Contributions by: Paul Larson <palarson@dal.mobil.com> Bruce
Boghosian <bruceb@bu.edu>

LGA stands for "lattice gas automaton." It is a particular type of cellular
automaton that is used for the simulation of viscous fluid flow.

LGA research is a highly developed subculture of general CA research. Some CA
researchers are completely immersed in it, while others have barely heard of it.

Rothman and Zaleski have a nice review article: Rev. Mod. Phys. 66
1417 (1994).

The October, 1995 issue of the Journal of Statistical Physics was devoted to
lattice gases. See the papers in that, and the secondary references. Also, do
you know how to access the electronic archives at LANL? If so, check out
comp-gas@xyz.lanl.gov, or find the comp-gas
archive. You can retrieve papers from there, do searches, etc. See also: Bogosian

Contributions by: Jeremy Henderson
<J.R.Henderson@durham.ac.uk> http://www.dur.ac.uk/~dgl0jrh From:
LUTHI Pascal <luthi@cui.unige.ch>

I've successfully implemented a cellular automata model to simulate 2-DIM
wave propagation (in a high heterogeneous macroscopic medium like an urban
microcell, for instance). The method is an application of the TLM (Transmission
Line Matrix) method for the time domain.

What's TLM? Read the folowing paper as an introduction:

Wolfgang J. R. Hoffer, IEEE Transaction on microwave theory and techniques,
Vol. MTT-33, No 10, october 1985

Dan Rothman had a paper on modeling the 2-D wave equation with CA, back about
1987 or 1988. His method was pure integer, and the application was intended to
be modeling of seismicity (I think).

Rothman's paper is in Geophysics Research Letters 14(1) 17-20 1987.

see also:

Rucker.waves
R. Rucker's project on modelling the wave equation.

For a lattice gas treatment of incompressible, convective flow in the
Boussinesq approximation, see C. Burges and S. Zaleski, "Buoyant mixtures of
cellular automaton gases," Complex Systems 1 (1987) 31.

Rucker: A simple way to make wind is to just have each cell copy the
cell to its right with each update. This makes a wind that blows to the left. In
the RC module of the CA LAB package, I ran this rule alternating with the heat
equation (rug rule). I also allowed there to be some stable blocks of cells in
the pattern which were *not* shifted to the left with each update - these were
copied in as masks with each update. The effect was of turbulence in the wind
behind the blocks, vortices, slipstreams, the whole shmear. Try it you'll like
it.

What are the commonly used rules of interaction between the cells in a lattice gas?

<bruceb@conx.bu.edu> Bruce Boghosian
<mikk0022@maroon.tc.umn.edu> Christoph L Mikkelson
<david@alpha1.csd.uwm.edu> Dave Stack Mohamed
<osman@eecs.wsu.edu>

There's a huge literature on this. Briefly, examples include two- and
three-dimensional Navier-Stokes flow, magnetohydrodynamics, immiscible fluids
with a surface tension interface, convection, two-phase liquid-gas flow,
Burgers' equation, and reaction-diffusion equations. For a summary of recent
works on the subject, see any of the reviews listed below; also there should be
a very recent (or imminent) Physics Reports devoted to lattice gases by Rothman
and Zaleski.

``Lattice Gas Methods for Partial Differential Equations,'' Proceedings of
the Workshop on Large Nonlinear Systems, held in Los Alamos, New Mexico, August,
1987, Proceedings Volume IV of the Santa Fe Institute Studies in the Sciences of
Complexity, Addison-Wesley (1990), Doolen, G.D., ed.

``Lattice Gas Methods for PDE's, Theory, Applications and Hardware,''
Proceedings of the NATO Advanced Research Workshop, held at Los Alamos National
Laboratory, September 6-8, 1989, North-Holland (1991), Doolen, G.D., ed.

Proceedings of the Workshop on ``Discrete Kinetic Theory, Lattice Gas
Dynamics and Foundations of Hydrodynamics,'' held in Torino, Italy, September
20-24, 1988, World Scientific (1989), Monaco, R., ed.

``Cellular Automata and Modeling of Complex Physical Systems,'' Proceedings
of the Winter School, Les Houches, France, February 21-28, 1989, Springer
(1989), Manneville, P., Boccara, N., Vichniac, G., Bidaux, R., eds.

Proceedings of the Colloquium Euromech No. 267 on ``Discrete Models of Fluid
Dynamics,'' held in Figueira da Foz, Portugal, September 19-22, 1990, World
Scientific (1991), Alves, A.S. ed.

Proceedings of the NATO Advanced Research Workshop on Lattice Gas Automata:
Theory, Implementation and Simulation, held at l'Observatoire de la Côte d'Azur,
Nice, France, June 25-28, 1991, to appear in J. Stat. Phys., Boon, J.-P.,
Lebowitz, J.L., eds.

One of the seminal papers is "Lattice Gas Hydrodynamics in Two and Three
Dimensions" by Frisch et. al. in Complex Systems Volumne 1 (1987) 649-707.

There is an excellent 3 part review paper by Brosl Hasslacher in a special
issue of Los Almos Science in 1987. The title is "discrete fluids" and the the
title of part I is "background for lattice gas automata". Also see Computer in
physics, Nov. 1991, page 585 for an article by B.M. Boghosian.

Relationship between the shear viscosity and the collision probabilities.

Stephen Wolfram, ``Cellular Automaton Fluids 1: Basic Theory'', Journal
of Statistical Physics45 Nos.~3/4 (1986) 471-526. (equation 4.6.8 )

U.~Frisch, D.~d'Humières, B.~Hasslacher, Y.~Pomeau, J.~Rivet, ``Lattice Gas
Hydrodynamics in Two and Three Dimensions'', Complex Systems1
(1987) 649-707. (equation 8.25)

From general kinetic theoretical arguments, the viscosity of a fluid goes as
the product of the thermal velocity and the mean free path. Now the thermal
velocity of the particles of a lattice gas automaton is fixed at one lattice
spacing per time step. It follows that the viscosity goes as the mean free path.
Thus, decreasing the collision probability clearly increases the mean free path,
and hence the viscosity.

Does the lack of symmetry in the HPP model have any obvious bad effect, other than to remove the inertial term?

Contributions by: Bruce Boghosian <bruceb@bu.edu>

I don't think it removes the inertial term. There is still a form of the
inertial term with HPP, though it is not isotropic. And, yes, it does have
another effect on the equation: The viscous term, like the inertial term, is
present but anisotropic.

Are there unphysical conservation laws with HPP?

Contributions by: Bruce Boghosian <bruceb@bu.edu>

Yes, HPP has several unphysical conservation laws. First, if you color the
sites white and black, like a checkerboard, you can convince yourself that the
dynamics on the white squares are completely independent of the dynamics on the
black squares. Thus, all conserved quantities (mass and momentum) are conserved
*separately* on the two checkerboard sublattices. More seriously, y-momentum is
conserved separately within each column, and x-momentum is conserved separately
within each row (assuming periodic b.c.'s).

What are the physical manifestations of anisotropy?

Contributions by: Bruce Boghosian <bruceb@bu.edu>

Here is a physical manifestation of the problem of anisotropy: If you tried
to do a Poiseuille flow simulation with HPP, you would find that the drag on the
plates depended on the angle of orientation of the plates with respect to the
underlying lattice. This problem would be present even at low Reynolds number.
With FHP, on the other hand, the drag would be independent of this orientation.