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THE HILBERT CONTRACTION (continued)
tions: The gas reaches local equilibrium in a collision time or so; the one-particle distribution function has a local MaxwellBoltzmann form (or whatever form is appropriate), call it fi; a second time scale is assumed where space gradients are still small, but collective modes develop at large distances and at times much greater than molecular collision times. Then one assumes a general functional perturbation expansion exists of the form
The solubility conditions for this are that L1 must be orthogonal to the five zero eigenmodes of C (()) = 0 (the solutions are 1, v, and v2). These solubility conditions are the Euler equation for p, v, and T and the ideal gas equation of state. In this way one derives a sequence of hydrodynamical equations with explicit forms for the transport coefficients. Order 0 gives the Euler equation, order 1 gives the Navier-Stokes equations, order 2 and greater give the generalized hydrodynamical equations, which have some validity only in special situations. The expansion which turns out to be explicitly a spatial is an asymptotic functional expansion, so gradient expansion: going beyond Navier-Stokes takes one away from ordinary fluids rather than closer to them. Solving explicitly for the various (") gives a way to evaluate the transport quantities (viscosity, etc.).
ƒ = ƒ✩(1 + §(1) + §(2) + · · ·),
ƒ =ƒ¿(1+c1(v)(AV) + c2(v)(AV)2 + · · ·)
where is the mean free path in the system and v is the macrovelocity.
The perturbation expansion is set up so that at nth order, the correction to f obeys an integral equation of the form fLC (()) = Ln, where C is the Boltzmann collision operator and L, is an operator that depends only on lower order spatial derivatives. This generates a recursive tower of relations (") whose solubility conditions at order n are the (n - 1)thorder hydrodynamical equations. For example, assume
f = fi (1 +¿1);
There are many other ways to do the same thing-multiple time expansions, dispersion methods, etc. We have developed everything so far within the conceptual frame of the Boltzmann transport equation. Within that framework the problem of deriving macrodynamical equations and associated transport coefficients reduces to tedious but straightforward linear algebra that has absorbed the best efforts of excellent technical people since the turn of the century. It is a problem best suited to a computer but only recently have algebraic processors of sufficient power been available.
This asymptotic perturbation expansion is a way to compute measurable quantities from microdynamical properties, but the physical insight one gains from doing it is small. The other methods mentioned, especially correlation-function techniques, are much more revealing. All of these comments and approaches carry over directly to the discrete case of the lattice gas. Nothing conceptually new arises in the totally discrete case, but explicit calculations are a great deal easier. ■
Boltzmann equation. The zero-order relation gives the Euler equations and the second-order relation gives the NavierStokes equations. However, Hilbert's method is an asymptotic functional expansion, so that the higher order terms take one away from ordinary fluids rather than closer to them. Nevertheless, solving explicitly for the terms in the functional expansion provides a way of evaluating transport coefficients such as viscosity. (See the "Hilbert Contraction" for more discussion.)
Summary of the Kinetic Theory Pic-
This review was intended to give a flavor for the chain of reasoning involved. We will use this chain again in the totally discrete lattice world. However, just as important as understanding the kinetic theory viewpoint is keeping in mind its limitations. In particular, notice that perturbation theory was the main tool used for going from the exact Boltzmann transport equation to the Navier-Stokes equations. We did not discover more pow
erful techniques for finding solutions to the Navier-Stokes equations than we had before. To go from the Boltzmann to the Navier-Stokes description, we made many smoothness assumptions in various probabilistic disguises; in other words, we recreated an approximation to the continuum. It is true one could compute (at least for relatively simple systems) the transport coefficients, but in a sense these coefficients are a property of microkinetics, not macrodynamics.
We are at a point where we can ask some questions about the emergence of macrodynamics from microscopic physics. It is clear by now that microscopic conservation laws, those of mass, momentum, and energy are crucial in fixing the form of large-scale dynamics. These are in a sense sacred. But one can question the importance of the description of individual collisions. How detailed must micromechanics be to generate the qualitative behavior predicted by the Navier-Stokes equations? Can it be done with simple collisions and very few classes of them? There exists a whole collection of equations whose functional form is very nearly that of the Navier-Stokes equations. What microworlds generate these? Do we have to be exactly at the Navier-Stokes equations to generate the qualitative behavior and numerical values that we derive from the Navier-Stokes equations or from real fluid experiments? Is it possible to design a collection of synthetic microworlds that could be considered local-interaction board games, all having Navier-Stokes macrodynamics? In other words, does the detailed microphysics of fluids get washed out of the macrodynamical picture under very rapid iteration of the deterministic system? If the microgame is simple enough to update it deterministically on a parallel machine, is the density of states required to see everything we see in ordinary Navier-Stokes simulations much smaller than the density of atoms in real physical fluids? If so, these synthetic
lection of equilibrium distribution functions whose macroscopic parameters are unconstrained. These distribution functions have a Maxwell-Boltzmann form, e-E(pv)/T. If these distribution functions are made to deviate slightly from equilibrium, then local conservation laws impose consistency conditions among their parameters, which become constrained variables. These consistency conditions are the macrodynamical equations necessary to put a consistent equilibrium function description onto the many-element system. In physical fluids they are the Navier-Stokes equations. This is the general setup that will guide us in creating a lattice model.
Evolution of Discrete Fluid Models
Continuous Network Models. The Navier-Stokes equations, however derived, are analytically intractable, except in a few special cases for especially clean geometries. Fortunately, one can avoid them altogether for many problems, such as shocks in certain geometries. The strategy is to rephrase the problem in a very simple phase space and solve the Boltzmann transport equation directly. If a single type of particle is constrained to move continuously only along a regular grid, the Boltzmann equation is so tightly constrained that it has simple analytic solutions. In the early 1960s Broadwell and others applied this simplified method of analysis to the dynamics of shock problems. Their numerical results agreed closely with much more elaborate computer modeling from the NavierStokes equations. However, there was no real insight into why such a calculation in such a simplified microworld should give such accurate answers. The accuracy of the limited phase-space approach was considered an anomaly.
Discrete Skeletal Models. The next development in discrete fluid theory was. a discrete modification of the continuousspeed network models of the Broadwell class. By forming a loose analogy to the structure of the Ising model (spins on a lattice), Hardy, de Pazzis, and Pomeau created the first minimalist fluid model on a two-dimensional square lattice. It was a simple, binary-valued, nearest-neighbor gas with a single species of molecule, limited to binary collisions. The new feature was a totally discrete velocity and state space for the gas. Particles hopped from one site to the next without a notion of continuous movement between sites. Particles were confined to the vertices of the network, and the velocity vector of each particle could point in only one. of four directions. Since there was no natural way to deal with bound states, these authors imposed the arbitrary rule that the maximum number of particles occupying any vertex be four.
This simple model possessed remarkable properties including local thermodynamic equilibrium and the emergence of a scale separation; that is, the typical collective motion scale L is much greater than the microscopic mean free path Im; L≫lm. However, the macrodynamics that emerged was not that of the Navier-Stokes equations but a more complex one with unphysical features. The square model was the first example of rich dynamics emerging wholly on a cellular space. It had all the right ingredients except one: isotropy under the rotation group of the lattice. The momentum flux tensor must reduce to a scalar for isotropy, but this is impossible with a square lattice. In two dimensions the neighborhood that has the minimal required symmetry and tiles the plane is a hexagonal neighborhood. In Part II we will present the simple hexagonal model, analyze it mathematically, and describe the simulations of fluid phenomena that have been done so far.
3. A lattice on which the particles reside only at the vertices. In the simplest case the lattice is regular and has a hexagonal neighborhood to guarantee an isotropic momentum flux tensor. We use a triangular lattice for convenience.
4. A minimum set of collision rules that define symmetric binary and triple collisions such that momentum and particle number are conserved (Fig. 4).
5. An exclusion principle so that at each vertex no two particles can have identical velocities. This limits the maximum number of particles at a vertex to six, each one having a velocity that points in one of the six directions defined by the hexagonal neighborhood.
The only way to make this hexagonal lattice gas simpler is to lower the rotation symmetry of the lattice, remove collision rules, or break a conservation law. In a two-dimensional universe with boundaries, any such modification will not give NavierStokes dynamics. Left as it is, the model will. Adding attributes to the model, such
as different types of particles, different speeds, enlarged neighborhoods, or weighted
At this point it is instructive to look at the complete table of allowed states for
Only a head-on collision of
Three particles colliding at 120°
In most configurations parti-