4. Evaluation of Transport Coefficients

4.1. Introduction

Section 2 gave a derivation of the general form of the hydrodynamic equations for a sample cellular automaton fluid model. This section considers the evaluation of the specific transport coefficients that appear in these equations. While these coefficients may readily be found by explicit simulation, as discussed in the second paper in this series, no exact mathematical procedure is known for calculating them. This section considers primarily an approximation method based on the Boltzmann transport equation. The results obtained are expected to be accurate for certain transport coefficients at low particle densities. (4)

4.2. Basis for Boltzmann Transport Equation

The kinetic equation (2.3.5) gives an exact result for the evolution of the one-particle distribution function . But the collision term in this equation depends on two-particle distribution functions, which in turn depend on higher order distribution functions, forming the BBGKY hierarchy of kinetic equations. To obtain explicit results for the one must close or truncate this hierarchy.

The simplest assumption is that there are no statistical correlations between the particles participating in any collision. In this case, the multiparticle distribution functions that appear in can be replaced by products of one-particle distribution functions , yielding an equation of the standard Boltzmann transport form, which can in principle be solved explicitly for the .

Even if particles were uncorrelated before a collision, they must necessarily show correlations after the collision. As a result, the factorization of multiparticle distribution functions used to obtain the Boltzmann transport equation cannot formally remain consistent. At low densities, it may nevertheless in some cases provide an adequate approximation.

Correlations produced by a particular collision are typically important only if the particles involved collide again before losing their correlations. At low densities, particles usually travel large distances between collisions, so that most collisions involve different sets of particles. The particles involved in one collision will typically suffer many other collisions before meeting again, so that they are unlikely to maintain correlations. At high densities, however, the same particles often undergo many successive collisions, so that correlations can instead be amplified.

In the Boltzmann transport equation approximation, correlations and deviations from equilibrium decay exponentially with time. Microscopic perturbations may, however, lead to collective, hydrodynamic, effects, which decay only as a power of time. Such effects may lead to transport coefficients that are nonanalytic functions of density and other parameters, as mentioned in Section 2.6.

4.3. Construction of Boltzmann Transport Equation

This subsection describes the formulation of the Boltzmann transport equation for the sample cellular automaton fluid model discussed in Section 2.

The possible classes of particle collisions in this model are illustrated in Fig. 3. The rules for different collisions within each class are related by lattice symmetries. But, as illustrated in Fig. 3, several choices of overall rules for each class are often allowed by conservation laws.

In the simplest case, the same rule is chosen for a particular class of collisions at every site. But it is often convenient to allow different choices of rules at different sites. Thus, for example, there could be a checkerboard arrangement of sites on which two-body collisions lead alternately to scattering to the left and to the right. In general, one may apply a set of rules denoted by at some fraction of the sites in a cellular automaton. (A similar procedure was mentioned in Section 3.5 as a means for obtaining isotropic behavior on three-dimensional cubic lattices.) The randomness of microscopic particle configurations suggests that the should serve merely to change the overall probabilities for different types of collisions.

[ Figure 3 ] Possible types of initial and final states for collisions in the cellular automaton fluid model of Section 2.

The term in the kinetic equation (2.3.5) for is a sum of terms representing possible collisions involving particles of type . Each term gives the change in the number of type particles due to a particular type of collisions, multiplied by the probability for the arrangement of particles involved in the collision to occur. In the Boltzmann equation approximation, the probability for a particular particle arrangement is taken to be a simple product of the densities for particles that should be present, multiplied by factors for particles that should be absent.

The complete Boltzmann transport equation for the model of Section 2 thus becomes

where

Here

where all indices on the are evaluated modulo , and in this case . Note that in Eq. (4.3.2), the index has been dropped on both and .

The Boltzmann transport equations for any cellular automaton fluid model have the overall form of Eqs. (4.3.1) and (4.3.2). In a more general case, the simple addition of constants to the indices in the definition of can be replaced by transformations with appropriate lattice symmetry group operations.

Independent of the values of the , is seen to satisfy the momentum and particle number constraints (2.4.4) and (2.4.5).

In the following calculations it is often convenient to maintain arbitrary values for the so as to trace the contributions of different classes of collisions. But to obtain a form for that is invariant under the complete lattice symmetry group, one must take

4.4. Linear Approximation to Boltzmann Transport Equation

In studying macroscopic behavior, one assumes that the distribution functions differ only slightly from their equilibrium values, as in the Chapman-Enskog expansion (2.5.1). The may thus be approximated as

With this approximation, the collision term in the Boltzmann transport equation may be approximated by a power series expansion in the :

The matrix here is analogous to the usual linearized collision operator (e.g., Ref. 26). Notice that for a cellular automaton fluid model with collisions involving at most particles, the expansion (4.4.2) terminates at .

Microscopic reversibility immediately implies that the tensors are all completely symmetric in their indices. The conservation laws (2.4.4) and (2.4.5) yield conditions on all the of the form

In the particular case of , the more stringent conditions

and

also apply.

In the model of Section 2, all particle types are equivalent up to lattice symmetry transformations. As a result, is always given simply by a cyclic shift of , so that the complete form of can be determined from the first row . The are thus circulant tensors (e.g., Ref. 50), and the values of their components depend only on numerical differences between their indices, evaluated modulo .

Expansion of (4.3.2) now yields

where . Taking for simplicity , , , , one finds

4.5. Approach to Equilibrium

In a spatially uniform system close to equilibrium, one may use a linear approximation to the Boltzmann equation (4.3.1):

This equation can be solved in terms of the eigenvalues and eigenvectors of the matrix . The circulant property of considerably simplifies the computations required.

An circulant matrix can in general be written in the form

where

is an cyclic permutation matrix, and is the identity matrix. From this representation, it follows that all circulants have the same set of right eigenvectors , with components given by

Writing

the corresponding eigenvalues are found to be

Using these results, the eigenvectors of for the model of Section 2 are found to be

where

and the corresponding eigenvalues are

Combinations of the corresponding to eigenvectors with zero eigenvalue are conserved with time according to Eq. (4.5.1). Three such combinations are associated with the conservation laws (2.4.1) and (2.4.2). corresponds to , which is the total particle number density. and correspond, respectively, to the and components of the momentum density .

The may always be written as sums of pieces proportional to each of the orthogonal eigenvectors of Eq. (4.5.7):

The coefficients , , and give the values of the conserved particle and momentum densities in this representation, and remain fixed with time.

The general solution of Eq. (4.5.1) is given in terms of Eq. (4.5.9) by

Equation (4.5.8) shows that for any positive choices of the , all nonzero have negative real parts. As a result, the associated must decay exponentially with time. Only the combinations of associated with conserved quantities survive at large times.

This result supports the local equilibrium assumption used for the derivation of hydrodynamic equations in Section 2. It implies that regardless of the initial average densities , collisions bring the system to an equilibrium that depends only on the values of the macroscopic conserved quantities (2.4.1) and (2.4.2). One may thus expect to be able to describe the local state of the cellular automaton fluid on time scales large compared to () solely in terms of these macroscopic conserved quantities. [Section 4.2 nevertheless mentioned some effects not accounted for by the Boltzmann equation (4.3.1) that can slow the approach to equilibrium.]

One notable feature of the results (4.5.8) is that they imply that the final equilibrium values of the are not affected by the choice of the parameters and , which determine the mixtures of two- and four-particle collisions with different chiralities. When the rate for collisions with different chiralities are unequal, however, and acquire imaginary parts, which lead to damped oscillations in the as a function of time.

When all the types of collisions illustrated in Fig. 3 can occur, Eq. (4.5.8) implies that momentum and particle number are indeed the only conserved quantities. If, however, only two-particle collisions are allowed, then there are additional conserved quantities. In fact, whenever symmetric three-particle collisions are absent, so that , Eq. (4.5.8) implies that the quantities

where the index is evaluated modulo , is conserved. Thus, independent of the value of , the total momenta on the two sides of any line (not along a lattice direction) through the cellular automaton must independently be conserved.

If three-particle symmetric collisions are absent, the cellular automaton thus exhibits a spurious additional conservation law, which prevents the attainment of standard local equilibrium, and modifies the hydrodynamic behavior discussed in Section 2. Section 4.8 considers some general conditions which avoid such spurious conservation laws.

4.6. Equilibrium Conditions and Transport Coefficients

Section 4.5 discussed the solution of the Boltzmann transport equation for uniform cellular automaton fluids. This section considers nonuniform fluids, and gives some approximate results for transport coefficients.

The Chapman-Enskog expansion (2.5.1) gives the general form for approximations to the microscopic distribution functions . The coefficients and that appear in this expansion can be estimated using the Boltzmann transport equation (4.3.1) from the microscopic equilibrium condition

In estimating , one must maintain terms in to the second order in , but one can neglect spatial variation in the . As a result, the Boltzmann equation (4.3.1) becomes

Substituting forms for the from the Chapman-Enskog expansion (2.5.1), one obtains

where according to Eq. (2.5.4). Using the forms for and determined by the expansion of Eq. (4.3.2), one finds that the two terms in (4.6.3) show exactly the same dependence on the . The final result for is thus independent of the , and is given by

In the Boltzmann equation approximation, this implies that the coefficient of the term in the hydrodynamic equation (2.6.1) is . Notice that, as discussed in Section 2.6, this coefficient is not in general equal to 1.

The value of the coefficient can be found by a slightly simpler calculation, which depends only on the linear part of the expansion of the collision term . Keeping now first-order spatial derivatives of the , one can determine from the equilibrium condition

which yields

With the approximations used, Eq. (2.4.7) implies that . Then Eq. (4.6.6) gives the result

Using Eq. (2.6.3), this gives the kinematic viscosity of the cellular automaton fluid in the Boltzmann equation approximation as

Some particular values are

For one obtains in these cases and , respectively, while for , and .

4.7. A General Nonlinear Approximation

At least for homogeneous systems, Boltzmann's theorem (e.g., Ref. 51) yields a general form for the equilibrium solution of the full nonlinear Boltzmann equation (4.3.1). The function can be defined as

where

This definition is analogous to that used for Fermi-Dirac particles (e.g., Refs. 51, 52): the factors account for the exclusion of more than one particle on each link, as in Eq. (4.3.1). The microscopic reversibility of (4.3.1) implies that when the equilibrium condition holds, all products must be equal for all initial and final sets of particles that can participate in collisions. As a result, the must be simple linear combinations of the quantities conserved in the collisions. If only particle number and momentum are conserved, and there are no spurious conserved quantities such as (4.5.11), the can always be written in the form

The one-particle distribution functions thus have the usual Fermi-Dirac form

where and are in general functions of the conserved quantities and .

For small , one may write

These expansions can be substituted into Eq. (4.7.4), and the results compared with the Chapman-Enskog expansion (2.7.1).

For , one finds immediately the ``fugacity relation''

Then, from the expansion (related to that for generating Euler polynomials)

together with the constraints (2.7.3)--(2.7.5) one obtains (for )

where it has been assumed that the form an isotropic set of unit vectors, satisfying Eq. (3.1.5). The complete Chapman-Enskog expansion (2.7.1) then becomes

where for the last term it has been assumed that .

The result (4.6.4) for follows immediately from this expansion. For cellular automaton fluid models with isotropic, the continuum equation (2.7.7) holds. The results for the coefficients that appear in this equation can be obtained from the approximation (4.7.9), and have the simple forms

These results allow an estimate of the importance of the next-order corrections to the Navier-Stokes equations included in Eq. (2.7.7). They suggest that the corrections may be important whenever is not small compared to 1. The corrections can thus potentially be important both at high average velocities and high particle densities.

The hexagonal lattice model of Section 2 yields a continuum equation of the form (2.7.6), with an anisotropic term. Equation (4.7.9) gives in this case

The term is as given in Eq. (2.5.11). The terms are anisotropic, and are not even invariant under exchange of and coordinates ( rotation). For small densities , Eqs. (4.7.12) and (4.7.13) become

The results (4.7.10) and (4.7.11) follow from the Fermi-Dirac particle distribution (4.7.4). If instead an arbitrary number of particles were allowed at each site, the equilibrium particle distribution (4.7.4) would take on the Maxwell-Boltzmann form

With this simpler form, more complete results for as a function of and can be found. Results which are isotropic to all orders in can be obtained only for an infinite set of possible particle directions, parametrized, say, by a continuous angle . In this case, the number and momentum densities (2.4.1) and (2.4.2) may be obtained as integrals

With the distribution (4.7.16), these integrals become

where , and the are modified Bessel functions (e.g., Ref. 53)

The rapid convergence of the series (4.7.21) means that Eqs. (4.7.19) and (4.7.20) provide highly accurate approximations even for a small number of discrete directions . [For example, with , , , and , the error in Eq. (4.7.19) is less than .]

For the simple distribution (4.7.16) the momentum flux density tensor (2.4.9) may be evaluated in direct analogy with Eqs. (4.7.19) and (4.7.20) as

Using the recurrence relation (4.7.23), and substituting the results (4.7.19) and (4.7.20), this may be rewritten in the form

Combining Eqs. (4.7.19) and (4.7.20), one finds that the function is independent of , and can be determined from the implicit equation

Expanding in powers of , as in Eq. (4.7.5), yields

Equation (4.7.19) then gives

In the limit , .

The above results immediately yield values for the transport coefficients in the Chapman-Enskog expansion:

independent of density. Equation (4.7.29) implies that the coefficient of the convective term in the Navier-Stokes equation (2.6.1) is equal to . The deviation from the Galilean invariant result is associated with the constraint of fixed speed particles.

Figure 4 shows the exact result for obtained from Eq. (4.7.26), compared with series expansions to various orders. Significant deviations from the ``Navier-Stokes'' approximation are seen for .

For Fermi-Dirac distributions of the form (4.7.4), the integrals (4.7.19) and (4.7.20) can only be expressed as infinite sums of Bessel functions.

4.8. Other Models

The results obtained so far can be generalized directly to a large class of cellular automaton fluid models.

[ Figure 4 ] Dependence of from Eq. (4.7.16) on the magnitude of the macroscopic velocity. The results are for Maxwell-Boltzmann particles with unit speeds and arbitrary directions in two dimensions. The function appears both in the microscopic distribution function (4.7.16) and in the macroscopic momentum flux tensor (4.7.25). The result for from an exact solution of the implicit equation (4.7.26) is given, together with results from the series expansion (4.7.27). The result corresponds to the Navier-Stokes approximation. Deviation from the exact result is seen for .

In the main case considered in Section 3, particles have velocities corresponding to a set of unit vectors . If this set is invariant under inversion, then both and always occur. As a result, two particles colliding head on with velocities and can always scatter in any directions and with . One simple possibility is to choose the rules at different sites so that each scattering direction occurs with equal probability. If only such two-particle collisions are possible (as in a low-density approximation), and only one particle is allowed on each link, then the Boltzmann transport equation becomes

where is the distribution function for particles with direction . To second order in the expansion (4.4.2) this gives

where denotes summation over the triangular region in which the indices form a strictly increasing sequence.

The form of the for a homogeneous system can be obtained from the general equilibrium conditions of Section 4.7. The coefficients in the Chapman-Enskog expansion are then given by Equations (4.7.10) and (4.7.11). The convective transport coefficient in the Navier-Stokes equation (2.6.1) is thus given by

The cannot be obtained by the methods of Section 4.7. But from Eq. (4.8.2) one may deduce immediately the linearized collision term

Then, in analogy with Eq. (4.6.8), the kinematic viscosity for the cellular automaton fluid is found to be

For an icosahedral set of , with and , this yields

Several generalizations may now be considered. First, one may allow not just one, but, say, up to particles on each link of the cellular automaton array. In the limit an arbitrary density of particles is thus allowed in each cell. The Boltzmann equation for this case is the same as (4.8.1), but with all factors omitted. The resulting transport coefficients are

Another generalization is to allow collisions that involve more than two particles. The simplest such collisions are ``composite'' ones, formed by superposing collisions involving two or less particles. The presence of such collisions changes the values of transport coefficients, but cannot affect the basic properties of the model. The four-particle and asymmetric three-particle collisions in the hexagonal lattice model of Section 2 are examples of composite collisions. They increase the total collision rate, and thus, for example, decrease the viscosity, but do not change the overall macroscopic behavior of the model.

In general, collisions involving particles can occur if the possible are such that

for some sets of incoming and outgoing particles and . Cases in which all the and are distinct may be considered ``elementary'' collisions. In the hexagonal lattice model of Section 2, only two-particle and symmetric three-particle collisions are elementary.

No elementary three-particle collisions are possible on primitive and body-centered cubic three-dimensional lattices, or with corresponding to the vertices of icosahedra or dodecahedra. For a face-centered cubic lattice, however, eight distinct triples of sum to zero [an example is ], so that elementary three-particle collisions are possible.

One feature of the hexagonal lattice model discussed in Section 2 is the existence of the conservation law (4.5.11) when elementary three-body symmetric collisions are absent. Such spurious conservation laws exist in any cellular automaton fluid model in which all particles have the same speed, and only two-particle collisions can occur. Elementary three-particle collisions provide one mechanism for avoiding these conservation laws and allowing the equilibrium of Section 4.7 to be attained.

4.9. Multiple Speed Models

A further generalization is to allow particles with velocities of different magnitudes. This generalization is significant not only in allowing two-particle collisions alone to avoid the spurious conservation laws of Section 4.5, but also in making it possible to obtain isotropic hydrodynamic behavior on cubic lattices, as discussed in Section 3.5.

One may define a kinetic energy that differs for particles of different speeds. In studies of processes such as heat conduction, one must account for the conservation of total kinetic energy. In many cases, however, one considers systems in contact with a heat bath, so that energy need not be conserved in individual collisions.

In a typical case, one may then take pairs of particles with speed colliding head on to give pairs of particles with some other speed . In general, different collision rules may be used on different sites, typically following some regular pattern, as discussed in Section 4.3. Thus, for example, collisions between speed particles may yield speed particles at a fraction of the sites.

The number of possible particles with speed that can occur at each site is determined by the structure of the lattice. The collision rules at different sites may be arranged, as in Section 4.8, to yield particles of a particular speed with equal probabilities in each of the possible directions.

In a homogenous system, the probability for a link with speed to be populated should satisfy the master equation

where it assumed that

and is the reduced particle density given by Eq. (4.7.2). With two speeds, becomes

The solutions of Eq. (4.9.1) can be found in terms of the eigenvalues and corresponding eigenvectors of this matrix:

In the large-time limit, only the equilibrium eigenvector (4.9.4) should survive, giving a ratio of reduced particle densities

For three particle speeds, one finds the equilibrium conditions

Different choices for the yield different equilibrium speed distributions. The probabilities give the weights that appear in Eq. (3.1.16). Equation (4.9.6) shows that by choosing

one obtains a ratio of weights for the model of Eq. (3.5.1) that satisfy the condition (3.5.5) for the isotropy of . [There is a small correction to equality in Eq. (4.9.8) associated with the difference between and .]

On a cubic lattice, one may similarly satisfy the condition (3.5.12) for the isotropy of simply by taking , and

In this way, one may obtain approximate isotropic hydrodynamic behavior on a three-dimensional cubic lattice.

4.10. Tagged Particle Dynamics

In the discussion above, all the particles in the cellular automaton fluid were assumed indistinguishable. This section considers the behavior of a small concentration of special ``tagged'' particles.

The density of tagged particles with direction satisfies an equation of the Fokker-Planck type (e.g., Ref. 26):

Assuming as in the Boltzmann equation approximation that there are no correlations between particles at different sites, the collision term of Eq. (4.10.1) may be written in the form

where gives the probability that a particle that arrives at a particular site from direction leaves in direction with . The probability is averaged over different arrangements of ordinary particles. Various deterministic rules may be chosen for collisions between ordinary and tagged particles. The simplest assumption is that on average the tagged particles take the place of any of the outgoing particles with equal probability.

Conservation of the total number of tagged particles implies

The total momentum of tagged particles is not conserved; the background of ordinary particles acts like a ``heat bath'' which can exchange momentum with the tagged particles through the noise term . Assuming a uniform background fluid, one may make an expansion for the of the form

The total number of tagged particles then satisfies the equation

where the collision term disappears as a result of Eq. (4.10.3). With the chosen so that is isotropic, Eq. (4.10.5) becomes the standard equation for self-diffusion,

with the diffusion coefficient given by

The value of must be found by solving Eq. (4.10.1) for using the approximation (4.10.4). The equilibrium condition for Eq. (4.10.1) in this case becomes

Thus is given in this approximation by the mean free path for particle scattering, so that the diffusion coefficient is given by the standard kinetic theory formula

For the hexagonal lattice model of Section 2,

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