Lattice cellular automata

Although LB therefore nowadays may be considered as a solver for the NS equations, there is definitely more behind it. The method originally stems from the lattice gas automaton (LGA), which is a cellular automaton. In a LGA, a fluid can be considered as a collection of discrete particles having interaction with each other via a set of simple collision rules, thereby taking into account that the number of particles and momentum is conserved. 

Several groups have developed cellular automata models for particular reaction-diffusion systems. In particular, the Belousov-Zhabotinsky oscillating reaction has been examined in a number of studies.84-86 Attention has also been directed at the A + B —> C reaction, using both lattice-gas models 87-90 and a generalized Margolus diffusion approach.91 We developed a simple, direct cellular automaton model92 for hard-sphere bimolecular chemical reactions of the form... 

Fig. 12.18. Schematic of the coupling between a finite element treatment of heat transfer and the cellular automaton treatment of the grain structure (adapted from Gandin and Rappaz (1994)). The top frame shows the finite element mesh and the lower frame shows the lattice of points used to do bookkeeping concerning the state (liquid or solid) and orientations of the associated points.

Perea-Reeves and Stockman (1997) applied a lattice-gas cellular automaton model to study solute dispersion, including the effect of fluid buoyancy arising from solution density differences, in a pocketed channel. They found good agreement with the indented capillary model discussed in the Variable Shape, Discrete Pore Models section. For Peclet numbers smaller than 3 however, they found that K was actually smaller than the molecular diffusion coefficient. They attributed this to the restriction to diffusion in the direction of flow imposed by the pocket walls. They also observed that density differences between the existing and introduced fluids... 

Another powerful tool is the cellular automata method invented by John (or Janos) von Neumann and Stanislaw Marcin Ulam (under the name of cellular spaces ). The cellular automata are mathematical models in which space and time both have a granular structure (similar to Monte Carlo simulations on lattices, in MD only time has such a structure). A cellular automaton consists of a periodic lattice of cells (nodes in space). In order to describe the system locally, we assume that every cell has its state representing a vector of N components. Each component is a Boolean variable i.e., a variable having a logical value (e.g., 0 for false and 1 for true ). 

Fig. 7.16. Operation of a cellular automaton-a model of gas. The particles occupy the lattice nodes (cellsl. Their displacement from the node symbolizes which direction they are heading on with the velocity equal to 1 length unit per 1 time step. In the left scheme (a), the initial situation is shown. In the right scheme the result of the one step propagation and one step collision is shown. Collision only took place in one case (at 03 2). collision rule has been applied (of the lateral outgoing). The game would...

The Lattice Boltzmann Method (LBM), including the method Cellular Automaton (AC), present a powerful alternative to standard apvproaches known like "of up toward down" and "of down toward up". The first approximation study a continuous description of macroscopic phenomenon given for a partial differential equation (an example of this, is the Navier-Stokes equation used for flow of incompressible fluids) some numerical techniques like finite difference and the finite element, they are used for the transformation of continuous description to discreet it permits solve numerically equations in the compniter. 

Lattice Boltzmann equation can be obtain through two ways, first is through of "cellular automaton" and second starting from Boltzmann equation, it was review previously, for carries out derivation of Boltzmann s lattice equation is necessary the space time discretization. Immediately presents brief description of second way, it shows by p>ace series. 

Two other approaches treat a spatially distributed system as consisting of a grid or lattice. The cellular automaton technique looks at the numbers of particles, or values of some other variables, in small regions of space that interact by set rules that specify the chemistry. It is a deterministic and essentially macroscopic approach that is especially useful for studying excitable media. Lattice gas automata are mesoscopic (between microscopic and macroscopic). Like their cousins, the cellular automata, they use a fixed grid, but differ in that individual particles can move and react through probabilistic rules, making it possible to study fluctuations. 

An interesting variant of the cellular automaton approach is the lattice gas automaton. Despite the name, lattice gas automata can be used to simulate reactions in condensed as well as gaseous media. They were introduced initially as an alternative to partial differential equations (PDE) for modeling complex problems in fluid flow (Hardy et al., 1976) and have been adapted to include the effects of chemical reaction as well (Dab et al., 1990). Lattice gas automata are similar to cellular automata in that they employ a lattice, but differ because they focus on the motions of individual particles along the lattice and because they can accoimt... 

Cellular automata and the related lattice gas automaton models provide less quantitative, more cost-effective, and often more intuitive alternatives to differential equation models. Although they can be constructed to include considerable detail, they are best used to provide qualitative insights into the behavior of carefully designed models of complex reaction-diffusion systems. 

A first attempt to get couect microscopic properties from a discrete (lattice) description was done by Broadwdl for a gas in which the possible partide vdodties are restricted to a finite set. Then, Frisch a devised a cellular automaton... 

Original developments in this area stem from the work of Frisch et al. (1986) who employed the technique of lattice gas hydrodynamics in which the fluid is modelled as a cellular automaton and the flow represented by the motion of particles on a lattice. More numerically efficient variants of this method, such as the lattice Boltzmann approach (McNamara and Zanetti, 1988), were subsequently developed. 

Formulated on a lattice and with discrete time, the WR model becomes a variant of a cellular automaton (it was later used in the construction of more complex cellular automata for excitable media, see [30-32]). However, Wiener and Rosenblueth actually assumed in [6] that both time and space were continuous. In their model smooth excitation fronts propagate into the regions where the medium is in the state of rest. The duration Tg of excitation is taken... 

In this chapter we shall examine some of the effects of molecular fluctuations on chemical oscillations, waves and patterns. There are many ways one can attempt to study fluctuation dynamics in reacting systems, the most familiar of which are master equation models [ 1 ]. Here we present results obtained using a specific class of cellular automaton models, termed lattice-gas cellular automata [2-4]. These cellular automaton models provide a mesoscopic description of the spatially-distributed reacting system and are constructed to model the microscopic collision dynamics. The modeling strategy and rule construction are different from those for traditional cellular automata and are based on lattice-gas cellular automaton models for hydrodynamics [5]. However, reactive lattice-gas models differ from the corresponding hydrodynamics models in a number of important respects and are closely related to master equation descriptions of the reactive dynamics. 

Cellular automata are abstract discrete dynamical systems introduced by Von Neumann in an attempt to model self-replication in biological systems [11]. A cellular automaton consists of a set of nodes, usually arranged on a regular lattice, each of which supports state variables that take on a finite number of possible values. The state variables are synchronously updated at discrete... 

Reactive lattice-gas cellular automaton models are constructed in the full position and velocity phase space of the system. In this phase space all variables are discrete. Time and position are discrete variables thus, the velocity is discrete. Since the dynamical description is in terms of the molecules of the system, the particle numbers representing these molecules are also discrete variables. 

The first model of this type was a cellular automaton, called the Lattice-Gas-Automaton (LG). The algorithm consists of particles which jump between nodes of... 

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