Journal ArticleDOI
Statistical mechanics of cellular automata
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Analysis is given of ''elementary'' cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to p definite rules involving the values of its nearest neighbors.Abstract:
Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions \ensuremath{\simeq} 1.59 or \ensuremath{\simeq} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.read more
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Journal ArticleDOI
Computation theoretic aspects of cellular automata
K. Culik,L. P. Hurd,S. Yu +2 more
TL;DR: This paper surveys results largely motivated by an attempt to answer questions posed by the approach to cellular automata by means of techniques from the former.
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A local activator-inhibitor model of vertebrate skin patterns
TL;DR: A model for vertebrate skin patterns is presented in which the differentiated pigment cells produce two diffusible morphogens, an activator and an inhibitor.
Journal ArticleDOI
Finite Size Scaling in Three-Dimensional Bootstrap Percolation
TL;DR: In this article, the authors consider the problem of bootstrap percolation on a three-dimensional lattice and study its finite size scaling behavior, showing that the scaling function for this problem is of the form f(L) = const/ In In L. They prove a conjecture proposed by A. C. van Enter.
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The attractor—basin portrait of a cellular automaton
TL;DR: In this article, the authors describe the attractor-basin portrait of nonlinear elementary cellular automata rule 18, whose global dynamics is largely determined by a single regular attracting domain.
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Applications of ergodic theory and sofic systems to cellular automata
TL;DR: It is shown that cellular automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group, and therefore their statistical and dynamical behavior can be told exactly by using Fourier analysis and ergodic theory.
References
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Book
Introduction to Automata Theory, Languages, and Computation
TL;DR: This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity, appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.
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The Chemical Basis of Morphogenesis
TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
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On Computable Numbers, with an Application to the Entscheidungsproblem
TL;DR: This chapter discusses the application of the diagonal process of the universal computing machine, which automates the calculation of circle and circle-free numbers.
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Metabolic stability and epigenesis in randomly constructed genetic nets
TL;DR: The hypothesis that contemporary organisms are also randomly constructed molecular automata is examined by modeling the gene as a binary (on-off) device and studying the behavior of large, randomly constructed nets of these binary “genes”.
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Diffusion-limited aggregation, a kinetic critical phenomenon
Abstract: A model for random aggregates is studied by computer simulation The model is applicable to a metal-particle aggregation process whose correlations have been measured previously Density correlations within the model aggregates fall off with distance with a fractional power law, like those of the metal aggregates The radius of gyration of the model aggregates has power-law behavior The model is a limit of a model of dendritic growth