311 research outputs found

    Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients

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    The foundations of the chaotic scattering theory for transport and reaction-rate coefficients for classical many-body systems are considered here in some detail. The thermodynamic formalism of Sinai, Bowen, and Ruelle is employed to obtain an expression for the escape-rate for a phase space trajectory to leave a finite open region of phase space for the first time. This expression relates the escape rate to the difference between the sum of the positive Lyapunov exponents and the K-S entropy for the fractal set of trajectories which are trapped forever in the open region. This result is well known for systems of a few degrees of freedom and is here extended to systems of many degrees of freedom. The formalism is applied to smooth hyperbolic systems, to cellular-automata lattice gases, and to hard sphere sytems. In the latter case, the goemetric constructions of Sinai {\it et al} for billiard systems are used to describe the relevant chaotic scattering phenomena. Some applications of this formalism to non-hyperbolic systems are also discussed.Comment: 35 pages, compressed file, follow directions in header for ps file. Figures are available on request from [email protected]

    Von Neumann Regular Cellular Automata

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    For any group GG and any set AA, a cellular automaton (CA) is a transformation of the configuration space AGA^G defined via a finite memory set and a local function. Let CA(G;A)\text{CA}(G;A) be the monoid of all CA over AGA^G. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element τCA(G;A)\tau \in \text{CA}(G;A) is von Neumann regular (or simply regular) if there exists σCA(G;A)\sigma \in \text{CA}(G;A) such that τστ=τ\tau \circ \sigma \circ \tau = \tau and στσ=σ\sigma \circ \tau \circ \sigma = \sigma, where \circ is the composition of functions. Such an element σ\sigma is called a generalised inverse of τ\tau. The monoid CA(G;A)\text{CA}(G;A) itself is regular if all its elements are regular. We establish that CA(G;A)\text{CA}(G;A) is regular if and only if G=1\vert G \vert = 1 or A=1\vert A \vert = 1, and we characterise all regular elements in CA(G;A)\text{CA}(G;A) when GG and AA are both finite. Furthermore, we study regular linear CA when A=VA= V is a vector space over a field F\mathbb{F}; in particular, we show that every regular linear CA is invertible when GG is torsion-free elementary amenable (e.g. when G=Zd, dNG=\mathbb{Z}^d, \ d \in \mathbb{N}) and V=FV=\mathbb{F}, and that every linear CA is regular when VV is finite-dimensional and GG is locally finite with Char(F)o(g)\text{Char}(\mathbb{F}) \nmid o(g) for all gGg \in G.Comment: 10 pages. Theorem 5 corrected from previous versions, in A. Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer, 201

    The Kinetic Basis of Self-Organized Pattern Formation

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    In his seminal paper on morphogenesis (1952), Alan Turing demonstrated that different spatio-temporal patterns can arise due to instability of the homogeneous state in reaction-diffusion systems, but at least two species are necessary to produce even the simplest stationary patterns. This paper is aimed to propose a novel model of the analog (continuous state) kinetic automaton and to show that stationary and dynamic patterns can arise in one-component networks of kinetic automata. Possible applicability of kinetic networks to modeling of real-world phenomena is also discussed.Comment: 8 pages, submitted to the 14th International Conference on the Synthesis and Simulation of Living Systems (Alife 14) on 23.03.2014, accepted 09.05.201

    Estimating topological entropy of multidimensional nonlinear cellular automata

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    Cellular automata are discrete dynamical systems whose configurations are determined by local rules acting on each cell in synchronous. Topological entoropy is a tool for measuring the complexity of these dynamical systems. In this paper we estimate topological entropy of a two-dimensional nonlinear cellular automaton. The method we use is that a one-dimensional cellular automaton with positive topological entoropy is “naturally” embedded into the twodimensional cellular automaton. Hence we obtain a multidimensional cellular automaton with infinite topological entoropy
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