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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Book
Analysis of Lattice-Boltzmann Methods: Asymptotic and numeric investigation of a singularly perturbed system
Abstract: Lattice-Boltzmann algorithms represent a quite novel class of numerical schemes, which are used to solve evolutionary partial differential equations (PDE). In contrast to finite difference and finite element schemes, lattice-Boltzmann methods rely on a mesoscopic (kinetic) approach. The essential idea consists in setting up an artificial, grid-based particle dynamics, which is chosen such that appropriate averages provide approximate solutions of a certain PDE, typically in the area of fluid dynamics. As lattice-Boltzmann schemes are closely related to finite velocity Boltzmann equations being singularly perturbed by special scalings, their consistency is not obvious, however. This work is concerned with the analysis of lattice-Boltzmann methods, where a particular interest lies in some numeric phenomena like initial layers, multiple time scales and boundary layers. As major analytic tool, regular expansions (Hilbert expansion) are employed to establish consistency. Exemplarily, two and three population algorithms are studied in one space dimension, mostly discretizing the advection-diffusion equation. It is shown how these ‘model schemes’ can be derived from two dimensional schemes in the case of special symmetries. The analysis of the schemes is preceded by an examination of the singular limit being characteristic of the corresponding scaled finite velocity Boltzmann equations. Convergence proofs are obtained using a Fourier series approach and alternatively a general regular expansion combined with an energy estimate. The appearance of initial layers is investigated by multiscale and irregular expansions. Among others, a hierarchy of equations is found which gives insight into the internal coupling of the initial layer and the regular part of the solution. Next, the consistency of the model algorithms is considered followed by a discussion of stability. Apart from proving stability for several cases entailing convergence as byproduct, the spectrum of the evolution operator is examined in detail. Based on this, it is shown that the CFL-condition is necessary and sufficient for stability in the case of a two population algorithm discretizing the advection equation. Furthermore, the presentation touches upon the question whether reliable stability statements can be obtained by rather formal arguments. To gather experience and prepare future work, numeric boundary layers are analyzed in the context of a finite difference discretization for the one-dimensional Poisson equation.
Journal ArticleDOI
Topological classification of cellular automata
TL;DR: A new classification of CA is proposed, complementary to that of Wolfram, in which attractor globality is important, and with the use of fixed boundary conditions he finds global periodic attractors in CA for the first time.
Journal ArticleDOI
Cellular Automata on Graphs: Topological Properties of ER Graphs Evolved towards Low-Entropy Dynamics
Carsten Marr,Marc-Thorsten Hütt +1 more
TL;DR: This work extends the investigation towards graphs obtained in a simulated-evolution procedure, starting from Erdő s–Renyi (ER) graphs and selecting for low entropies of the CA dynamics, finding a strong association of low Shannon entropy with a broadening of the graph’s degree distribution.
Journal ArticleDOI
Cellular-automata-array-based diagnosis of board level faults
TL;DR: A novel scheme for board level fault diagnosis based on cellular automata array (CAA) is presented, which results in a simple and modular test structure that is well suited for VLSI implementation.
Book ChapterDOI
Theory of Composing Non-linear Machines with Predictable Cyclic Structures
TL;DR: It is shown that with appropriate choice of the rules of the linear CA the authors may obtain invertible, balanced Boolean mappings with strong non-linearity, and it is proved that such a scheme generates machines with state transitions having predictable cyclic properties.
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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”.
Journal ArticleDOI
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