3. Symmetry Considerations

The form of the macroscopic equations (2.4.7) and (2.5.11) depends on few specific properties of the hexagonal lattice cellular automaton model. The most important properties relate to the symmetries of the tensors

These tensors are determined in any cellular automaton fluid model simply from the choice of the basic particle directions . The momentum flux tensor (2.4.9) is given in terms of them by

where repeated indices are summed, and to satisfy the conditions (2.4.1) and (2.4.2)

The basic condition for standard hydrodynamic behavior is that the tensors for which appear in (3.1.2) should be isotropic. From the definition (3.1.1), the tensors must always be invariant under the discrete symmetry group of the underlying cellular automaton array. What is needed is that they should in addition be invariant under the full continuous rotation group.

The definition (3.1.1) implies that the must be totally symmetric in their space indices. With no further conditions, the could have independent components in space dimensions. Symmetries in the underlying cellular automaton array provide constraints which can reduce the number of independent components.

Tensors that are invariant under all rotations and reflections (or inversions) can have only one independent component. Such invariance is obtained with a continuous set of vectors uniformly distributed on the unit sphere. Invariance up to finite can also be obtained with certain finite sets of vectors .

Isotropic tensors obtained with sets of vectors in space dimensions must take the form

where

and in general consists of a sum of all the possible products of Kronecker delta symbols of pairs of indices, given by the recursion relation

The form of the can also be specified by giving their upper simplicial components (whose indices form a nonincreasing sequence). Thus, in two dimensions,

where the 1111, 2111, 2211, 2221, and 2222 components are given. In three dimensions,

Similarly,

and

in two and three dimensions, respectively.

For isotropic sets of vectors , one finds from (3.1.5)

so that for

while for

Similarly,

In the model of Section 2, all the particle velocities are fundamentally equivalent, and so are added with equal weight in the tensor (3.1.1). In some cellular automaton fluid models, however, one may, for example, allow particle velocities with unequal magnitudes (e.g., Ref. 31). The relevant tensors in such cases are

where the weights are typically determined from coefficients in the Chapman-Enskog expansion.

3.2. Polygons

As a first example, consider a set of unit vectors corresponding to the vertices of a regular -sided polygon:

For sufficiently large , any tensor constructed from these must be isotropic. Table 1 gives the conditions on necessary to obtain isotropic . In general, it can be shown that is isotropic if and only if does not divide any of integers {} Thus, for example, must be isotropic whenever .

[ Table 1 ] Conditions for the tensors of Eq. (3.1.1) to be isotropic with the lattice vectors chosen to correspond to the vertices of regular -sided polygons.

In the case , corresponding to the hexagonal lattice considered in Section 2, the are isotropic up to . The macroscopic equations obtained in this case thus have the usual hydrodynamic form. However, a square lattice, with , yields an anisotropic , given by

where is the Kronecker delta symbol with indices. The macroscopic equation obtained in this case is

which does not have the standard Navier-Stokes form. (3)

On a hexagonal lattice, is isotropic, but has the component form

which differs from the isotropic result (3.1.11). The corrections (2.7.6) to the Navier-Stokes equation are therefore anisotropic in this case.

3.3. Polyhedra

As three-dimensional examples, one can consider vectors corresponding to the vertices of regular polyhedra. Only for the five Platonic solids are all the equal. Table 2 gives results for the isotropy of the in these cases. Only for the icosahedron and dodecahedron is found to be isotropic, so that the usual hydrodynamic equations are obtained. As in two dimensions, the for the cube are all proportional to a single Kronecker delta symbol over all indices.

[ Table 2 ] Isotropy of the tensors with chosen as the vertices of regular polyhedra. In the forms for (which are given without normalization), the notation ``cyc:'' indicates all cyclic permutations. (All possible combinations of signs are chosen in all cases.) is the golden ratio .

In five and higher dimensions, the only regular polytopes are the simplex, and the hypercube and its dual. These give isotropic only for , and for and , respectively.

In four dimensions, there are three additional regular polytopes, specified by Schlafi symbols , , and . (The elements of these lists give the number of edges around each vertex, face, and 3-cell, respectively.) The polytope has 24 vertices with coordinates corresponding to permutations of . It yields that are isotropic up to . The polytope has 120 vertices corresponding to , all permutations of , and even-signature permutations of , where . The polytope is the dual of . Both yield that are isotropic up to .

3.4. Group Theory

The structure of the was found above by explicit calculations based on particular choices for the . The general form of the results is, however, determined solely by the symmetries of the set of . A finite group of transformations leaves the invariant. (For the hexagonal lattice model of Section 2, it is the hexagonal group .) In general is a finite subgroup of the -dimensional rotation group .

The form the basis for a representation of , as do their products . If the representation carried by the is irreducible, then the can have only one independent component, and must be rotationally invariant. But is in general reducible. The number of irreducible representations that it contains gives the number of independent components of allowed by invariance under .

This number can be found using the method of characters (e.g., Refs. 35 and 36). Each class of elements of in a particular representation has a character that receives a fixed contribution from each irreducible component of . Characters for the representation of can be found by first evaluating them for arbitrary rotations, and then specializing to the particular sets of rotations (typically through angles of the form ) that appear in . To find characters for arbitrary rotations, one writes the as sums of completely traceless tensors which form irreducible representations of (e.g., Ref. 37):

The characters of the are then sums of the characters for the irreducible tensors . For proper rotations through an angle , the are given by (e.g., Ref. 37)

The resulting characters for the representations formed by the are given in Table 3.

The number of irreducible representations in can be found as usual by evaluating the characters for each class in (e.g., Ref. 35). Consider as an example the case of with the octahedral group . This group has classes , , , , where represents the identity, and represents a proper rotation by about a -fold symmetry axis. The characters for these classes in the representation can be found from Table 3. Adding the results, and dividing by the total number of classes in , one finds that contains exactly two irreducible representations of . Rank 4 symmetric tensors can thus have up to two independent components while still being invariant under the octahedral group.

[ Table 3 ] Characters of transformations of totally symmetric rank tensors in dimensions. , where is the rotation angle. For improper rotations in three dimensions, must be used.

In general, one may consider sets of vectors that are invariant under any point symmetry group. Typically, the larger the group is, the smaller the number of independent components in the can be. In two dimensions, there are an infinite number of point groups, corresponding to transformations of regular polygons. There are only a finite of nontrivial additional point groups in three dimensions. The largest is the group of symmetries of the icosahedron (or dodecahedron). Second largest is the cubic group . As seen in Table 2, only guarantees isotropy of all tensors up to (compare Ref. 39).

It should be noted, however, that such group-theoretic considerations can only give upper bounds on the number of independent components in the . The actual number of independent components depends on the particular choice of the , and potentially on the values of weights such as those in Eq. (3.1.16).

3.5. Regular Lattices

If the vectors correspond to particle velocities, then the possible displacements of particles at each time step must be of the form . In discrete velocity gases, particle positions are not constrained. But in a cellular automaton model, they are usually taken to correspond to the sites of a regular lattice.

Only a finite number of such ``crystallographic'' lattices can be constructed in any space dimension (e.g., Refs. 40 and 41). As a result, the point symmetry groups that can occur are highly constrained. In two dimensions, the most symmetrical lattices are square and hexagonal ones. In three dimensions, the most symmetrical are hexagonal and cubic. The group-theoretic arguments of Section 3.4 suffice to show that in two dimensions, hexagonal lattices must give tensors that are isotropic up to , and so yield standard hydrodynamic equations (2.5.11). In three dimensions, group-theoretic arguments alone fail to establish the isotropy of for hexagonal and cubic lattices. A system with icosahedral point symmetry would be guaranteed to yield an isotropic , but since it is not possible to tesselate three-dimensional space with regular icosahedra, no regular lattice with such a large point symmetry group can exist.

Crystallographic lattices are classified not only by point symmetries, but also by the spatial arrangement of their sites. The lattices consist of ``unit cells'' containing a definite arrangement of sites, which can be repeated to form a regular tesselation. In two dimensions, five distinct such Bravais lattice structures exist; in three dimensions, there are 14 (e.g., Refs. 40 and 41).

Sites in these lattices can correspond directly to the sites in a cellular automaton. The links which carry particles in cellular automaton fluid models are obtained by joining pairs of sites, usually in a regular arrangement. The link vectors give the velocities of the particles.

In the simplest cases, the links join each site to its nearest neighbors. The regularity of the lattice implies that in such cases, all the are of equal length, so that all particles have the same speed.

For two-dimensional square and hexagonal lattices, the with this nearest neighbor arrangement have the form (3.2.1). The results of Section 3.2 then show that with hexagonal lattices, such give that are isotropic up to , and so yield the standard hydrodynamic continuum equations (2.6.1).

Table 4 gives the forms of for the most symmetrical three-dimensional lattices with nearest neighbor choices for the . None yield isotropic (compare Ref. 38).

The hexagonal and face-centered cubic lattices, which have the largest point symmetry groups in two and three dimensions, respectively, are also the lattices that give the densest packings of circles and spheres (e.g., Ref. 42). One suspects that in more than three dimensions (compare Ref. 43) the lattices with the largest point symmetry continue to be those with the densest sphere packing. The spheres are placed on lattice sites; the positions of their nearest neighbors are defined by a Voronoi polyhedron or Wigner-Seitz cell. The densest sphere packing is obtained when this cell, and thus the nearest neighbor vectors , are closest to forming a sphere. In dimensions , it has been found that the optimal lattices for sphere packing are those based on the sets of root vectors for a sequence of simple Lie groups (e.g., Ref. 44). Results on the isotropy of the tensors for these lattices are given in Table 5.

More isotropic sets of can be obtained by allowing links to join sites on the lattice beyond nearest neighbors. On a square lattice, one may, for example, include diagonal links, yielding a set of vectors

Including weights as in Eq. (3.1.16), this choice of yields

If the ratio of particles on diagonal and orthogonal links can be maintained so that

then Eq. (3.5.3) shows that will be isotropic. This choice effectively weights the individual vectors and with a factor . As a result, the vectors (3.5.1) are effectively those for a regular octagon, given by Eq. (3.2.1) with .

[ Table 4 ] Forms of the tensors for the most symmetrical three-dimensional Bravais lattices. The basic vectors (used here without normalization) are taken to join each site with its nearest neighbors. represents the Kronecker delta symbol of indices; represents the rotationally invariant tensor defined in Eqs. (3.1.6)--(3.1.8). is the sum of all possible products of pairs of Kronecker delta symbols with and indices, respectively.

[ Table 5 ] Sequence of simple Lie groups whose sets of root vectors yield optimal lattices for sphere packing in dimensions. These lattices may also yield maximal isotropy for the tensors . Results are given for the maximum even at which the are found to be isotropic. The root vectors are given in Ref. 45.

Including all 24 with components on a square lattice, one obtains

With , and are isotropic if

They cannot both be isotropic if also vanishes.

In three dimensions, one may consider a cubic lattice with sites at distances , , and joined. The in this case contain all those for primitive, face-centered, and body-centered cubic lattices, as given in Table 4. The can then be deduced from the results of Table 4, and are given by

Isotropy of is obtained when

and of when

Notice that (3.5.12) and (3.5.13) cannot simultaneously be satisfied by any nonzero choice of weights. Nevertheless, so long as (3.5.12) holds, isotropic hydrodynamic behavior is obtained in this three-dimensional cellular automaton fluid. Isotropic can be obtained by including in addition vectors of the form (and permutations), and choosing

The weights in Eq. (3.1.17) give the probabilities for particles with different speeds to occur. These probabilities are determined by microscopic equilibrium conditions. They can potentially be controlled by using different collision rules on different time steps (as discussed in Section 4.9). Each set of collision rules can, for example, be arranged to yield each particle speed with a certain probability. Then the frequency with which different collision rules are used can determine the densities of particles with different speeds.

3.6. Irregular Lattices

The general structure of cellular automaton fluid models considered here requires that particles can occur only at definite positions and with definite discrete velocities. But the possible particle positions need not necessarily correspond with the sites of a regular lattice. The directions of particle velocities should be taken from the directions of links. But the particle speeds may consistently be taken independent of the lengths of links.

As a result, one may consider constructing cellular automaton fluids on quasilattices (e.g., Ref. 46), such as that illustrated in Fig. 2. Particle velocities are taken to follow the directions of the links, but to have unit magnitude, independent of the spatial lengths of the links. Almost all intersections involve just two links, and so can support only two-particle interactions. These intersections occur at a seemingly irregular set of points, perhaps providing a more realistic model of collisions in continuum fluids.

[ Figure 2 ] Lattices and quasilattices constructed from grids oriented in the directions of the vertices of regular -sided polygons. An appropriate dual of the pattern is the Penrose aperiodic tiling.

The possible on regular lattices are highly constrained, as discussed in Section 3.5. But it is possible to construct quasilattices which yield any set of . Given a set of generator vectors , one constructs a grid of equally spaced lines orthogonal to each of them. The directions of these lines correspond to the .

If the tangent of the angles between the are rational, then these lines must eventually form a periodic pattern, corresponding to a regular lattice. But if, for example, the correspond to the vertices of a pentagon, then the pattern never becomes exactly periodic, and only a quasilattice is obtained. A suitable dual of the quasilattice gives in fact the standard Penrose aperiodic tiling.

In three dimensions, one may form grids of planes orthogonal to generator vectors . Possible particle positions and velocities are obtained from the lines in which these planes intersect.

Continuum equations may be derived for cellular automaton fluids on quasilattices by the same methods as were used for regular lattices above. But by appropriate choices of generator vectors, three-dimensional quasilattices with effective icosahedral point symmetry may be obtained, so that isotropic fluid behavior can be obtained even with a single particle speed.

Quasilattices yield an irregular array of particle positions, but allow only a limited number of possible particle velocities. An entirely random lattice would also allow arbitrary particle velocities. Momentum conservation cannot be obtained exactly with discrete collision rules on such a lattice, but may be arranged to hold on average.

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