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single speed for all particles. The sound where speed in such models can be shown to be about two-thirds of the particle speed. Hence flows in which the Mach number (flow speed divided by sound speed) is greater than 1.5 cannot be simulated. This difficulty is avoided by adding particles with a variety of speeds.

The limited range of velocities also restricts the allowed range of Reynolds numbers. For small Reynolds numbers (0 to 1000) the flow is smooth, for moderate Reynolds numbers (2000 to 6000) some turbulence is observed, and for high Reynolds numbers (10,000 to 10,000,000) extreme turbulence occurs. Since the effective viscosity, v, is typically about 0.2 in two-dimensional problems, the Reynolds number scales with the characteristic length, I, allowed by computer memory. Currently the upper bound on I is of the order of 100,000.

conse

The velocity dependence of the equation of state is unusual and is quence of the inherent Fermi-Dirac distribution of the lattice gas (see the section on Theoretical Analysis of the Dis

crete Lattice Gas in the main text). The

low-velocity equation of state for a lattice gas can be written as p = p(1 - v2), where p is the pressure, p is the density, and v is the flow speed. Thus, for constant-pressure flows, regions of higher velocity flows have higher densities.

The velocity dependence of the equation of state is related to the fact that lattice gas models lack Galilean invariance. The standard Navier-Stokes equation for incompressible fluids is

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is the average number of particles per cell. The extra factor g(p) requires special treatment. The conventional way to adjust for the fact that g(p) Navier-Stokes equation) is to simply scale does not equal unity (as it does in the the time, t, and the viscosity, v, by the factor g(p) as follows: t' g(p)t and v' = v/g(p). (The pressure must also be scaled.) Hence a density-dependent scaling of the time, the viscosity, and the pressure is required to bring the lattice gas model into a form that closely approximates the hydrodynamics of incompressible fluids in the low-velocity limit.

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Finally, the discreteness of the lattice gas approximation introduces noise into the results. One method of smoothing the results for comparison with other methods is to average in space and time. In practice, spatial averages are taken over 64, 256, 512, or 1024 neighboring cells for time-dependent flows in two dimensions. For steady-state flows, time averaging is done. The details of noise reduction are complicated, but they must be addressed in each comparison calculation. The presence of noise is both a virtue and a defect.

Noise ensures that only robust (that is, physical) singularities survive, whereas in standard codes, which are subject to less noise, mathematical artifacts can produce

singularities. On the other hand, the noise in the model can trigger instabilities.

Conclusion

In the last few years lattice gas methods have been shown to simulate the qualitative features of hydrodynamic flows in

two and three dimensions. Precise comparisons with other methods of calculation remain to be done, but it is believed that the accuracy of the lattice gas method

can be increased by making the models more complicated. But how complicated they have to be to obtain the desired accuracy is an unanswered question.

Calculations based on the simple models are extremely fast and can be made several orders-of-magnitude faster by us ing special-purpose computers, but the models must be extended to get quantitative results with an accuracy greater than 1 percent. Significant research remains to be done to determine the accuracy of a given lattice gas method for a given flow problem. ■

Note added in proof: Recently Kadanoff, McNamara, and Zanetti reported precise comparisons between theoretical predictions and lattice gas simulations (University of Chicago preprint, October 1987). They used a seven-bit hexagonal model on a small automaton universe to simulate forced two-dimensional channel flow for long times. Three tests were used to probe the hydrodynamic and statisticalmechanical behavior of the model. The tests determined (1) the profile of momentum density in the channel, (2) the equation of state given by the statistical mechanics of the system, and (3) the logarithmic divergence in the viscosity (a famous effect in two-dimensional hydrodynamics and a deep test of the accuracy of the model in the strong nonlinear regime).

The results were impressive. First, to within the accuracy of the simulation, there is no discrepancy between the parabolic velocity profile predicted by macroscopic theory and the lattice gas simulation data. Second, the equation of state derived from theory fits the simulation data to better than 1 percent. Finally, the measured logarithmic divergence in the viscosity as a function of channel] width agrees with prediction. These results are at least one order of magnitude more accurate than any previously reported calculations.

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L

n the sidebar "Calculations Using Lattice Gas Techniques" we displayed the results of generalizing the simple hexagonal model to three dimensions. Here, in the last part of the article, we will discuss numerous ways to extend and adapt the simple model. In particular, we emphasize its role as a paradigm for parallel computing.

Adjusting the Model To Fit the Phenomenon

There are several reasons for altering the geometry and rule set of the fundamental hexagonal model. To understand the mathematical physics of lattice gases, we need to know the class of functionally equivalent models, namely those models. with different geometries and rules that produce the same dynamics in the same. parameter range.

To explore turbulent mechanisms in fluids, the Reynolds number must be significantly higher than for smooth flow, so models must be developed that increase the Reynolds number in some way. The most straightforward method, other than increasing the size of the simulation universe, is to lower the effective mean free path in the gas. This lowers the viscosity and the Reynolds number rises in inverse proportion. Increasing the Reynolds number is also important for practical applications. In "Reynolds Number and Lattice Gas Calculations"

we discuss the computational storage and work needed to simulate high-Reynoldsnumber flows with cellular automata.

To apply lattice gas methods to systems such as plasmas, we need to develop models that can support widely separated time scales appropriate to, for example, both photon and hydrodynamical modes. The original hexagonal model on a single lattice cannot do so in any natural way but must be modified to include several lattices or the equivalent (see below).

Within the class of fluids, problems involving gravity on the gas, multi-component fluids, gases of varying density, and gases that undergo generalized chemical reactions require variations of the hexagonal model. Once into the subject of applications rather than fundamental statistical mechanics, there is an endless industry in devising clever gases that can simulate the dynamics of a problem effectively.

We outline some of the possible extensions to the hexagonal gas, but do so only to give an overview of this developing field. Nothing fundamental changes by making the gas more complex. This model is very much like a language. We can build compound sentences and paragraphs out of simple sentences, but it does not change the fundamental rules by which the language works.

The obvious alterations to the hexagonal model are listed below. They comprise almost a complete list of what can be done in two dimensions, since a lattice

gas model contains only a few adjustable structural elements.

Indistinguishable particles can be colored to create distinguishable species in the gas, and the collision rules can be appropriately modified. Rules can be weighted to different outcomes; for example, one can create a chiral gas (leftor right-handed) by biasing collisions. to make them asymmetric. In three dimensions there is an instability at any Reynolds number caused by lack of microscopic parity, so the chiral gas is an important model for simulating this instability.

At the next order of complexity, multispeed particles can be introduced, either alone or with changes in geometry. The simplest example is a square neighborhood in two dimensions in which the collision domain is enlarged to include nextto-nearest neighbors, and a diagonal particle with speed √2 is introduced to force an isotropic lattice gas. In general, any lattice model with only two-body collisions and a single speed will contain spurious conservation laws. But if multiple speeds are allowed, models with binary collisions can maintain isotropy. In other words, models with multiple speeds are equivalent to single-speed models with a higher order rotation group and extended collision sets. Many variations are possible and each can be designed to a problem where it has a special advantage.

Finally, colored multiple-speed mod

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els are in general equivalent to singlespecies models operating on separate lattices. Colored collision rules couple the lattices so that information can be transferred between them at different time scales. Certain statistical-mechanical phenomena such as phase transitions can be done this way.

By altering the rule domain and adding gas species with distinct speeds, it is possible to add independent energy conservation. This allows one to tune gas models to different equations of state. Again, we gain no fundamental insight into the development of large collective models by doing so, but it is useful for applications.

In using these lattice gas variations to construct models of complex phenomena, we can proceed in two directions. The first direction is to study whether or not complex systems with several types of coupled dynamics are described by skeletal gases. Can complex chemical reactions in fluids and gases, for example, be simulated by adding collision rules operating on colored multi-speed lattice gases? Complex chemistry is set up in the gas in outline form, as a gross scheme of closed sets of interaction rules. The same idea might be used for plasmas. From a theoretical viewpoint one wants to study how much of the known dynamics of such systems is reproduced by a skeletal gas; consequently both qualitative and quantitative results are important.

Exploring Fundamental Questions. Models of complex gas or fluid systems, like other lattice gas descriptions, may either be a minimalist description of microphysics or simply have no relation to microphysics other than a mechanism for carrying known conservation laws and reactions. We can always consider such gas models to be pure computers, where we fit the wiring, or architecture, to the problem, in the same fashion that ordinary discretization schemes have no relation to the microphysics of the problem. However for lattice gas models, or cellularcontinued on page 214

cellular automata, then discuss some possible ways out, and finally estimate the seriousness of the situation for a realistic large-scale simulation.

The turbulent regime has many length scales, bounded above by the length of the simulation box and below by the scale at which turbulent dynamics degenerates into pure dissipation, the so-called dissipation scale. We focus on these extreme scales and, with a few definitions, derive a bound on the computational storage and work needed for simulating highReynolds-number flows with cellular au

tomata.

The Reynolds number R is usually defined not in terms of time but simply as R = VL/v where L is a characteristic length, v is characteristic speed, and

is the kinematic shear viscosity. One sees immediately why calculating viscosity functions for particular models is important. It is the only variable one can adjust in a flow problem, given a fixed flow in a fixed geometry. First, we calculate a rough upper bound on Reynolds numbers attainable with lattice models. If the speed of sound in the lattice gas is cs and the spacing between lattice nodes is , then by definition the kinematic viscosity vcl. Now viscosity estimated this way must agree with that fixed by the scale of hydrodynamic modes. Given a global length L and a global velocity V associated with these modes, R = VL/V at best. In terms of the Mach number (M V/c,), the Reynolds number is equal to ML/l. But M also characterizes fluid flow, and L and are modeldependent. In a lattice gas we can relate the ratio L/l to the number of nodes in the gas simulator, namely n = (L/ed, where d is the space dimension of the model. Therefore, the number of nodes in a lattice model must grow at least as n~ (R/M). Computational work is the number of lattice nodes per time step multiplied by the number of time steps required to resolve hydrodynamical fea

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tures. This is L/M steps (to cross the hydrodynamical feature at the given Mach number), and so we find the computational work is of order Rd+1/Md+2. For a so-called normal simulation based on the usual ways of discretizing the NavierStokes equation, the growth in storage is roughly proportional to one power lower in the Reynolds number than the growth in storage for the lattice gas. So at first it seems that simulating high-Reynoldsnumber flows by lattice gas techniques is costly compared to ordinary methods.

This argument is not only approximate; it is also tricky and must be applied with great care. The normal way of simulating flows escapes power-law penalties by cutting off degrees of freedom at the turbulence-dissipation scale, which the lattice gas does not do. The gas computes within these scales and so wastes computational resources for some problems. Actually computation of these very small scales is the source of the noisy character of the gas and is responsible for its power to avoid spurious mathematical singularities. One way around this is to find an effective gas with new collision rules for which the dissipation length scales are averaged out. A possible technique uses the renormalization group, but it is useful only if the effective gas is not too complex and has the attributes that made the original gas attractive, including locality. Work is going on at present to explore this possibility, and it seems likely that some such method will be developed.

The more serious consideration is what happens in a realistic large-scale simulation, and here we will find the lattice gas does very well indeed.

First, we note that a dissipation length la with the behavior la → ∞ as R → ∞ is actually required to guarantee the scale separation between the lattice spacing and separation between the lattice spacing and the hydrodynamic modes that is necessary to develop hydrodynamic behavior.

The actual Reynolds number in lat

tice gas models is much more complex than in normal fluid models. An accurate form is R = Lvg (po)/(po), where v is an averaged velocity and the fundamental unit of distance (the lattice spacing ) and the fundamental unit of time (the speed required to traverse the lattice spacing ) have been set to 1. To remain nearly incompressible, the velocities in the model should remain small compared to the speed of sound c,, but c, in lattice gases is model-dependent. So we factor the Reynolds number into modeldependent and invariant factors this way: we define R(po) = cs (g(po)/v(po)) so that R = MLR(po). The value of Ê depends critically on the model used. In two dimensions it ranges from 0.39 to about 6 times that, depending on the amount of the state table we want to include. For the three-dimensional projection of the fourdimensional model, it is known that  is

about 9.

By repeating essentially the same dimensional arguments, only more carefully, we find that the dissipation length la = (MR)-1R-1/2 for two dimensions and la = (MR)-R-1/4 for three dimensions.

For a typical simulation in three dimensions, we take M = 0.3 for incompressibility, R ~ 9, and L = 103, which is a large simulation, possible only on the largest Cray-class machines. Then la is about three lattice spacings, and the simulation wastes very little computational power. The subtle point is that the highly model-dependent factor R is not of order 1, as is usually estimated. It depends critically on the complexity of the collision set, going up a factor of 20 from the elementary hexagonal model in two dimensions to the projected four-dimensional case with an optimal collision table.

There is a great deal of work to be done on the high-Reynolds-number problem, but it is clear that the situation is complicated and rich in possibilities for evading simple dimensional arguments.■

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