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Cellular Automata And Lattice Gases

Deterministic Chaos in Tropical Atmospheric Dynamics

We examine an 11-year data set from the tropical weather station of Tlaxcala, Mexico. We find that mutual information drops quickly with the delay, to a positive value which relaxes to zero with a time scale of 20 days. We also examine the mutual dependence of the observables and conclude that the data set gives the equivalent of 8 variables per day, known to a precision of 2% . We determine the effective dimension of the attractor to be D eff ≈11.7 at the scale 3.5%<R/ R max <8% . We find evidence that the effective dimension increases as R/ R max →0 , supporting a conjecture by Lorenz that the climate system may consist of a large number of weakly coupled subsystems, some of which have low-dimensional attractors. We perform a local reconstruction of the dynamics in phase space; the short-term predictability is modest and agrees with theoretical estimates. Useful skill in predictions of 10-day rainfall accumulation anomalies reflects the persistence of weather patterns, which follow the 20-day decay rate of the mutual information.

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Cellular Automata And Lattice Gases

Diffusion in Lorentz Lattice Gas Cellular Automata: the honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found ranging from propagation, super-diffusion, normal, quasi-normal, anomalous to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far, is summarized.

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Cellular Automata And Lattice Gases

Diffusion in a multi-component Lattice Boltzmann Equation model

Diffusion phenomena in a multiple component lattice Boltzmann Equation (LBE) model are discussed in detail. The mass fluxes associated with different mechanical driving forces are obtained using a Chapman-Enskog analysis. This model is found to have correct diffusion behavior and the multiple diffusion coefficients are obtained analytically. The analytical results are further confirmed by numerical simulations in a few solvable limiting cases. The LBE model is established as a useful computational tool for the simulation of mass transfer in fluid systems with external forces.

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Cellular Automata And Lattice Gases

Discretization of the velocity space in solution of the Boltzmann equation

We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to be equivalent to truncating the Hermite expansion of the distribution function to the corresponding order. The truncated part of the distribution has no contribution to the moments of low orders and is negligible at small Mach numbers. Higher order approximations to the Boltzmann equation can be achieved by using more velocities in the quadrature.

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Cellular Automata And Lattice Gases

Domain Growth, Wetting and Scaling in Porous Media

The lattice Boltzmann (LB) method is used to study the kinetics of domain growth of a binary fluid in a number of geometries modeling porous media. Unlike the traditional methods which solve the Cahn-Hilliard equation, the LB method correctly simulates fluid properties, phase segregation, interface dynamics and wetting. Our results, based on lattice sizes of up to 4096×4096 , do not show evidence to indicate the breakdown of late stage dynamical scaling, and suggest that confinement of the fluid is the key to the slow kinetics observed. Randomness of the pore structure appears unnecessary.

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Cellular Automata And Lattice Gases

Don't bleach chaotic data

A common first step in time series signal analysis involves digitally filtering the data to remove linear correlations. The residual data is spectrally white (it is ``bleached''), but in principle retains the nonlinear structure of the original time series. It is well known that simple linear autocorrelation can give rise to spurious results in algorithms for estimating nonlinear invariants, such as fractal dimension and Lyapunov exponents. In theory, bleached data avoids these pitfalls. But in practice, bleaching obscures the underlying deterministic structure of a low-dimensional chaotic process. This appears to be a property of the chaos itself, since nonchaotic data are not similarly affected. The adverse effects of bleaching are demonstrated in a series of numerical experiments on known chaotic data. Some theoretical aspects are also discussed.

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Cellular Automata And Lattice Gases

Dynamic Predictions from Time Series Data- An Artificial Neural Network Approach

A hybrid approach, incorporating concepts of nonlinear dynamics in artificial neural networks (ANN), is proposed to model time series generated by complex dynamic systems. We introduce well known features used in the study of dynamic systems - time delay τ and embedding dimension d - for ANN modelling of time series. These features provide a theoretical basis for selecting the optimal size for the number of neurons in the input layer. The main outcome for the number of neurons in the input layer. The main outcome of the new approach for such problems is that to a large extent it defines the ANN architecture and leads to better predictions. We illustrate our method by considering computer generated periodic and chaotic time series. The ANN model developed gave excellent quality of fit for the training and test sets as well as for iterative dynamic predictions for future values of the two time series. Further, computer experiments were conducted by introducing Gaussian noise of various degrees in the two time series, to simulate real world effects. We find rather surprising results that upto a limit introduction of noise leads to a smaller network with good generalizing capability.

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Cellular Automata And Lattice Gases

Effect of Boundary Conditions on Cellular Automata that Classify Density

The properties of two-state nearest-neighbour cellular automata (CA) that are capable of density classification are discussed. It is shown that these CA actually conserve the total density, rather than merely classifying it. This is also the criterion for any CA simulation of DFT. The effect of boundary and periodicity conditions upon the evolution of such CA are elaborated by considering linear and cyclic lattices and boundary conditions consistent with the conservation criterion. In a bounded linear lattice, it is possible to achieve a configuration with a single 01 and no 10 domain wall, or vice versa, but this is not possible in a cyclic lattice, where these two domain walls have to appear in pairs. This determines the final stable state of the automaton.

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Cellular Automata And Lattice Gases

Elimination of Nonlinear Deviations in Thermal Lattice BGK Models

Abstracet: We present a new thermal lattice BGK model in D-dimensional space for the numerical calculation of fluid dynamics. This model uses a higher order expansion of equilibrium distribution in Maxwellian type. In the mean time the lattice symmetry is upgraded to ensure the isotropy of 6th order tensor. These manipulations lead to macroscopic equations free from nonlinear deviations. We demonstrate the improvements by conducting classical Chapman-Enskog analysis and by numerical simulation of shear wave flow. The transport coefficients are measured numerically, too.

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Cellular Automata And Lattice Gases

Estimating Functions of Distributions from A Finite Set of Samples, Part 2: Bayes Estimators for Mutual Information, Chi-Squared, Covariance and other Statistics

We present estimators for entropy and other functions of a discrete probability distribution when the data is a finite sample drawn from that probability distribution. In particular, for the case when the probability distribution is a joint distribution, we present finite sample estimators for the mutual information, covariance, and chi-squared functions of that probability distribution.

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