Journal of Statistical Physics

, Volume 48, Issue 5–6, pp 1031–1044 | Cite as

The search for Turing structures

  • P. Borckmans
  • G. Dewel
  • D. Walgraef
  • Y. Katayama


Albeit their prediction by Turing, chemical periodic stationary patterns under far from equilibrium conditions, which are considered as prototypes of dissipative structures, have not yet been observed in a clear-cut way. We here assess the situation and discuss the perturbative influence of convection on their formation and selection mechanisms.

Key words

Chemical instabilities double-diffusion convection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Benard,Rev. Gen. Sci. Pure Appl. 11:1261 (1900).Google Scholar
  2. 2.
    G. I. Taylor,Phil. Trans. R. Soc. Lond. A 223:289 (1923).Google Scholar
  3. 3.
    T. Tanaka, S. Sun, Y. Hirokawa, S. Katayama, J. Kucera, Y. Hirose, and T. Amiya,Nature 325:796 (1987); J. S. Preston, H. M. van Driel, and J. E. Sipe,Phys. Rev. Lett. 58:289 (1987).Google Scholar
  4. 4.
    A. T. Winfree,J. Chem. Ed. 61:661 (1984).Google Scholar
  5. 5.
    P. Glansdorff and I. Prigogine,Thermodynamic Theory of Structures, Stability and Fluctuations (Wiley, New York, 1971).Google Scholar
  6. 6.
    R. J. Field and M. Burger (eds.),Oscillations and Traveling Waves in Chemical Systems (Wiley, New York, 1984).Google Scholar
  7. 7.
    R. Imbihl, M. P. Cox, G. Ertl, H. Müller, and W. Brenig,J. Chem. Phys. 83:1578 (1985); L. F. Razon and R. A. Schmitz,Catal. Rev. Sci. Eng. 28:89 (1986).Google Scholar
  8. 8.
    A. M. Turing,Phil. Trans. R. Soc. Lond. B 237:37 (1952).Google Scholar
  9. 9.
    G. Nicolis and I. Prigogine,Self Organization in Nonequilibrium Systems (Wiley, New York, 1977); D. Walgraef, G. Dewel, and P. Borckmans,Adv. Chem. Phys. 49:311 (1982).Google Scholar
  10. 10.
    P. K. Becker and R. Field,J. Phys. Chem. 89:118 (1985).Google Scholar
  11. 11.
    A. Boiteux and B. Hess,Ber. Bunsenges. Phys. Chem. 84:392 (1980).Google Scholar
  12. 12.
    A. M. Zhabothinsky and A. N. Zaikin,J. Theor. Biol. 40:45 (1973).Google Scholar
  13. 13.
    K. Showalter,J. Chem. Phys. 73:3735 (1980).Google Scholar
  14. 14.
    M. Orban,J. Am. Chem. Soc. 102:4311 (1980).Google Scholar
  15. 15.
    M. Marek, inModelling of Patterns in Space and Time, W. Jäger and J. D. Murray, eds. (Springer, Berlin, 1984), p. 214; K. Bar-Eli and W. Geiseler,J. Phys. Chem. 85:3461 (1981).Google Scholar
  16. 16.
    P. Möckel,Naturwissenschaften 64:224 (1977).Google Scholar
  17. 17.
    M. Kagan, A. Levi, and D. Avnir,Naturwissenschaften 69:548 (1982);70:144 (1983).Google Scholar
  18. 18.
    M. Gimenez and J.-C. Micheau,Naturwissenschaften 70:90 (1983).Google Scholar
  19. 19.
    D. Avnir and M. Kagan,Nature 307:717 (1984).Google Scholar
  20. 20.
    J.-C. Micheau, M. Gimenez, P. Borckmans, and G. Dewel,Nature 305:43 (1983).Google Scholar
  21. 21.
    S. Müller and Th. Plesser, inModelling of Patterns in Space and Time, W. Jäger and J. D. Murray, eds. (Springer, Berlin, 1984); S. Müller, Th. Plesser, and B. Hess, inTemporal Order, Synergetics No. 29, L. Rensing and N. Jaeger, eds. (Springer, Berlin, 1985).Google Scholar
  22. 22.
    I. R. Epstein, M. Morgan, C. Steel, and O. Valdes-Aguilar,J. Phys. Chem. 87:3955 (1983).Google Scholar
  23. 23.
    M. Gimenez, J.-C. Micheau, D. Lavabre, and J.-P. Laplante,J. Phys. Chem. 89:1 (1985).Google Scholar
  24. 24.
    V. Croquette, P. Le Gal, A. Pocheau, and R. Guglielmetti,Europhys. Lett. 1:393 (1986).Google Scholar
  25. 25.
    J. Bdzil and H. Frisch,Phys. Fluids 14:476, 1077 (1971).Google Scholar
  26. 26.
    J. K. Platten and J.-C. Legros,Convection in Liquids (Springer, Berlin, 1984), Part D.Google Scholar
  27. 27.
    G. Dewel, D. Walgraef, and P. Borckmans,Proc. Natl. Acad. Sci. USA 80:6429 (1983).Google Scholar
  28. 28.
    T. J. McDougall,J. Fluid Mech. 126:379 (1983).Google Scholar
  29. 29.
    M. Kagan and D. Avnir, inInterfacial Phenomena: Proceedings NATO ASI and EPS Summerschool and Conference on Physicochemical Hydrodynamics, Spain 1986, M. G. Velarde and B. Nichols, eds. (Plenum Press, New York, in press).Google Scholar
  30. 30.
    V. Steinberg and E. Moses, inPatterns, Defects and Microstructures in Nonequilibrium Systems: NATO ASI (Series E: Applied Sciences No. 121), D. Walgraef, ed. (Martinus Nijhoff, Dordrecht, 1987); C.M. Surko, P. Kolodner, A. Passner, and R. W. Waiden,Physica D 23:220 (1986).Google Scholar
  31. 31.
    E. Moses, J. Fineberg, and V. Steinberg,Phys. Rev. A 35:2757 (1987); R. Heinrichs, G. Ahlers, and D. S. Cannell,Phys. Rev. A 35:2761 (1987).Google Scholar
  32. 32.
    G. I. Taylor,Proc. R. Soc. Lond. A 219:186 (1953);225:473 (1954).Google Scholar
  33. 33.
    F. Sagues and W. Horsthemke,Phys. Rev. A 34:4136 (1986).Google Scholar
  34. 34.
    J. P. Gollub and T. H. Solomon, inChaos and Related Nonlinear Phenomena: Where Do We Go from Here?, I. Procaccia, ed. (Plenum Press, New York, in press).Google Scholar
  35. 35.
    G. Dewel, P. Borckmans, and D. Walgraef, inChemical Instabilities: NATO ASI (Series C: Mathematical and Physical Sciences No. 120), G. Nicolis and F. Baras, eds. (Reidel, Dordrecht, 1984).Google Scholar
  36. 36.
    M. Marek and E. Svobodova,Biophys. Chem. 3:263 (1975).Google Scholar
  37. 37.
    E. A. Spiegel and S. Zalesky,Phys. Lett. 106A:335 (1984).Google Scholar
  38. 38.
    S. Kai, S. Müller, and J. Ross,J. Chem. Phys. 76:1392 (1982).Google Scholar
  39. 39.
    G. Martin,Phys. Rev. B 21:2122 (1980).Google Scholar
  40. 40.
    K. Krishan,Nature 287:420 (1980).Google Scholar
  41. 41.
    D. J. Scalapino and B. A. Huberman,Phys. Rev. Lett. 39:1365 (1977).Google Scholar
  42. 42.
    K. F. Berggren and B. A. Huberman,Phys. Rev. B 18:3369 (1978).Google Scholar
  43. 43.
    D. Walgraef and E. C. Aifantis,J. Appl. Phys. 58:688 (1985).Google Scholar
  44. 44.
    L. A. Lugiato and R. Lefever,Phys. Rev. Lett. 58:2209 (1987).Google Scholar
  45. 45.
    L. W. Anacker and R. Kopelman,Phys. Rev. Lett. 58:289 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • P. Borckmans
    • 1
  • G. Dewel
    • 1
  • D. Walgraef
    • 1
  • Y. Katayama
    • 2
  1. 1.Service de Chimie-Physique IIUniversité Libre de BruxellesBruxellesBelgium
  2. 2.College of EngineeringNihon UniversityFukushimaJapan

Personalised recommendations