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## 8.3.1 Lattice Gas Cellular Automata

The first example of cellular automata was John H. Conway's computer game Life'': some initial pixel patterns on a screen are set, and at each iterative step every pixel is set or erased depending on the status of the neighboring pixels [EIGEN 82].

Generally, cellular automata are one- or two-dimensional bit patterns that evolve in (discrete) time according to certain simple rules. They have come to play a role in such diverse fields as informatics [WOLFRAM 86], evolution theory, and the mathematical theory of complexity [WOLFRAM 84].

A model representing a two-dimensional flow field in terms of bit patterns was introduced by Hardy, Pomeau and de Pazzis. Their HPP model'' is defined as follows [HARDY 73]:

- Represent the flow region by a point lattice
- Populate each grid point with up to four particles'' whose velocities must point into different compass directions
- Absolute value of each velocity is always

There are possible states'' of a grid point.
Representation: Use a 4-bit (half-byte) computer word to describe the empty'' or full'' status of the compass directions E,N,W,S by one bit each (see Fig. 8.3).
Alternative representation: Combine the bits referring to the same direction at several successive grid points into one computer word.
Example: In a grid each compass direction described by a set of 32 words of 1 byte each (see Fig. 8.4).

Evolution law: by a deterministic rule comprised of two substeps, Free flight and Scattering.
Free flight: Each particle moves on by one vertex in its direction of flight.
In the representation of Figure 8.4 each north'' bit in the second row (in words and ), if at time would be reset to ; the respective bit above (in words and ) would be set to . Similar translations take place for the south'' bits.
The -bits in the east'' and west'' words are right- and left-shifted, respectively, by one position.

In most programming languages logical operations may be performed not only with single bits but also with byte-words or even integers made up of several bytes. In the above example the new word could be computed as
 (8.81)

with denoting the bitwise or-operation.
Analogous commands apply to the -words.
The and -words have to be handled, in this representation, in a bit-by-bit manner. By combining the east and west bits column-wise the translation may then be formulated as for north and south.

Free flight and boundaries:
Reflection: Transform all -bits in the top row into -bits before the translation takes place.
Periodic boundary conditions: To describe part of a longer tube etc.

Note: Periodic boundary conditions preserve momentum and energy exactly, while in the presence of reflexion the conservation laws can hold only on the average.

Scattering:
- If after the translation step a grid point is inhabited by two particles, change its state according to Fig. 8.5
- Otherwise: the state remains unaltered

Momentum and energy are conserved by this scattering rule. We may write the HPP scattering rule in a concise, computer-adapted way as follows:
 (8.82)

where is the state of grid point before scattering (but after translation), and
 (8.83)

, and are the logical operators and, exclusive or, and not. ( differs from in that .)

Lookup table: A fast way to implement the scattering rules is the use of a table containing all possible pre- an post-scattering states of a site.

Reviews of the performance of this simple model: [FRISCH 86,WOLFRAM 86B].

Application:
- Population number at a grid point = density at that position
- Sum of velocities at = local velocity density
- Coarse-graining in space and time averaged dynamics obeys equations similar to the flow equations
- However: only logical operations calculations are much faster!

Improvement: the FHP model
Frisch, Hasslacher, and Pomeau [FRISCH 86] suggested to introduce hexagonal cells in place of the simple quadratic lattice:
- Six possible flight directions per grid point
- More scattering rules (see Figure 8.6)

Refinements: Possibility of particles at rest richer microdynamics.

Advantage over HPP: Rotational symmetry of the flow distribution, in spite of the discretized velocities.

FHP in 3 dimensions: To retain rotational symmetry, a four-dimensional face centered hypercubic (FCHC) lattice is set up and used in the propagation and collision steps. The results are then mapped onto three dimensions following a rule due to [D'HUMIERES].

Note: In the basic HPP and FHP models the particles lose their identity in the process of scattering (see Figs. 8.5).
To determine single particle properties, like velocity autocorrelations, `tag'' some particles and pass the tags in the scattering step in a unique way.
Application: Long time behavior of the velocity autocorrelation function [FRENKEL 90,ERNST 91]. Using a two-dimensional FHP simulation with tagging, Frenkel et al. were able to produce proof for the expected long time tail.

Next: 8.3.2 The Lattice Boltzmann Up: 8.3 Lattice Gas Models Previous: 8.3 Lattice Gas Models
Franz J. Vesely Oct 2005