Cellular Automata And Lattice Gases

We show that predicting the HPP or FHP III lattice gas for finite time is equivalent to calculating the output of an arbitrary Boolean circuit, and is therefore P-complete: that is, it is just as hard as any other problem solvable by a serial computer in polynomial time. It is widely believed in computer science that there are inherently sequential problems, for which parallel processing gives no significant speedup. Unless this is false, it is impossible even with highly parallel processing to predict lattice gases much faster than by explicit simulation. More precisely, we cannot predict t time-steps of a lattice gas in parallel computation time O(log^k t) for any k, or O(t^\alpha) for \alpha < 1/2, unless the class P is equal to the class NC or SP respectively.

We review the class of cellular automata known as lattice gases, and their applications to problems in physics and materials science. The presentation is self-contained, and assumes very little prior knowledge of the subject. Hydrodynamic lattice gases are emphasized, and non-lattice-gas cellular automata -- even those with physical applications -- are not treated at all. We begin with a review of lattice gases as the term is understood in equilibrium statistical physics. We then discuss the various methods that have been proposed to simulate hydrodynamics with a lattice gas, leading up to the 1986 discovery of a lattice gas for the isotropic Navier-Stokes equations. Finally, we discuss variants of lattice-gas models that have been used for the simulation of complex fluids.

The recent development of the lattice gas automata method and its extension to the lattice Boltzmann method have provided new computational schemes for solving a variety of partial differential equations and modeling chemically reacting systems. The lattice gas method, regarded as the simplest microscopic and kinetic approach which generates meaningful macroscopic dynamics, is fully parallel and can, as a result, be easily programmed on parallel machines. In this paper, we introduce the basic principles of the lattice gas method and the lattice Boltzmann method, their numerical implementations and applications to chemically reacting systems. Comparisons of the lattice Boltzmann method with the lattice gas technique and other traditional numerical schemes, including the finite difference scheme and the pseudo-spectral method, for solving the Navier-Stokes hydrodynamic fluid flows will be discussed. Recent developments of the lattice gas and the lattice Boltzmann method and their applications to pattern formation in chemical reaction-diffusion systems, multiphase fluid flows and polymeric dynamics will be presented.

We present an extension of a simple automaton model to incorporate non-local interactions extending over a spatial range in lattice gases. {}From the viewpoint of Statistical Mechanics, the lattice gas with interaction range may serve as a prototype for non-ideal gas behavior. {}From the density fluctuations correlation function, we obtain a quantity which is identified as a potential of mean force. Equilibrium and transport properties are computed theoretically and by numerical simulations to establish the validity of the model at macroscopic scale.

A simple extension of the Lattice Boltzmann equation is proposed, which permits to handle reactive flow dynamics in the limit of fast chemistry at virtually no extra-cost with respect to the purely hydrodynamic scheme.

The lattice Boltzmann model is a simplified kinetic method based on the particle distribution function. We use this method to simulate problems in MEMS, in which the velocity slip near the wall plays an important role. It is demonstrated that the lattice Boltzmann method can capture the fundamental behavior in micro-channel flow, including velocity slip and nonlinear pressure drop along the channel. The Knudsen number dependence of the position of the vortex center and the pressure contour in micro-cavity flows is also demonstrated.

We develop our existing two-dimensional lattice-gas model to simulate the flow of single-phase, binary-immiscible and ternary-amphiphilic fluids. This involves the inclusion of fixed obstacles on the lattice, together with the inclusion of ``no-slip'' boundary conditions. Here we report on preliminary applications of this model to the flow of such fluids within model porous media. We also construct fluid invasion boundary conditions, and the effects of invading aqueous solutions of surfactant on oil-saturated rock during imbibition and drainage are described.

We investigate the dynamical behavior of both binary fluid and ternary microemulsion systems in two dimensions using a recently introduced hydrodynamic lattice-gas model of microemulsions. We find that the presence of amphiphile in our simulations reduces the usual oil-water interfacial tension in accord with experiment and consequently affects the non-equilibrium growth of oil and water domains. As the density of surfactant is increased we observe a crossover from the usual two-dimensional binary fluid scaling laws to a growth that is {\it slow}, and we find that this slow growth can be characterized by a logarithmic time scale. With sufficient surfactant in the system we observe that the domains cease to grow beyond a certain point and we find that this final characteristic domain size is inversely proportional to the interfacial surfactant concentration in the system.

LS is a particular type of computational processes simulating living tissue. They use an unlimited branching process arising from the simultaneous substitutions of some words instead of letters in some initial word. This combines the properties of cellular automata and grammars. It is proved that 1) The set of languages, computed in a polynomial time on such LS that all replacing words are not empty, is exactly NP- languages. 2) The set of languages, computed in a polynomial time on arbitrary LS, contains the polynomial hierarchy. 3) The set of languages, computed in a polynomial time on a nondeterministic version of LS, strictly contains the set of languages, computed in a polynomial time on Turing Machines with a space complexity n a , where a is positive integer. In particular, the last two results mean that Lindenmayer systems may be even more powerful tool of computations than nondeterministic Turing Machine.

In systems removed from equilibrium, intrinsic microscopic fluctuations become correlated over distances comparable to the characteristic macroscopic length over which the external constraint is exerted. In order to investigate this phenomenon, we construct a microscopic model with simple stochastic dynamics using lattice gas automaton rules that satisfy local detailed balance. Because of the simplicity of the automaton dynamics, analytical theory can be developed to describe the space and time evolution of the density fluctuations. The exact equations for the pair correlations are solved explicitly in the hydrodynamic limit. In this limit, we rigorously derive the results obtained phenomenologically by fluctuating hydrodynamics. In particular, the spatial algebraic decay of the equal-time fluctuation correlations predicted by this theory is found to be in excellent agreement with the results of our lattice gas automaton simulations for two different types of boundary conditions. Long-range correlations of the type described here appear generically in dynamical systems that exhibit large scale anisotropy and lack detailed balance.