Lattice gas automata for reactive systems

Physics Report

Volumn 273, Issue 2, 1996, Pages 55-147

  Boon, Jean Pierre ^a   David, Dab ^b   Kapral, Raymond ^c   Lawniczak, Anna ^d  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^c UNIVERSITY OF TORONTO   (Canada)

^d UNIVERSITY OF GUELPH   (Canada)

Author keywords

Chemical kinetics; Lattice gas automata; Reaction diffusion; Reactive systems; Stochastic processes

Indexed keywords

EID: 0030211243     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/0370-1573(95)00080-1     Document Type: Review

References (161)

1. [1] J. Hardy, O. de Pazzis and Y. Pomeau, Phys. Rev. A 13 1949 (1976).
2. [2] U. Frisch, B. Hasslacher and Y. Pomeau, Phys. Rev. Lett. 56, 1505 (1986); D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau and J.P. Rivet, Complex Systems 1, 648 (1987).
3. [2] U. Frisch, B. Hasslacher and Y. Pomeau, Phys. Rev. Lett. 56, 1505 (1986); D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau and J.P. Rivet, Complex Systems 1, 648 (1987).
4. [3] Lattice Gas Methods for Partial Differential Equations, ed. G. Doolen (Addison-Wesley, NY, 1990); Lattice Gas Automata: Theory, Implementation, Simulation, ed. J.P. Boon, J. Stat. Phys. 68, 3/4 (1992); R. Benzi, S. Succi and M. Vergassola, Phys. Reports 222, 145 (1992); D.H. Rothman and S. Zaleski, Rev. Mod. Phys. 66, 1417 (1994); Pattern Formation and Lattice Gas Automata, eds. A.T. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
5. [3] Lattice Gas Methods for Partial Differential Equations, ed. G. Doolen (Addison-Wesley, NY, 1990); Lattice Gas Automata: Theory, Implementation, Simulation, ed. J.P. Boon, J. Stat. Phys. 68, 3/4 (1992); R. Benzi, S. Succi and M. Vergassola, Phys. Reports 222, 145 (1992); D.H. Rothman and S. Zaleski, Rev. Mod. Phys. 66, 1417 (1994); Pattern Formation and Lattice Gas Automata, eds. A.T. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
6. [3] Lattice Gas Methods for Partial Differential Equations, ed. G. Doolen (Addison-Wesley, NY, 1990); Lattice Gas Automata: Theory, Implementation, Simulation, ed. J.P. Boon, J. Stat. Phys. 68, 3/4 (1992); R. Benzi, S. Succi and M. Vergassola, Phys. Reports 222, 145 (1992); D.H. Rothman and S. Zaleski, Rev. Mod. Phys. 66, 1417 (1994); Pattern Formation and Lattice Gas Automata, eds. A.T. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
7. [3] Lattice Gas Methods for Partial Differential Equations, ed. G. Doolen (Addison-Wesley, NY, 1990); Lattice Gas Automata: Theory, Implementation, Simulation, ed. J.P. Boon, J. Stat. Phys. 68, 3/4 (1992); R. Benzi, S. Succi and M. Vergassola, Phys. Reports 222, 145 (1992); D.H. Rothman and S. Zaleski, Rev. Mod. Phys. 66, 1417 (1994); Pattern Formation and Lattice Gas Automata, eds. A.T. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
8. [3] Lattice Gas Methods for Partial Differential Equations, ed. G. Doolen (Addison-Wesley, NY, 1990); Lattice Gas Automata: Theory, Implementation, Simulation, ed. J.P. Boon, J. Stat. Phys. 68, 3/4 (1992); R. Benzi, S. Succi and M. Vergassola, Phys. Reports 222, 145 (1992); D.H. Rothman and S. Zaleski, Rev. Mod. Phys. 66, 1417 (1994); Pattern Formation and Lattice Gas Automata, eds. A.T. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
9. [4] V.G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, New Jersey, 1962).
10. [5] P. Clavin, P. Lallemand, Y. Pomeau and G. Searby, J. Fluid Mech. 188, 437 (1989); V. Zehnle and G. Searby, J. Phys. (Paris) 50, 1083 (1989).
11. [5] P. Clavin, P. Lallemand, Y. Pomeau and G. Searby, J. Fluid Mech. 188, 437 (1989); V. Zehnle and G. Searby, J. Phys. (Paris) 50, 1083 (1989).
12. [6] P. Ortoleva and S. Yip, J. Chem. Phys. 65, 2045 (1976).
13. [7] J. Boissonade, Phys. Lett. A 74, 285 (1979).
14. [8] F. Baras and M. Malek-Mansour, J. Chem. Phys 96, 2000 (1992); M. Malek-Mansour and F. Baras, Physica A 188, 253 (1992); J. Gorecki, Phys. Rev. Lett., submitted (1994); J. Gorecki, K. Kitahara, K. Yoshikawa and I. Hanazaki, Physica A, submitted (1994); J. Gorecki, in Far-From-Equilibrium Dynamics of Chemical Systems, Proceedings of the Third International Symposium, Borki, Poland, eds. J. Gorecki, A.S. Cukrowski, A.L. Kawczynski and B. Nowakowski (World Scientific, Singapore, 1994).
15. [8] F. Baras and M. Malek-Mansour, J. Chem. Phys 96, 2000 (1992); M. Malek-Mansour and F. Baras, Physica A 188, 253 (1992); J. Gorecki, Phys. Rev. Lett., submitted (1994); J. Gorecki, K. Kitahara, K. Yoshikawa and I. Hanazaki, Physica A, submitted (1994); J. Gorecki, in Far-From-Equilibrium Dynamics of Chemical Systems, Proceedings of the Third International Symposium, Borki, Poland, eds. J. Gorecki, A.S. Cukrowski, A.L. Kawczynski and B. Nowakowski (World Scientific, Singapore, 1994).
16. [8] F. Baras and M. Malek-Mansour, J. Chem. Phys 96, 2000 (1992); M. Malek-Mansour and F. Baras, Physica A 188, 253 (1992); J. Gorecki, Phys. Rev. Lett., submitted (1994); J. Gorecki, K. Kitahara, K. Yoshikawa and I. Hanazaki, Physica A, submitted (1994); J. Gorecki, in Far-From-Equilibrium Dynamics of Chemical Systems, Proceedings of the Third International Symposium, Borki, Poland, eds. J. Gorecki, A.S. Cukrowski, A.L. Kawczynski and B. Nowakowski (World Scientific, Singapore, 1994).
17. [8] F. Baras and M. Malek-Mansour, J. Chem. Phys 96, 2000 (1992); M. Malek-Mansour and F. Baras, Physica A 188, 253 (1992); J. Gorecki, Phys. Rev. Lett., submitted (1994); J. Gorecki, K. Kitahara, K. Yoshikawa and I. Hanazaki, Physica A, submitted (1994); J. Gorecki, in Far-From-Equilibrium Dynamics of Chemical Systems, Proceedings of the Third International Symposium, Borki, Poland, eds. J. Gorecki, A.S. Cukrowski, A.L. Kawczynski and B. Nowakowski (World Scientific, Singapore, 1994).
18. [8] F. Baras and M. Malek-Mansour, J. Chem. Phys 96, 2000 (1992); M. Malek-Mansour and F. Baras, Physica A 188, 253 (1992); J. Gorecki, Phys. Rev. Lett., submitted (1994); J. Gorecki, K. Kitahara, K. Yoshikawa and I. Hanazaki, Physica A, submitted (1994); J. Gorecki, in Far-From-Equilibrium Dynamics of Chemical Systems, Proceedings of the Third International Symposium, Borki, Poland, eds. J. Gorecki, A.S. Cukrowski, A.L. Kawczynski and B. Nowakowski (World Scientific, Singapore, 1994).
19. [9] See, for example, R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56 , 2553 (1986); B. Chopard and M. Droz, J. Phys. A 21, 205 (1988); R. Kopelman, Science 241, 1620 (1988); A.F. Voter, Phys. Rev. B 34, 6819 (1986).
20. [9] See, for example, R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56 , 2553 (1986); B. Chopard and M. Droz, J. Phys. A 21, 205 (1988); R. Kopelman, Science 241, 1620 (1988); A.F. Voter, Phys. Rev. B 34, 6819 (1986).
21. [9] See, for example, R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56 , 2553 (1986); B. Chopard and M. Droz, J. Phys. A 21, 205 (1988); R. Kopelman, Science 241, 1620 (1988); A.F. Voter, Phys. Rev. B 34, 6819 (1986).
22. [9] See, for example, R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56 , 2553 (1986); B. Chopard and M. Droz, J. Phys. A 21, 205 (1988); R. Kopelman, Science 241, 1620 (1988); A.F. Voter, Phys. Rev. B 34, 6819 (1986).
23. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
24. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
25. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
26. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
27. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
28. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
29. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
30. [10] See for instance : D. Ben-Avraham, J. Stat. Phys. 48, 315 (1987); D. Ben-Avraham, and C. Doering, Phys. Rev. A 37 5007 (1988); D. Ben-Avraham, and S. Redner, Phys. Rev. A. 34, 501 (1986); K. Kang, and S. Redner, Phys. Rev. Lett. 52, 955 (1984); K. Kang, and S. Redner, Phys. Rev. A. 32, 435 (1985); M.A. Burschka, C.A. Doering, and D. Ben-Avraham, Phys. Rev. Lett. 63, 700 (1989); K. Lindenberg, B. West, and R. Kopelman, Phys. Rev. Lett. 60, 1777 (1988); B.J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).
31. [11] G. Nicolis and I. Prigogine, Self Organization in Nonlinear Systems (Wiley, NY, 1977).
32. [12] N.G. van Kampen, Stochastic Processes in Chemistry and Physics (North-Holland, New York, 1981).
33. [13] C.W. Gardiner, Handbook of Stochastic Processes, (Springer-Verlag, New York, 1985).
34. [14] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987).
35. [15] K. Kawasaki, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S. Green (Academic Press, New York, 1972) Vol. 2, p.443.
36. [16] D. Dab, A. Lawniczak, J.-P. Boon and R. Kapral, Phys. Rev. Lett. 64, 2462 (1990); A. Lawniczak, D. Dab, R. Kapral and J.P. Boon, Physica D 47, 132 (1991).
37. [16] D. Dab, A. Lawniczak, J.-P. Boon and R. Kapral, Phys. Rev. Lett. 64, 2462 (1990); A. Lawniczak, D. Dab, R. Kapral and J.P. Boon, Physica D 47, 132 (1991).
38. [17] F. Schlögl, Z. Phys. 253, 147 (1972).
39. [18] R. Kapral, A. Lawniczak and P. Masiar, Phys. Rev. Lett. 66, 2539 (1991); J. Chem. Phys. 96, 2762 (1992).
40. [18] R. Kapral, A. Lawniczak and P. Masiar, Phys. Rev. Lett. 66, 2539 (1991); J. Chem. Phys. 96, 2762 (1992).
41. [19] D. Dab, J.P. Boon and Y.-X. Li, Phys. Rev. Lett. 66, 2535 (1991).
42. [20] G. Zanetti, Phys. Rev. A 40, 1539 (1989).
43. [21] D. Bernardin, J. Stat. Phys. 68, 457 (1992).
44. [22] D. Dab, Automates de gaz sur réseaux: une approche microcscopique des systèmes réactifs, Thèse de Doctorat, Université Libre de Bruxelles ( 1992).
45. [23] W. Feller, An introduction to Probability Therory and its Applications, Vols I and II (Wiley, NY, 1968 and 1971).
46. [24] B. Chopard, M. Droz and L. Frachebourg, Int. J. Phys. C (to appear).
47. [25] I. Prigogine, and E. Xhrouct, Physica (Utrecht) 15, 913 (1949); I. Prigogine, and M. Mahieu, Physica (Utrecht) 16, 51 (1950);
48. [25] I. Prigogine, and E. Xhrouct, Physica (Utrecht) 15, 913 (1949); I. Prigogine, and M. Mahieu, Physica (Utrecht) 16, 51 (1950);
49. See also J. Ross and P. Mazur, J. Chem. Phys. 35, 19 (1961);
50. F. Baras, and M. Malek Mansour, Phys. Rev. Lett. 63, 2429 (1989).
51. [26] R. Kapral, J. Chem. Phys. 56, 1842 (1972).
52. [27] R. Kapral, Adv. Chem. Phys. 48, 71 (1981).
53. [28] D. Toussaint, and F. Wilczek, J. Chem. Phys. 78, 2642 (1983).
54. [29] B.M. Boghosian and W. Taylor, Phys. Rev. E 52, 510 ( 1995).
55. [30] B.M. Boghosian and W. Taylor, J. Stat. Phys. 81, 295 (1995).
56. [31] M. Hooghe, Travail de fin d'études, Faculté des Sciences Appliquées, Université Libre de Bruxelles (1992).
57. [32] D. Gruner, R. Kapral and A. Lawniczak, J. Chem. Phys. 99, 3938 (1993).
58. [33] J.D. Gunton, M. San Miguel and P.S. Sahni, in Phase Transitions and Critical Phenomena, eds. C. Domb and J.L. Lebowitz (Academic Press, NY, 1983), Vol. 8; J.D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States (Springer-Verlag, NY, 1983).
59. [33] J.D. Gunton, M. San Miguel and P.S. Sahni, in Phase Transitions and Critical Phenomena, eds. C. Domb and J.L. Lebowitz (Academic Press, NY, 1983), Vol. 8; J.D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States (Springer-Verlag, NY, 1983).
60. [34] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
61. [35] R.H. Harding and J. Ross, J. Chem. Phys. 92, 1936 (1990); R.H. Harding, P. Paoli and J. Ross, ibid. 1947 (1990).
62. [35] R.H. Harding and J. Ross, J. Chem. Phys. 92, 1936 (1990); R.H. Harding, P. Paoli and J. Ross, ibid. 1947 (1990).
63. [36] G.-L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); R. Kapral and G.-L. Oppo, Physica D 23, 455 (1986).
64. [36] G.-L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); R. Kapral and G.-L. Oppo, Physica D 23, 455 (1986).
65. [37] K. Diemer, A. Lawniczak and R. Kapral, in Pattern Formation and Lattice Gas Automata, edited by A. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
66. [38] W.Y. Tam, W. Horsthemke, Z. Noszticzius and H.L. Swinney, J. Chem. Phys. 88, 3395 (1988).
67. [39] J.R. Weimar, D. Dab, J.P. Boon and S. Succi, Europhys. Lett. 20, 627 (1992).
68. [40] P. Grosfils, J.P. Boon and P. Lallemand, Phys. Rev. Lett. 68, 1077 (1992).
69. [41] J.R. Weimar and J.P. Boon, unpublished (1994).
70. [42] A.T. Winfree, When Time Breaks Down: The Three Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton University Press, Princeton, 1987).
71. [43] G. Ertl, Adv. Catal. 40, 231 (1990).
72. [44] B.P. Belousov, in Sbornik Referatov po Radiatsionni Meditsine (Medgiz, Moscow, 1958); A.M. Zhabotinsky, Biofizika 9, 306 (1964).
73. [44] B.P. Belousov, in Sbornik Referatov po Radiatsionni Meditsine (Medgiz, Moscow, 1958); A.M. Zhabotinsky, Biofizika 9, 306 (1964).
74. [45] Oscillations and Traveling Waves in Chemical Systems, R.J. Field and M. Burger, eds.,(Wiley, NY, 1986).
75. [46] Chemical Waves and Patterns, R. Kapral and K. Showalter, eds. (Kluwer, Dordrecht, 1995).
76. [47] E.E. Selkov, Eur. J. Biochem. 4, 79 (1968).
77. [48] P.M. Richter, P. Rehmus and J. Ross, Prog. Theor. Phys. 66, 385 (1981).
78. [49] J.M. Greenberg and S.P. Hastings, SIAM J. Appl. Math. 34, 515 (1978).
79. [50] R. Kapral, J. Math. Chem. 6, 113 (1991).
80. [51] M. Markus and B. Hess, Nature 347, 56 (1990);
81. H.E. Scheppers and M. Marcus, Physica A 188, 337 (1992).
82. [52] M. Gerhardt, H. Schuster and J.J. Tyson, Science 247, 1563 (1990); Physica D 46, 392 (1990).
83. [52] M. Gerhardt, H. Schuster and J.J. Tyson, Science 247, 1563 (1990); Physica D 46, 392 (1990).
84. [53] J.D. Murray, Mathematical Biology (Springer, NY, 1989).
85. [54] A.T. Winfree, SIAM Review 32, 1 (1990); A.T. Winfree, Chaos 1, 303 (1991); A.S. Mikhailov and VS. Zykov, Physica D 52, 379 (1991); V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester U.P., Manchester, 1988); J.J. Tyson and J.P. Keener, Physica D 32, 327 (1988); J.P. Keener and J.J. Tyson, SIAM Review 34, 1 (1992).
86. [54] A.T. Winfree, SIAM Review 32, 1 (1990); A.T. Winfree, Chaos 1, 303 (1991); A.S. Mikhailov and VS. Zykov, Physica D 52, 379 (1991); V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester U.P., Manchester, 1988); J.J. Tyson and J.P. Keener, Physica D 32, 327 (1988); J.P. Keener and J.J. Tyson, SIAM Review 34, 1 (1992).
87. [54] A.T. Winfree, SIAM Review 32, 1 (1990); A.T. Winfree, Chaos 1, 303 (1991); A.S. Mikhailov and VS. Zykov, Physica D 52, 379 (1991); V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester U.P., Manchester, 1988); J.J. Tyson and J.P. Keener, Physica D 32, 327 (1988); J.P. Keener and J.J. Tyson, SIAM Review 34, 1 (1992).
88. [54] A.T. Winfree, SIAM Review 32, 1 (1990); A.T. Winfree, Chaos 1, 303 (1991); A.S. Mikhailov and VS. Zykov, Physica D 52, 379 (1991); V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester U.P., Manchester, 1988); J.J. Tyson and J.P. Keener, Physica D 32, 327 (1988); J.P. Keener and J.J. Tyson, SIAM Review 34, 1 (1992).
89. [54] A.T. Winfree, SIAM Review 32, 1 (1990); A.T. Winfree, Chaos 1, 303 (1991); A.S. Mikhailov and VS. Zykov, Physica D 52, 379 (1991); V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester U.P., Manchester, 1988); J.J. Tyson and J.P. Keener, Physica D 32, 327 (1988); J.P. Keener and J.J. Tyson, SIAM Review 34, 1 (1992).
90. [54] A.T. Winfree, SIAM Review 32, 1 (1990); A.T. Winfree, Chaos 1, 303 (1991); A.S. Mikhailov and VS. Zykov, Physica D 52, 379 (1991); V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester U.P., Manchester, 1988); J.J. Tyson and J.P. Keener, Physica D 32, 327 (1988); J.P. Keener and J.J. Tyson, SIAM Review 34, 1 (1992).
91. [55] A. Turing, Phil. Trans. Roy. Soc. (London) B 237, 37 (1952).
92. [56] V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Phys. Rev. Lett. 64, 2953 (1990); P. De Kepper, V. Castes, E. Dulos and J. Boissonade, Physica D 49, 161 (1991).
93. [56] V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Phys. Rev. Lett. 64, 2953 (1990); P. De Kepper, V. Castes, E. Dulos and J. Boissonade, Physica D 49, 161 (1991).
94. [57] Q. Ouyang, H.L. Swinney, Nature 352, 610 (1991); Q. Ouyang and H. Swinney, Chaos 1, 411 (1991).
95. [57] Q. Ouyang, H.L. Swinney, Nature 352, 610 (1991); Q. Ouyang and H. Swinney, Chaos 1, 411 (1991).
96. [58] J.J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel and P. Borckmans, Phys. Rev. Lett. 71, 1272 (1993).
97. [59] R.D. Vigil, Q. Ouyang and H.L Swinney, Physica A 188, 17 (1992).
98. [60] I. Lengyel and I. Epstein, Science 251, 650 (1991); I. Lengyel and I. Epstein, Proc. Natl. Acad. Sci. US, 89, 3977 (1992).
99. [60] I. Lengyel and I. Epstein, Science 251, 650 (1991); I. Lengyel and I. Epstein, Proc. Natl. Acad. Sci. US, 89, 3977 (1992).
100. [61] A. Hunding and P.G. Sorensen, J. Math. Biol. 26, 27 (1988).
101. [62] B. Hasslacher, R. Kapral and A. Lawniczak, Chaos 3, 7 (1993).
102. [63] K. Maginu, Math. Biosci. 27, 17 (1975); J. Diff. Eqs., 31, 130 (1978).
103. [63] K. Maginu, Math. Biosci. 27, 17 (1975); J. Diff. Eqs., 31, 130 (1978).
104. [64] Y.-X. Li, Phys. Lett. 147 A, 204 (1990)
105. [65] S.K. Scott, Chemical Chaos, (Oxford University Press, Oxford, 1991); P. Bergé, Y. Pomeau and C. Vidai, Order Within Chaos (Wiley, New York, 1986).
106. [65] S.K. Scott, Chemical Chaos, (Oxford University Press, Oxford, 1991); P. Bergé, Y. Pomeau and C. Vidai, Order Within Chaos (Wiley, New York, 1986).
107. [66] X.-G. Wu and R. Kapral, Phys. Rev. Lett. 70, 1940 (1993); R. Kapral and X.-G. Wu, Annals N.Y. Acad. Sci. 706, 186 (1993); X.-G. Wu and R. Kapral, J. Chem. Phys. 100, 5936 (1994).
108. [66] X.-G. Wu and R. Kapral, Phys. Rev. Lett. 70, 1940 (1993); R. Kapral and X.-G. Wu, Annals N.Y. Acad. Sci. 706, 186 (1993); X.-G. Wu and R. Kapral, J. Chem. Phys. 100, 5936 (1994).
109. [66] X.-G. Wu and R. Kapral, Phys. Rev. Lett. 70, 1940 (1993); R. Kapral and X.-G. Wu, Annals N.Y. Acad. Sci. 706, 186 (1993); X.-G. Wu and R. Kapral, J. Chem. Phys. 100, 5936 (1994).
110. [67] K.-D. Willamowski and O.E. Rössler, Z. Naturforsch. 35a, 317 (1980).
111. [68] B.D. Aguda and B.L. Clarke, J. Chem. Phys. 89, 7428 (1988).
112. [69] See, for example, J.P. Crutchfield, J.D. Farmer and B.A. Huberman, Phys. Repts. 92, 45 (1982).
113. [70] T. Kapitaniak, Chaos in Systems with Noise (World Scientific, Singapore, 1988).
114. [71] R.F. Fox and J.E. Keizer, Phys. Rev. Lett. 64, 249 (1990); Phys. Rev. A 43, 1709 (1991); R.F. Fox, Phys. Rev. A 41, 2969 (1990);
115. [71] R.F. Fox and J.E. Keizer, Phys. Rev. Lett. 64, 249 (1990); Phys. Rev. A 43, 1709 (1991); R.F. Fox, Phys. Rev. A 41, 2969 (1990);
116. [71] R.F. Fox and J.E. Keizer, Phys. Rev. Lett. 64, 249 (1990); Phys. Rev. A 43, 1709 (1991); R.F. Fox, Phys. Rev. A 41, 2969 (1990);
117. J.E. Keizer and R.F. Fox, Phys. Rev. A 46, 3572 (1992).
118. [72] G. Nicolis and V. Balakrishnan, Phys. Rev. A 46, 3569 (1992); P. Peters and G. Nicolis, Physica A 188, 426 (1992); P. Geysermans and G. Nicolis, J. Chem. Phys. 99, 8964 ( 1993).
119. [72] G. Nicolis and V. Balakrishnan, Phys. Rev. A 46, 3569 (1992); P. Peters and G. Nicolis, Physica A 188, 426 (1992); P. Geysermans and G. Nicolis, J. Chem. Phys. 99, 8964 ( 1993).
120. [72] G. Nicolis and V. Balakrishnan, Phys. Rev. A 46, 3569 (1992); P. Peters and G. Nicolis, Physica A 188, 426 (1992); P. Geysermans and G. Nicolis, J. Chem. Phys. 99, 8964 ( 1993).
121. [73] J. Güémez and M.A. Matías, Phys. Rev. E 48, R2351 (1993);
122. M.A. Matías and J. Güémez, ibid, to be published.
123. [74] X.-G. Wu and R. Kapral, Phys. Rev. E 50, 3560 (1994).
124. [75] B. Shraiman, C.E. Wayne and P.C. Martin, Phys. Rev. Lett. 46, 935 (1981).
125. [76] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1980).
126. [77] B. Chopard, M. Droz, S. Cornell and L. Frachebourg, in Cellular Automata for Astrophysics, J. Perdang and A. Lejeune, eds. (World Scientific, Singapore, 1994) .
127. [78] T. Karapiperis and B. Blankleider, Physica D 78, 30 (1994).
128. [79] X.-G. Wu and R. Kapral, unpublished.
129. [80] D.H. Zanette, Phys. Rev. A 46, 7573 (1992).
130. [81] R. Kapral, in Theory and Applications of Coupled Map Lattices, K. Kaneko, ed. (Wiley, New York, 1993), p. 135.
131. [82] G. McNamara and G. Zanetti, Phys. Rev. Lett. 61 2332 (1988); F. Higuera and J. Jimenez, Europhys. Lett. 9 663 (1989); S. Chen, H. Chen, D. Martinez and W.H. Matthaeus Phys. Rev. Lett. 67 3776 (1991); S. Chen, G.D. Doolen, X. Shan and J. Wang, J. Stat. Phys. 68 379 (1992).
132. [82] G. McNamara and G. Zanetti, Phys. Rev. Lett. 61 2332 (1988); F. Higuera and J. Jimenez, Europhys. Lett. 9 663 (1989); S. Chen, H. Chen, D. Martinez and W.H. Matthaeus Phys. Rev. Lett. 67 3776 (1991); S. Chen, G.D. Doolen, X. Shan and J. Wang, J. Stat. Phys. 68 379 (1992).
133. [82] G. McNamara and G. Zanetti, Phys. Rev. Lett. 61 2332 (1988); F. Higuera and J. Jimenez, Europhys. Lett. 9 663 (1989); S. Chen, H. Chen, D. Martinez and W.H. Matthaeus Phys. Rev. Lett. 67 3776 (1991); S. Chen, G.D. Doolen, X. Shan and J. Wang, J. Stat. Phys. 68 379 (1992).
134. [82] G. McNamara and G. Zanetti, Phys. Rev. Lett. 61 2332 (1988); F. Higuera and J. Jimenez, Europhys. Lett. 9 663 (1989); S. Chen, H. Chen, D. Martinez and W.H. Matthaeus Phys. Rev. Lett. 67 3776 (1991); S. Chen, G.D. Doolen, X. Shan and J. Wang, J. Stat. Phys. 68 379 (1992).
135. [83] S. Ponce Dawson, S. Chen, and G. Doolen, J. Chem Phys. 98 1514 (1993); S. Ponce Dawson, A. Lawniczak and R. Kapral, J. Chem. Phys. 100, 5211 (1994).
136. [83] S. Ponce Dawson, S. Chen, and G. Doolen, J. Chem Phys. 98 1514 (1993); S. Ponce Dawson, A. Lawniczak and R. Kapral, J. Chem. Phys. 100, 5211 (1994).
137. [84] J. Von Neumann, Theory of Self-Reproducing Automata, A.W. Birks ed. (University of Illinois, Urbana, 1966).
138. [85] M Hénon, in Discrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrodynamics, R. Monaco, ed. (World Scientific, Singapore, 1989), p. 160.
139. [86] Theory and Applications of Cellular Automata, S. Wolfram, ed. (World Scientific, Singapore, 1986).
140. [87] Cellular Automata. Theory and Experiment, H. Gutowitz ed., Physica D 45 (1990).
141. [88] Wiener N. and Rosenblueth A. (1946), Arch. Inst. Card. Mex. 16, 205.
142. [89] Farley, B.G. (1964), in Proceedings Ojai Symposium on Neural Theory and Modeling, Reiss, R.F., ed. (Stanford University Press, Palo Alto, CA, 1964), p. 43.
143. [90] R. Fisch, J. Gravner and D. Griffeath, Statistics and Computing 1, 23 (1991); Ann. Applied Probability, 3, 935 (1993); J. Gravner, and D. Griffeath, Trans. Amer. Math. Soc., 340, 837 (1993).
144. [90] R. Fisch, J. Gravner and D. Griffeath, Statistics and Computing 1, 23 (1991); Ann. Applied Probability, 3, 935 (1993); J. Gravner, and D. Griffeath, Trans. Amer. Math. Soc., 340, 837 (1993).
145. [90] R. Fisch, J. Gravner and D. Griffeath, Statistics and Computing 1, 23 (1991); Ann. Applied Probability, 3, 935 (1993); J. Gravner, and D. Griffeath, Trans. Amer. Math. Soc., 340, 837 (1993).
146. [91] R. Durrett and D. Griffeath, Asymptotic Behavior of Excitable Cellular Automata, J. Experimental Math., 2, 183 (1993); R. Durrett and J. Steif, J. Theor. Prob. (1991).
147. [91] R. Durrett and D. Griffeath, Asymptotic Behavior of Excitable Cellular Automata, J. Experimental Math., 2, 183 (1993); R. Durrett and J. Steif, J. Theor. Prob. (1991).
148. [92] R. Durrett, Nonlinear Science Today 1, 1 (1991).
149. [93] J.R. Weimar and J.P. Boon, Phys. Rev. E 49, 1749 (1994).
150. [94] J.R. Weimar, J.J. Tyson, and L.T. Watson, Physica D55, 309 (1992).
151. [95] J.R. Weimar, Cellular automaton methods for reactive systems, Ph. D. thesis (Université Libre de Bruxelles, 1995).
152. [96] P.S. Hagan, SIAM J. Appl. Math. 42, 762 (1982).
153. [97] X.-G. Wu, M.N. Chee, and R. Kapral, Chaos 1, 421 (1991).
154. [98] C. Brown and L. Van Pelt, Phys. Rev. Postcard 1, 1 (1993).
155. [99] J.R. Weimar and J.P. Boon, in Pattern Formation and Lattice Cas Automata, edited by A. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
156. [100] J.R. Weimar and J.P. Boon, Physica A 224, 207 (1995).
157. [101] D. Dab, O. Tribel, G. Nicolis and M. Promel, Phys. Rev. Lett. 74, 824-827 (1995).
158. [102] N. Margolus, in Pattern Formation and Lattice Gas Automata, edited by A. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
159. [103] X.-G. Wu and R. Kapral, in Pattern Formation and Lattice Gas Automata, edited by A. Lawniczak and R. Kapral, Fields Institute Communications, vol. 6 (Amer. Math. Soc., 1995).
160. [104] F. Siegert and C.J. Weijer, Physica D 49, 224 (1991).
161. [105] A. Deutsch, Proceedings of the 2nd ECMBM, Lyon, Dec., 1993.