with
monodisperse size distribution.Comment: 8 pages, LaTeX, 5 figures, to appear in Phys.Rev.
Chaos for Liouville probability densities
Using the method of symbolic dynamics, we show that a large class of
classical chaotic maps exhibit exponential hypersensitivity to perturbation,
i.e., a rapid increase with time of the information needed to describe the
perturbed time evolution of the Liouville density, the information attaining
values that are exponentially larger than the entropy increase that results
from averaging over the perturbation. The exponential rate of growth of the
ratio of information to entropy is given by the Kolmogorov-Sinai entropy of the
map. These findings generalize and extend results obtained for the baker's map
[R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992)].Comment: 26 pages in REVTEX, no figures, submitted to Phys. Rev.
Fractal formation and ordering in random sequential adsorption
We reveal the fractal nature of patterns arising in random sequential
adsorption of particles with continuum power-law size distribution, , . We find that the patterns become more and
more ordered as increases, and that the Apollonian packing is obtained
at limit. We introduce the entropy production rate as a
quantitative criteria of regularity and observe a transition from an irregular
regime of the pattern formation to a regular one. We develop a scaling theory
that relates kinetic and structural properties of the system.Comment: 4 pages, RevTex, 4 postscript figures. To appear in Phys.Rev.Let
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
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