Notes

[1] See for example S. Wolfram, ``Cellular automata as models of complexity'', Nature 311, 419 (1984) where applications to thermodynamics and hydrodynamics were mentioned but not explored.

[2] D. Hillis, The Connection Machine (MIT press, 1985). This application is discussed in S. Wolfram, ``Scientific computation with the Connection Machine'', Thinking Machines Corporation report (March 1985).

[3] More detailed results of theory and simulation will be given in a forthcoming series of papers.

[4] J. Hardy, Y. Pomeau and O. de Pazzis, ``Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions'', J. Math. Phys. 14, 1746 (1973); J. Hardy, O. de Pazzis and Y. Pomeau, ``Molecular dynamics of a classical lattice gas: transport properties and time correlation functions'', Phys. Rev. A13, 1949 (1976).

[5] U. Frisch, B. Hasslacher and Y. Pomeau, ``A lattice gas automaton for the Navier-Stokes equation'', Los Alamos preprint LA-UR-85-3503.

[6] The expansion rate gives the Lyapunov exponent as defined in N. Packard and S. Wolfram, ``Two-dimensional cellular automata'', J. Stat. Phys. 38, 901 (1985). Note that the effect involves many particles, and does not arise from instability in the motion of single particles, as in the case of hard spheres with continuous position variables (e.g. O. Penrose, ``Foundations of statistical mechanics'', Rep. Prog. Phys. 42, 129 (1979).)

[7] S. Wolfram, ``Origins of randomness in physical systems'', Phys. Rev. Lett. 55, 449 (1985); ``Random sequence generation by cellular automata'', Adv. Appl. Math. 7, 123 (1986).

[8] Simple patterns are obtained with very simple or symmetrical initial conditions. On a hexagonal lattice, the motion of an isolated particle in a rectangular box is described by a linear congruence relation, and is ergodic when the side lengths are not commensurate.

[9] N. Margolus, ``Physics-like models of computation'', Physica 10D, 81 (1984) shows this for some similar CA.

[10] S. Wolfram, ``Undecidability and intractability in theoretical physics'', Phys. Rev. Lett. 54, 735 (1985).

[11] cf. T. Toffoli, ``CAM: A high-performance cellular automaton machine'', Physica 10D, 195 (1984).

[12] e.g. A. Sommerfeld, Thermodynamics and statistical mechanics, (Academic Press, 1955).

[13] J. C. Maxwell, Scientific Papers II, (Cambridge University Press, 1890); J. Broadwell, ``Shock structure in a simple discrete velocity gas'', Phys. Fluids 7, 1243 (1964); S. Harris, The Boltzmann Equation, (Holt, Reinhart and Winston, 1971); J. Hardy and Y. Pomeau, ``Thermodynamics and hydrodynamics for a modeled fluid'', J. Math. Phys. 13, 1042 (1972); R. Gatignol, Theorie cinetique des gaz a repartition discrete de vitesse, (Springer, 1975).

[14] On a square lattice, the total momentum in each row is separately conserved, and so cannot be convected by velocity in the orthogonal direction [4]. Symmetric three particle collisions on a hexagonal lattice remove this spurious conservation law.

[15] The symmetric rank four tensor which determines the nonlinear and viscous terms in the Navier-Stokes equations is isotropic for a hexagonal but not a square lattice (cf. [5]). Higher order coefficients are anisotropic in both cases. In two dimensions, there can be logarithmic corrections to the Newtonian fluid approximation: these can apparently be ignored on the length scales considered, but yield a formal divergence in the viscosity (cf. [4]).

[16] e.g. D. J. Tritton, Physical fluid dynamics, (Van Nostrand, 1977).

[17] They can also treat microscopic boundary effects beyond the hydrodynamic approximation.

[18] Icosahedral symmetry yields isotropic fluid behaviour, and can be achieved with a quasilattice, or approximately by periodic lattices (cf. D. Levine et al., ``Elasticity and dislocations in pentagonal and icosahedral quasicrystals'', Phys. Rev. Lett. 54, 1520 (1985); P. Bak, ``Symmetry, stability, and elastic properties of icosahedral incommensurate crystals'', Phys. Rev. B32, 5764 (1985)).

[19] e.g. P. Roache, Computational fluid dynamics, (Hermosa, Albuquerque, 1976).

[20] S. Orszag and V. Yakhot, ``Reynolds number scaling of cellular automaton hydrodynamics'', Princeton University Applied and Computational Math. report (November 1985).

[21] In simple cases the resulting model is analogous to a deterministic microcanonical spin system (M. Creutz, ``Deterministic Ising dynamics'', Ann. Phys., to be published.)

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