Next: 8.3.2 The Lattice Boltzmann
Up: 8.3 Lattice Gas Models
Previous: 8.3 Lattice Gas Models
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).
Figure 8.3:
HPP model
|
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).
Figure 8.4:
Storage methods in the HPP model
|
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.
Figure 8.5:
Scattering law for the HPP model
|
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'HUMI`ERES].
Figure 8.6:
Scattering rules in the FHP model
|
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
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001