Cellular Automata And Lattice Gases

We propose an extension to multivariate time series of the phase-randomized Fourier-transform algorithm for generating surrogate data. Such surrogate data sets must mimic not only the autocorrelations of each of the variables in the original data set, they must mimic the cross-correlations {\em between} all the variables as well. The method is applied both to a simulated example (the three components of the Lorenz equations) and to data from a multichannel electroencephalogram.

We use nonlinear signal processing techniques, based on artificial neural networks, to construct an empirical mapping from experimental Rayleigh-Benard convection data in the quasiperiodic regime. The data, in the form of a one-parameter sequence of Poincare sections in the interior of a mode-locked region (resonance horn), are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied Rayleigh number. The dynamic phenomena apparent in the data include period doublings, complex intermittent behavior, secondary Hopf bifurcations, and chaotic dynamics. We use the fitted map to reconstruct the experimental dynamics and to explore the associated local and global bifurcation structures in phase space. Using numerical bifurcation techniques we locate the stable and unstable periodic solutions, calculate eigenvalues, approximate invariant manifolds of saddle type solutions and identify bifurcation points. This approach constitutes a promising data post-processing procedure for investigating phase space and parameter space of real experimental systems; it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings.

The hydrodynamic effects on the late stage kinetics in spinodal decomposition of multicomponent fluids are examined using a lattice Boltzmann scheme with stochastic fluctuations in the fluid and at the interface. In two dimensions, the three and four component immiscible fluid mixture (with a 1024 2 lattice) behaves like an off-critical binary fluid with an estimated domain growth of t 0.4±.03 rather than t 1/3 as previously predicted, showing the significant influence of hydrodynamics. In three dimensions (with a 256 3 lattice), we estimate the growth as t 0.96±0.05 for both critical and off-critical quenching, in agreement with phenomenological theory.

The thermal lattice BGK model is a recently suggested numerical tool which aims to simulate thermohydrodynamic problems. We investigate the quality of the lattice BGK simulation by calculating the temperature profiles in the Couette flow under different Eckert and Mach numbers. A revised lower order model is proposed and the higher order model is proved once again to be advantageous.

We examine the effects of hydrodynamics on the late stage kinetics in spinodal decomposition. From computer simulations of a lattice Boltzmann scheme we observe, for critical quenches, that single phase domains grow asymptotically like t α , with α≈.66 in two dimensions and α≈1.0 in three dimensions, both in excellent agreement with theoretical predictions.

Time-delay mappings constructed using neural networks have proven successful in performing nonlinear system identification; however, because of their discrete nature, their use in bifurcation analysis of continuous-time systems is limited. This shortcoming can be avoided by embedding the neural networks in a training algorithm that mimics a numerical integrator. Both explicit and implicit integrators can be used. The former case is based on repeated evaluations of the network in a feedforward implementation; the latter relies on a recurrent network implementation. Here the algorithms and their implementation on parallel machines (SIMD and MIMD architectures) are discussed.

A new approach of implementing initial and boundary conditions for the lattice Boltzmann method is presented. The new approach is based on an extended collision operator that uses the gradients of the fluid velocity. The numerical performance of the lattice Boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method. The discretization error of the lattice Boltzmann method decreases quadratically with finer resolution both in space and in time. The roundoff error of the lattice Boltzmann method creates problems unless double precision arithmetic is used.

We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving in each lattice direction. For this generalization we derive the equilibrium distribution function and the hydrodynamic equations, including the equation of state and the prefactor of the inertial term that arises from the breaking of galilean invariance in these models. We show that this prefactor can be set to unity in the generalized model, therby effectively restoring galilean invariance. Moreover, we derive an expression for the kinematic viscosity, and show that it tends to decrease with the maximum number of particles allowed in each direction, so that higher Reynolds numbers may be achieved. Finally, we derive expressions for the statistical noise and the Boltzmann entropy of these models.

A thermohydrodynamic lattice-BGK model for the ideal gas was derived by Alexander et al. in 1993, and generalized by McNamara et al. in the same year. In these works, particular forms for the equilibrium distribution function and the transport coefficients were posited and shown to work, thereby establishing the sufficiency of the model. In this paper, we rederive the model from a minimal set of assumptions, and thereby show that the forms assumed for the shear and bulk viscosities are also necessary, but that the form assumed for the thermal conductivity is not. We derive the most general form allowable for the thermal conductivity, and the concomitant generalization of the equilibrium distribution. In this way, we show that it is possible to achieve variable (albeit density-dependent) Prandtl number even within a single-relaxation-time lattice-BGK model. We accomplish this by demanding analyticity of the third moments and traces of the fourth moments of the equilibrium distribution function. The method of derivation demonstrates that certain undesirable features of the model -- such as the unphysical dependence of the viscosity coefficients on temperature -- cannot be corrected within the scope of lattice-BGK models with constant relaxation time.

We study an interface-capturing two-phase fluid model in which the interfacial tension is modelled as a volumetric stress. Since these stresses are obtainable from a Van der Waals-Cahn-Hilliard free energy, the model is, to a certain degree, thermodynamically realistic. Thermal fluctuations are not considered presently for reasons of simplicity. The utility of the model lies in its momentum-conservative representation of surface tension and the simplicity of its numerical implementation resulting from the volumetric modelling of the interfacial dynamics. After validation of the model in two spatial dimensions, two prototypical applications---instability of an initially high-Reynolds-number liquid jet in the gaseous phase and spinodal decomposition in a liquid-gas system--- are presented.