Phys i ca D 45 (1990) 4"/7-4"/9 North-Hol land

A P P E N D I X II MAPS OF RECENT CELLULAR AUTOMATA AND LATTICE GAS AUTOMATA LITERATURE

Howard A. GUTOWITZ Center for Nonlinear Studies and Complex Systems Group, Los Alamos National Laboratory, MS-B258, Los Alamos, NM 87545, USA

The purpose of this appendix is to give those new to cellular automata a quick overview of some of the interconnections between subdisciplines in the field.

The tool used for this is a set of maps which show the relationship between the literature in subdisciplines of the fields of cellular automata and lattice gas automata in the scientific literature during the years 1986-1988 (fig. 1). These maps

were prepared by David Pendelbury and staff at the Institute for Scientific Information (ISI) in Philadelphia. Each node in a map represents a book or research article which is often cited. The length of the lines which connect these "core docu- ments" give their distance in a co-citation metric. Two documents are close in this metric if they are often cited together. The size of the circle sur- rounding each core document indicates the rela-

'~k~t~a'n 84A

Wolfram 83 Wolfram 84B

~ e , ~ n 84 Franks 82

Denker 76

Ei~n 79

1986

rllne¢ 79

BeaJe 85

~ GalJgno~ 75

Wolfram 84A

Vichniac 84

Ka~frnan 69

~ )vonNeumann ~ Den~fTG

W~r~n86

1987

Cercig nan~ 15B

Cef0gnani 7~

CP, ang S4

IChapcr~ 60

C,,eutz 83

Ha-dy

hearna~n 86

Wolfram 83

Matin 84 Wolfr;~n 86

1988

Fig. 1. Citat ion maps of CA and L G A literature in the years 1986-1988 . Courtesy of the Inst i tute for Scientific In- formation, 3501 Market Street , Phi ladelphia , Pennsy lvan ia 19104.

0 1 6 7 - 2 7 8 9 / 9 0 / $ 03.50 (~) 1990 - Elsevier Sc ience Publ i shers B.V. (North-Hol land)

478 H.A. Gutowitz / Maps of CA literature

tive n u m b e r of c i ta t ions tha t art icle received in the given year. The full c i ta t ions of the articles which appear as nodes in these maps are listed in the bibl iography. Some addi t iona l , more recent,

reviews and books are listed at the end of the bib- liography.

It is evident tha t these maps do not direct ly in- dicate conceptual connec t ions between documents . They only reflect the c i ta t ion behavior of scientists working in the field. It is equal ly evident , however, tha t art icles which are often cited together may be expected to bear on s imi lar subjects . This expec- t a t ion is suppor t ed by research conduc ted at the ISI [44]. In any event , these core documen t s are useful keys for b ibl iographic da tabase searches.

Some of the m a i n features of these maps to ob- serve are:

In 1986, the field of lat t ice gas a u t o m a t a (rep- resented e.g. by the papers by Frisch et al. and Hardy et al. 1 had jus t emerged. It was not con- nected to cellular a u t o m a t a and hence not shown in 1986. The field of cellular a u t o m a t a (repre- sented e.g. by papers by Wolfram and yon Neu- m a n n ) was connected in 1986 on the right to work in pure ma thema t i c s on "shift dynamica l systems" (cellular a u t o m a t a ) , via the paper by Mar t i n et al. (algebraic propert ies of cellular a u t o m a t a ) . Cellu- lar a u t o m a t a were connected on the left th rough the field of r a n d o m Boolean networks (Kauffman) to s tudies on the origin of life (Eigen).

In 1987, research in lat t ice gas a u t o m a t a and cellular a u t o m a t a were s t rongly in te rconnec ted , and closely connected as well with r a n d o m Bool- ean networks. At a fur ther remove, they were con- nected to shift dynamica l systems on the right, and kinet ic theory (Broadwell , Gat ignol ) above.

In 1988, la t t ice gas a u t o m a t a and cellular au- t o m a t a had diverged somewhat , though they re- ma ined in terconnected . R a n d o m Boolean nets (not shown) had diverged to form a separa te sub- field. Lat t ice gas a u t o m a t a were connected to spin sys tems (Creutz) above, and kinetic theory to the left.

R e f e r e n c e s

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[2] R.L. Adler, D. Coppersmith and M. Hassner, Algo- rithms for sliding block codes: An application of sym- bolic dynamics to information theory, IEEE Trans. Information Theory 29 (1983) 5.

[3] P.W. Anderson, Suggested model for prebiotic evolu- tion: The use of chaos, Proc. Natl. Aead. Sci. US 80 (1983) 3386.

[4] R.A. Bagnold, Experiments on a gravity-free disper- sion of large solid spheres in a Newtonian fluid under sheer, Proc. R. Soc. London Ser. A 225 (1954) 49.

[51 J.T. Beale, Large=time behavior of the Broadwell model of a discrete velocity gas, Commun. Math. Phys. 102 (1985) 217.

[6] N. Bellomo and G. Toscani, On the Cauchy problem for the nonlinear Boltzmann equation: global exis- tence, uniqueness and asymptotic stability, J. Math. Phys. 26 (1985) 334:

[7] L. Block, Periodic points and topological entropy of one-dimensional maps, in: Global theory of dynami- cal systems, Lecture Notes in Mathemathics No. 819 (1980) p. 18.

[81 J. Broadwell, Shock structure in a simple discrete ve- locity gas, Phys. Fluids 7 (1964) 1243-1247.

[9[ C. Cercignanni, Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1969).

[101 C. Cercignanni, Theory and Applications of the Boltz- mann equation (Scottish Academic Press, Edinburgh, 1975).

(11] C.S.W. Chang, G.E. Uhlenbeck and J. de Boer, The head conductivity and viscosity of polyatomic gases, in: Studies in Statistical Mechanics , Vol. II, eds. J. de Boer and G.E. Uhlenbeck (North-Holland, Amster- dana, 1964).

[12] S. Chapman and T.G. Cowling, Mathematical Theory of Nonuniform Gases (Cambridge Univ. Press, Cam- bridge, 1960, 1970).

[13] C. Conley, Isolated invariant sets and the Morse in- dex, Am. Math. Soc., Regional Conference Series 38 (1978).

[14] M. Creutz, Microcanonical Monte Carlo simulation, Phys. Rev. Lett. 50 (1983) 1411.

[15] M. Creutz, Deterministic Ising dynamics, Ann. Phys. 167 (1986) 62.

[161 M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math- ematics No. 527 (Springer, Berlin, 1976). ,

[171 B. Derrida, J. Phys. (Paris) 47 (1986) 1297; : ' [18] M. Eigen, Selforganization of matter and the

evolution of biological macromolecules, Naturwis- senschaften 58 (1971) 465.

]19] oM. Eigen, The hypercycle, part A: Emergence of the "hypercycle, Naturwissenschaften 64 (1977) 541.

[20] M. Eigen, The hypercycle, part C: The realistic hy- lYercycle, Naturwissenschaften 65 (1978) 341.

[21] M. Eigen and P. Schuster, The Hypercycle: A prin- ciple of Natural Self-Organization (Springer, Berlin, 1979).

I221 M.H. Ernst, Nonlinear model - Boltzmann equations and exact solutions, Phys. Rep. 78 (1981) 1.

H.A. Gutowitz / Maps of CA literature 479

[23] J.D. Farmer, T. Toffoli and S. Wolfram, Cellular Au- tomata , Proceedings of an Interdisciplinary Work- shop, Physica D 10 (1984).

[24] J.H. Ferzinger, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972).

[251 J.M. Franks, Homology and dynamical systems, Am. Math. Soc. Conf. Board 49 (1982).

[26] U. Frish, B. Hasslacher and Y. Pomeau, Lattice gas au toma ta for the Navier-Stokes equation, Phys. Rev. Lett. 56 (1986) 1505.

[27] R. Gatignol, Th6orie Cin~tique des Gaz/L Repart i t ion Discrete de Vitesses, Lecture Notes in Physics No. 36 (Springer, Berlin, 1975).

[28] P.K. Haft, Grain flow as a fluid-mechanical phe- nomenon, J. Fluid Mech. 134 (1983) 401.

[29] J. Hardy, Y. Pomeau and O. de Pazzis, Time evolu- tion of a two-dimensional model system. I. Invariant s tates and time correlation functions, J. Math. Phys. 14 (1973) 1746.

[30] J. Hardy, O. de Pazzis and Y. Pomeau, Molecular dynamics of a classical lattice gas: t ranspor t proper- ties and t ime correlation functions, Phys. Rev. A 13 (1976) 1949. G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theor. 3 (1969) 320.

[32] H.J. Herrmann, Fast algori thm for the simulation of Ising models, J. Stat . Phys. 45 (1986) 145.

[33] R. Illner, Global existence for two-velocity models of the Bol tzmann equation, Math. Meth. Appl. Sci. 1 (1979) 187.

[34] B.L. Jones, R.H. Enns and Rangnekar, On the theory of selection of coupled macromolecular systems, Bull. Math. Biol. 38 (1976) 15.

[35] S. Kaniel and M. Shinbrot, The Bol tzmann equation, 1. Uniqueness and local existence, Commun. Math Phys. 58 {1978} 65.

[36] S.A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol. 22 (1969) 437.

[37] M.N. Kogan, Rarefied gas dynamics (Plenum Press, New York, 1969).

[38] M. Krook and T.T. Wu, Formation of Maxwellian tails, Phys. Rev. Lett. 36 (1976) 1107.

[39] C.K.K. Lun, S.B. Savage, D.J. Jeffrey and N. Chep- urny, Kinetic theories for granular flow: inelastic par- ticles in Couette flow and slightly inelastic particles in a general flowfield, J. Fluid Mech. 140 (1984) 223.

[40] O. Martin, A. Odlyzko and S. Wolfram, Algebraic properties of cellular automata , Commun. Math Phys. 93 (1984) 219. R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces (Wiley, New York, 1976) (Reissue: Krieger, New York, 1987). L. J. Monchick and E.A. Mason, Transport properties of polar gases, Chem. Phys. 35 (1984) 1676.

[31]

I41]

[42]

[43] M. Newman, Integral Matrices (Academic Press, New York, 1972).

[44] H. Small and E. Garfield, The geography of science: disciplinary and nat ional mappings, J. Info. Sci. 11 (1985) 147-159.

[45] D.R. Smart, Fixed Point Theorems (Cambridge Univ. Press, Cambridge, 1974).

[46] C. Truesdell and R.G. Muncaster, Fundamentals of Maxwell 's Kinetic Theory of a Simple Monatomic Gas; Treated as a Branch of Rational Mechanics (Aca- demic Press, New York, 1980).

[47] G.Y. Vichniac, Simulating physics with cellular au- tomata, Physica D 10 (1984) 96.

[48] J. yon Neumann, Theory of Self-reproducing Au- tomata, ed. A.W. Burks (Univ. of Illinois Press, Champaign, IL, 1966).

[49] B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math. 77 (1973) 462.

[50] R.F. Williams, Classification of shifts of finite type, Ann. Math. 98 (1973) 120.

[51] S. Wolfram, Statistical mechanics of cellular au- tomata , Rev. Mod. Phys. 55 (1983) 601.

[52] S. Wolfram, Cellular au tomata as models of complex- ity, Nature 311 (1984) 419 [Wolfram 84a in 1986 and 1987].

[53] S. Wolfram, Computa t ion theory of cellular au- tomata , Commun. Math. Phys. 96 (1984) 15 [Wolfram 84a in 1988].

[54] S. Wolfram, Universality and complexity in cellular au tomata , Physica D 10 (1984) 1 [Wolfram 84b in 1986 and 1987, Wolfram 84a in 1988].

[55] S. Wolfram, Theory and Applications of Cellular Au- tomata (World Scientific, Singapore, 1986).

S o m e a d d i t i o n a l b o o k s a n d r e v i e w s

[56] H.J. Herrmann, Cellular automata , in: Nonlinear Phe- nomena in Complex Systems, ed. A.N Proto (North- Holland, Amsterdam, 1989).

I57] C.G. Langton, Artificial life, Santa Fe Inst i tute Stud- ies in the Sciences of Complexity (Addison-Wesley, New York, 1988).

[581 R. Livi, S. Ruffo, S. Ciliberto and M. Buiatti , Chaos and Complexity (World Scientific, Singapore, 1988).

[59] P. Manneville, N. Boccara, G.Y. Vichniac and R. Bidaux, Cellular au tomata and modeling of complex physical systems, Springer Proceedings in Physics No. 46 (Springer, Berlin, 1989).

[60] P. Rujan, Cellular au tomata and statistical mechani- cal models, J. Stat. Phys. 49 (1987) 139.

[611 T. Toffoli and N. Margolus, Cellular Automata Ma- chines - A New Environment for Modeling (MIT Press, Cambridge, MA, 1987).