Abstract
We present a fully nonlinear stochastic theory of chemical reactions closely below, at and above instability points. As explicit example we treat the Prigogine-Lefever-Nicolis model of two interacting kinds of molecules, generalizing it to two and three dimensions, to a mode continuum, and taking into account fluctuations. Adopting a description by means of birth and death processes we establish the master-equation, and proceed to the Fokker-Planck equation. This is transformed to new coordinates connected with unstable and stable modes. After adiabatic elimination of the stable modes, we obtain a functional Fokker-Planck equation for a continuous set of “unstable” modes. This final equation can now be treated by standard methods. In one dimension our results reveal striking analogies to the Ginzburg-Landau theory of superconductivity, to the continuous mode laser and to small-band excitations at hydrodynamical instabilities, while in three dimensions, in thin layers a hexagonal structure similar to Bénard cells occurs.
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Turing, A.M.: Phil. Trans. R. Soc. B237, 37 (1952)
Belousov, B.: Sb. ref. radiats. med. za. Medzig, Moscow (1959)
Prigogine, I., Nicolis, G.: J. Chem. Phys.46, 3542 (1967)
Prigogine, I., Lefever, R.: J. Chem. Phys.48, 1695 (1968)
Prigogine, I., Lefever, R., Goldbeter, A., Hershkowitz-Kaufman, M.: Nature223, 913 (1969)
Zaikin, A.N., Zhabotinskij, A.M.: Nature225, 535 (1970)
Noyes, R.M., Field, R.J., Körös, E.: J. Am. Chem. Soc.94, 1394 (1972)
Noyes, R.M., Field, R.J., Körös, E.: J. Am. Chem. Soc.94, 8649 (1972)
Winfree, A.T.: Science175, 634 (1972)
Winfree, A.T.: A. Katchalsky Memorial Symposium Berkeley 1973
Field, R.J., Noyes, R.M.: Nature237, 390 (1972)
Busse, H.G.: J. Phys. Chem.73, 750 (1969)
Kopell, N., Howard, L.N.: Science180, 1171 (1973)
Ross, J., Ortoleva, P.: J. Chem. Phys.58 (1973)
Glansdorff, P., Prigogine, I.: Structure, Stability and Fluctuations, London: Wiley Interscience 1971, with many further references
Prigogine, I., Lefever, K.: In: Synergetics, ed. H. Haken. Stuttgart: Teubner 1973, where further references are given
Nicolis, G.: In: Cooperative Effects, Progress in Synergetics, ed. H. Haken. Amsterdam: North Holland, 1974, where further references are given
Schlögl, F.: Z. Physik253, 147 (1972);248, 446 (1971)
Czaijkowski, G.: Z. Physik,270, 25 (1974)
Janssen, H.K.: Z. Physik270, 67 (1974)
McNeil, K.J., Walls, D.F.: J. Stat. Phys.10, 439 (1974)
Faraday Symposium 9, Physical Chemistry of Oscillatory Phenomena, London, December 1974, with contributions of Nicolis, G., Prigogine, I., Field, R.J., Noyes, R.M., Körös, E., Burger, M., Friedrich, V., Ladányi, L., Nagy, Zs., Orbán, M., Winfree, A.T., DePoy, P.E., Mason, D.M., Matsuzaki, J., Nakajima, T., Liebhafsky, H.A., Franck, U.F., Balslev, I., Degn, H., Nitzan, A., Ortoleva, P., Ross, J., Texter, J. and others
Graham, R., Haken, H.: Z. Physik213, 420 (1968);237, 31 (1970)
Haken, H.: In: Festkörperprobleme, Vol. X, ed. O. Madelung. Marburg: Vieweg 1970
DeGiorgio, V., Scully, M.O.: Phys. Rev. A2, 1170 (1970)
Grossmann, S., Richter, P.H.: Z. Physik242, 458 (1971)
Dohm, V.: Solid State Comm.11, 1273 (1972)
Haken, H., Wöhrstein, H.G.: Optics Comm.9, 2, 123 (1973)
Graham, R.: Springer Tracts in Mod. Phys.66, 1 (1973)
Haken, H.: Rev. Mod. Phys. January 1975
Landauer, R.: IBM J. Res. and Developm.5, 3 (1961), for a review see R. Landauer in: Synergetics, ed. H. Haken. Stuttgart: Teubner 1973
Tomita, K., Ohto, T., Tomita, M.: Irreversible Circulation and Orbital Revolution: Hard Mode Instability in far from Equilibrium Situation, preprint 1974
Kopell, N., Howard, L.N.: Studies in Appl. Math. LII, No. 4, 291 (1973)
Hopf, E.: Ber. Math. Phys. Kl. Sächs. Akad. Wiss. Leipzig94, 1 (1942)
Haken, H.: Encyclopedia of Physics, XXV/2c. Berlin-Heidelberg-New York: Springer 1970, for a detailed list of references
Haken, H.: Phys. Lett. A46, 193 (1973)
Synergetics, ed. H. Haken. Stuttgart: B. G. Teubner 1973
Cooperative Effects: Progress in Synergetics, ed. H. Haken. Amsterdam: North Holland 1974
For a critical discussion of birth-death processes see Kuramoto, Y. Progr. Theoret. Physics49, 1782 (1973);52, 711 (1974) Nicolis, G., and Prigogine, I., Proc. Nat. Acad. Sc.68, 2102 (1971)
Van Kampen, N.G.: Canad. J. Physics39, 351 (1961)
See e.g. Haken, H.: Quantenfeldtheorie des Festkörpers. Stuttgart: B. G. Teubner 1973
Haken, H.: Z. Physik219, 246 (1969) Graham, R., Haken, H.: Z. Physik243, 289 (1971)
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Haken, H. Statistical physics of bifurcation, spatial structures, and fluctuations of chemical reactions. Z Physik B 20, 413–420 (1975). https://doi.org/10.1007/BF01313213
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DOI: https://doi.org/10.1007/BF01313213