Bifurcations, Fluctuations and Dissipative Structures

  • G. Nicolis
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 75)

Abstract

The purpose of the present lectures is to discuss the emergence of structures from dissipative processes in macroscopic systems. This class of phenomena, generally studied by means of nonequilibrium thermodynamics, is quite different from solitons and other structures appearing in nonlinear Hamiltonian systems, which were covered by many lecturers at this school.

Keywords

Entropy Convection Manifold Enthalpy Covariance 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

A. General References

  1. De Groot, S., and Mazur, P., 1962, “Nonequilibrium Thermodynamics,” North-Holland, Amsterdam.Google Scholar
  2. Glansdorff, P., and Prigogine, I., 1971, “Thermodynamic Theory of Structure, Stability and Fluctuations,” Wiley-Interscience, London.MATHGoogle Scholar
  3. Nicolis, G., and Prigogine, I., 1977, “Self-Organization in Non-equilibrium Systems,” Wiley-Interscience, New York.Google Scholar

For the mathematical analyses, we mainly refer to

  1. Feller, W., 1967 (first edition), 1971 (second edition), “An Introduction to Probability Theory and its Applications,” Wiley, New York.Google Scholar
  2. Minorski, N., 1962, “Nonlinear Oscillations,” Van Nostrand, Princeton.Google Scholar
  3. Sattinger, D., 1973, “Topics in Stability and Bifurcation Theory,” Lecture Notes in Mathematics 309, Springer-Verlag, Berlin.Google Scholar
  4. Sattinger, D., 1979, “Group Theoretic Methods in Bifurcation Theory,” Lecture Notes in Mathematics 762, Springer-Verlag, Berlin.Google Scholar

B. Specialized References

  1. Abramowitz, M., and Stegun, I., 1964, “Handbook of Mathematical Functions,” Dover Publications, New York.MATHGoogle Scholar
  2. Ahlers, G., and Waiden, R.W., 1980, Turbulence near the onset of convection, Phys. Rev. Lett., 44: 445.ADSCrossRefGoogle Scholar
  3. Andronov, A., Vitt, A., and Khaikin, O., 1966, “Theory of Oscillators,” Pergamon Press, Oxford.MATHGoogle Scholar
  4. Arnold, L., 1973, “Stochastic Differential Equations,” Wiley, New York.Google Scholar
  5. Arnold, L., 1979, in: “Dynamics of Synergetic Systems,” H. Haken, ed., Springer-Verlag, Berlin.Google Scholar
  6. Bedeaux, D., Mazur, P., and Pasmanter, R., 1977, The ballast resistor; an electrothermal instability in a conducting wire I; the nature of the stationary states, Physica, 86 A:355.Google Scholar
  7. Bell, G., and Primbley, P., 1978, “Theoretical Immunology,” Dekker, New York.Google Scholar
  8. Berge, P., Dubois, M., Manneville, P., and Pomeau, Y., 1980, Intermittency in Rayleigh-Benard convection, Jour, de Physique — Lettres, 41: 341.CrossRefGoogle Scholar
  9. Berggren, K., and Huberman, B., 1978, Peierls state far from equilibrium, Phys. Rev. B, 18: 3369.ADSCrossRefGoogle Scholar
  10. Boiteux, A., Goldbeter, A., and Hess, B., 1975, Control of oscilla-ting glycolysis of yeast by stochastic, periodic and steady source of substrate: A model and experimental study, Proc. of the Nat. Acad, of Sciences of the USA, 72: 3829.ADSCrossRefGoogle Scholar
  11. Borckmans, P., Dewel, G., and Walgraef, D., 1980, Concepts in the theory of nonequilibrium phase transitions, Jour, de Physique — Collogue C4, 41: 101.MathSciNetGoogle Scholar
  12. Briggs, T., and Rauscher, W., 1973, An oscillating iodine clock, Jour, of Chem. Educ., 50: 496.ADSCrossRefGoogle Scholar
  13. Caroli, B., Caroli, C., and Roulet, B., 1979, Diffusion in a bistabl potential: a systematic WKB treatment, Jour, of Stat. Phys., 21: 415.ADSCrossRefGoogle Scholar
  14. Chandrasekhar, S., 1961, “Hydrodynamic and Hydromagnetic Stability,” Oxford University Press.MATHGoogle Scholar
  15. Chorin, A., Marsden, J., and Smale, S., 1977, “Turbulence Seminar (Berkeley 1976/77),” Lecture Notes in Mathematics 615, Springer- Verlag, Berlin.Google Scholar
  16. Clavin, P., and Guyon, E., 1978, La flamme, La Recherche, 94: 954.Google Scholar
  17. Cole, J., 1968, “Perturbation Methods in Applied Mathematics,” Blaisdell Publishing Company, Waltham, Mass.Google Scholar
  18. Creel, C.L., and Ross, J., 1976, Multiple stationary states and hysteresis in a chemical reaction, Jour, of Chem. Phys., 65: 3779ADSCrossRefGoogle Scholar
  19. Decker, D., and Keller, H., 1980, Multiple limit point bifurcation, Jour, of Math. Anal, and its Appl., 75: 417.MathSciNetMATHCrossRefGoogle Scholar
  20. Degn, H., 1972, Oscillating chemical reactions in homogeneous phase, Jour, of Chem. Educ., 49: 302.ADSCrossRefGoogle Scholar
  21. de Pasquale, F., and Tombesi, P., 1979, The decay of an unstable equilibrium state near a “critical point”, Phys. Lett., 72 A: 7.Google Scholar
  22. Dewel, G., Walgraef, D., and Borckmans, P., 1977, Renormalization group approach to chemical instabilities, Zeits. fur Phys., B 28: 235.ADSCrossRefGoogle Scholar
  23. Dubois, M., and Berge, P., 1980, Experimental evidence for the oscillators in a convective biperiodic regime, Phys. Lett., 76 A: 53.Google Scholar
  24. Eigen, M., and Schuster, P., 1979, “The Hypercycle, a Principle of Natural Self-Organization,” Springer-Verlag, Berlin.Google Scholar
  25. Ermentrout, G., and Cowan, J., 1978, Large scale spatially organized activity in neuronal nets, SIAM Jour, on Appl. Math., 38: 1.MathSciNetCrossRefGoogle Scholar
  26. Erneux, T., and Herschkowitz-Kaufman, M., 1977, Rotating waves as asymptotic solutions of a model chemical reaction, Jour, of Chem. Phys. 66: 248.ADSCrossRefGoogle Scholar
  27. Erneux, T., and Hiernaux, J., 1980, Transition from polar to dupli-cate patterns, Jour, of Math. Biol., 9: 193.MathSciNetMATHCrossRefGoogle Scholar
  28. Field, R., 1975, Limit cycle oscillations in the reversible oregonator, Jour, of Chem. Phys., 63: 2289.ADSCrossRefGoogle Scholar
  29. Fife, P., 1978, Asymptotic states for equations of reacting and diffusion, Bull, of the Amer. Math. Soc., 84: 693.MathSciNetMATHCrossRefGoogle Scholar
  30. Fraedrich, K., 1978, Structural and stochastic analysis of a o-dimensional climate system, Quart. Jour, of the Royal Meteor. Soc., 104: 461.ADSCrossRefGoogle Scholar
  31. Gardiner, C., 1976, A comment on chemical Langevin equations, Jour. of Stat. Phys., 15: 451.MathSciNetADSCrossRefGoogle Scholar
  32. Gerisch, A., and Hess, B., 1974, Cyclic-AMP-controlled oscillations in suspended Dictyostelium cells: their relation to morphogenetic cell interactions, Proc. of thé Nat. Acad, of Sci. of the USA, 71: 2118.ADSCrossRefGoogle Scholar
  33. Gierer, A., and Meinhardt, H., 1972, A theory of biological pattern foundation, Kybernetik, 12: 30.CrossRefGoogle Scholar
  34. Goldbeter, A., 1980, Models for oscillations and excitability in biochemical systems, in: “Mathematical Models in Molecular and Cellular Biology,” L.A. Segel, ed., Cambridge Univ. Press.Google Scholar
  35. Golubitsky, M., and Schaeffer, D., 1979, A theory for imperfect bifurcation via singularity theory, Commun, on Pure and Appl. Math., 32: 21.MathSciNetMATHCrossRefGoogle Scholar
  36. Grossman, S., 1976, Langevin forces in chemically reacting multi- component fluids, Jour, of Chem. Phys., 65: 2007.Google Scholar
  37. Haberman, R., 1977, On the behavior of spatially dependent nonlinear wave envelopes, SIAM Jour, on Appl. Math., 32: 154.MathSciNetMATHCrossRefGoogle Scholar
  38. Haberman, R., 1979, Slowly varying jump and transition phenomena ssociated with algebraic bifurcation problems, SIAM Jour. on Appl. Math., 57: 69.MathSciNetCrossRefGoogle Scholar
  39. Haken, H., 1978, “Synergetics, an Introduction,” Springer-Verlag, Berlin.MATHGoogle Scholar
  40. Haken, H., 1980, editor, “Dynamics of Synergetic Systems,” Proceedings of the International Symposium of Synergetics (Bielefeld 1979 ), Springer-Verlag, Berlin.Google Scholar
  41. Hentschel, H., 1978, Application of the dynamic renormalization group to non-equilibrium instabilities: The k = o hard mode transition, Zeits. fur Phys., B 31: 401.Google Scholar
  42. Horsthemke, 1980, “Proceedings of the XVIIth Solvay Conference on Physics (Washington 1980),” G. Nicolis, G. Dewel and J.W. Turner, ed., Wiley, N.Y.Google Scholar
  43. Jacobs, J.A., 1975, “The Earth’s Core,” Academic Press, London.Google Scholar
  44. Joseph, D., 1973, Remarks about bifurcation and stability of quasi- periodic solutions which bifurcate from periodic solutions of the Navier-Stokes equation, in: “Nonlinear Problems in the Physical Sciences and Biology,” I. Stakgold, D. Joseph, and D. Sattinger, ed., Springer-Verlag, Berlin.Google Scholar
  45. Joseph, D., and Sattinger, D., 1972, Bifurcating time-periodic solutions and their stability, Archive for Rational Mechanics and Analysis, 45: 79.MathSciNetADSMATHCrossRefGoogle Scholar
  46. Karlin, S., and Mc Gregor, J., 1957, The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans, of the Amer. Math. Soc., 85: 489.MathSciNetMATHCrossRefGoogle Scholar
  47. Kauffman, S., and Wille, J., 1975, The mitotic oscillator in Physarum polycephalum, Jour, of Theor. Biol., 55: 47.CrossRefGoogle Scholar
  48. Kuramoto, Y., and Tsuzuki, T., 1975, On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach, Prog, of Theor. Phys., 54: 687.ADSCrossRefGoogle Scholar
  49. Kurtz, T.G., 1976, Limit theorems and diffusion approximations for density dependent Markov chains, Math. Prog. Study, 5: 67.MathSciNetCrossRefGoogle Scholar
  50. Kurtz, T.G., 1978, Strong approximation theorems for density dependent Markov chains, Stoch. Proc. and Their Appl., 6: 223.MathSciNetMATHCrossRefGoogle Scholar
  51. Landau, L., and Lifschitz, E., 1959, “Fluid Mechanics,” Addison- Wesley, Reading, Mass.Google Scholar
  52. Landauer, R., 1962, Fluctuations in bistable tunnel diode circuits, Jour, of Appl. Phys., 33: 2209.ADSCrossRefGoogle Scholar
  53. Lanford III, 0., 1973, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, in: “Nonlinear Problems in the Physical Sciences and Biology,” I. Stakgold, D. Joseph and D. Sattinger, ed., Springer-Verlag, Berlin.Google Scholar
  54. Langer, J.S., 1980, Eutectic solidification and marginal stability, Phys. Rev. Lett., 44: 1023.Google Scholar
  55. Langer, J.S., 1980, Instabilities and pattern formation in crystal growth, Rev, of Mod. Phys., 52: 1.ADSCrossRefGoogle Scholar
  56. Lefever, R., Herschkowitz-Kaufman, M. and Turner, J.W., 1977, Dissipative structures in a soluble non-linear reaction-diffusion system, Phys. Lett., 60 A: 389.Google Scholar
  57. Lefever, R., and Garay, R., 1978, A mathematical model of the immune surveillance against cancer, in: “Theoretical Immunology,” G. Bell and P. Primblely, ed., Dekker, N.Y.Google Scholar
  58. Lorenz, E., 1963, Deterministic nonperiodic flow, Jour, of Atmos. Sci., 20: 130.Google Scholar
  59. Lovett, R., Ortoleva, P., and Ross, J., 1978, Kinetic instabilities in first order phase transitions, Jour, of Chem. Phys., 69: 947.ADSCrossRefGoogle Scholar
  60. Ma, S.K., 1976, “Modern Theory of Critical Phenomena,” Benjamin, Reading, Mass.Google Scholar
  61. Malek Mansour, M., and Nicolis, G., 1975, A master equation description of local fluctuations, Jour, of Stat. Phys., 13: 197.MathSciNetADSCrossRefGoogle Scholar
  62. Malek Mansour, M., 1978, Sur la description stochastique des systemes de non-equilibre, Jour, de Phys. — Collogue C5, 39: 79.Google Scholar
  63. Malek Mansour, M. and Van Den Broeck, C., 1980a, Inhomogeneous fluctuations in reaction-diffusion systems, Preprint.Google Scholar
  64. Malek Mansour, M., Van Den Broeck, C., Nicolis, G., and Turner, J.W., 1980, Asymptotic properties of markovian master equations, Ann, of Phys., in press.Google Scholar
  65. Mandel, P., 1979, Lasers with saturable absorbers driven by a coherent field, Zeits. für Phys., B 33: 205.Google Scholar
  66. Manneville, P., and Pomeau, Y., 1980, Different ways to turbulence in dissipative dynamical systems, Physica, 1 D:219.Google Scholar
  67. Matijevic, E., 1978, “Surface and Colloid Science,” Vol. 10, Plenum Publishing Corporation.CrossRefGoogle Scholar
  68. Matsuo, K., 1977, Relaxation mode analysis of nonlinear birth and death processes, Jour, of Stat. Phys., 16: 169.MathSciNetADSCrossRefMATHGoogle Scholar
  69. May, R., 1976, Simple mathematical models with very complicated dynamics, Nature, 261: 459.ADSCrossRefGoogle Scholar
  70. Miller, C.A., 1978, Stability of interfaces, in: “Surface and Colloid Science,” Vol. 10, E. Matijevic, ed., Plenum, N.Y.Google Scholar
  71. Misra, B., Courbage, M., and Prigogine, I., 1979, From deterministic dynamics to probabilistic descriptions, Proc. of thè Nat. Acad. of Sci. (USA), 76: 3607.MathSciNetADSMATHCrossRefGoogle Scholar
  72. Moreau, M., 1980, Note on the reaction rate in non-ideal mixtures, Physica, 102 A:389.Google Scholar
  73. Mori, H., 1965, Transport, collective motion and brownian motion, Prog, of Theor. Phys., 33: 423.ADSMATHCrossRefGoogle Scholar
  74. Mori, H., 1975, Stochastic processes of macroscopic variables, Prog, of Theor. Phys., 53: 1617.Google Scholar
  75. Nakamura, K., 1977, Nonlinear fluctuations associated with instabilities in dissipative systems, Prog, of Theor. Phys., 57: 1874.ADSCrossRefGoogle Scholar
  76. Nakaya, J., 1954, “Snow Crystals,” Harvard University Press, Cambridge.Google Scholar
  77. Nicolis, G., and Turner, J.W., 1977, Stochastic analysis of a non-equilibrium phase transition: some exact results, Physica, 89 A:326.Google Scholar
  78. Nicolis, G., and Turner, J.W., 1979, Effects of fluctuations on bifurcation phenomena, Ann, of the N.Y. Acad, of Sci., 316: 251.MathSciNetADSCrossRefGoogle Scholar
  79. Nicolis, G., and Malek Mansour, M., 1980, Systematic analysis of the multivariate master equation for a reaction-diffusion system, Jour, of Stat. Phys., 22: 495.MathSciNetADSMATHCrossRefGoogle Scholar
  80. Nicolis, G., Dewel, G., and Turner, J.W., 1980, ed., “Proceedings of the XVIIth Solvay Conference on Physics (Washington 1980),” Wiley, New York.Google Scholar
  81. Normand, C., Pomeau, Y., and Velarde, M., 1977, Convective instabilities: a physicist’s approach, Rev, of Mod. Phys., 49: 581.MathSciNetADSCrossRefGoogle Scholar
  82. Prigogine, I., and Lefever, R., 1968, Symmetry breaking instabilities in dissipative systems. II, Jour, of Chem. Phys., 48: 1695.ADSCrossRefGoogle Scholar
  83. Resibois, P., and de Leener, M., 1977, “Classical Kinetic Theory of Fluids,” Wiley-Interscience, New York.Google Scholar
  84. Rice, S., Freed, K., and Light, J., 1972, ed., “Statistical Mechanics: New Concepts, New Problems, New Applications,” Proceedings of the Sixth IUPAP Conference on Statistical Mechanics ( Chicago 1971 ), University of Chicago Press.Google Scholar
  85. Richter, F., 1973, Sea floor spreading, Rev. of Geophys. and Space Phys., pp. 233-287.Google Scholar
  86. Risken, H., and Nummedal, K., 1968, Self-pulsing in lasers, Jour. of Appi. Phys., 39: 4662.ADSCrossRefGoogle Scholar
  87. Roux, J.C., Rossi, A., Bachelart, S., and Vidal, C., 1980, Representation of a strange attractor from an experimental study of chemical turbulence, Phys. Lett., 77 A: 391.Google Scholar
  88. Ruschin, S. and Bauer, S., 1979, Bistability, hysteresis and critical behavior of a CO2 laser, with SFg intracavity as a saturable absorber, Chem. Phys. Lett., 66: 100.ADSCrossRefGoogle Scholar
  89. Scalapino, D., and Huberman, B., 1977, Onset of an inhomogeneous state in a nonequilibrium superconducting film, Phys. Rev. Lett., 39: 1365.ADSCrossRefGoogle Scholar
  90. Schloegl, F., 1971, On stability of steady states, Zeits. fur Phys., 243: 303.ADSCrossRefGoogle Scholar
  91. Schloegl, F., 1972, Chemical reaction models for nonequilibrium phase transitions, Zeits. fur Phys., 253: 147.ADSCrossRefGoogle Scholar
  92. Schnakenberg, J., 1976, Network theory of microscopic and macroscopic behavior of master equation systems, Rev, of Mod. Phys., 48: 571.MathSciNetADSCrossRefGoogle Scholar
  93. Schuss, Z., and Matkowsky, B., 1979, The exit problem: A new approach to diffusion across potential barriers, SIAM Jour. on Appl. Math., 36: 604.MathSciNetMATHCrossRefGoogle Scholar
  94. Segel, LA., 1980, ed., “Mathematical Models in Molecular and Cellular Biology,” Cambridge University Press.Google Scholar
  95. Stakgold, I., Joseph, D., and Sattinger, D., 1973, ed., “Nonlinear Problems in the Physical Sciences and Biology,” Proceedings of a Battelle Summer Institute, Seattle, July 3-28, 1972, Lecture Notes in Mathematics 322, Springer-Verlag, Berlin.Google Scholar
  96. Suzuki, M., 1976, Scaling theory of nonequilibrium systems near the instability point. I. General aspects of transient phenomena, Prog, of Theor. Phys., 56: 77.ADSMATHCrossRefGoogle Scholar
  97. Suzuki, M., 1976, Scaling theory of nonequilibrium systems near the instability point. II. Anomalous fluctuation theorems in the extensive region, Prog, of Theor. Phys., 56: 477.ADSCrossRefGoogle Scholar
  98. Suzuki, M., 1980, in: “Proc. of the XVIIth Conf. on Phys.,” G. Nicolis, G. Dewel and J.W. Turner, ed., Wiley, New York.Google Scholar
  99. Thorn, R., 1972, “Stabilite Structurelle et Morphogenese,” Benjamin, New York.Google Scholar
  100. Turner, J.S., 1980, Private communication.Google Scholar
  101. Turner, J.W., 1979, Stationary and time-dependent solutions of master equations in several variables, jLn: “Dynamics of Synergetic Systems,” H. Haken, ed., Springer-Verlag, Berlin.Google Scholar
  102. Tyson, J., and Kauffman, S., 1975, Control of mitosis by a continuous biochemical oscillation: Synchronization; Spatially inhomo-geneous oscillations, Jour, of Math. Biol., 1: 289.MATHCrossRefGoogle Scholar
  103. Tyson, J., 1976, “The Belousov-Zhabotinski Reaction,” Lecture Notes in Biomathematics 10, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  104. Van Den Broeck, C., Horsthemke, W., and Malek Mansour, M., 1977, On the diffusion operator of the multivariate master equation, Physica, 89 A:339.Google Scholar
  105. Van Den Broeck, C., Houard, J., and Malek Mansour, M., 1980, Chapman-Enskog development of the multivariate master equation, Physica, 101 A:167.Google Scholar
  106. Van Der Pol, B., 1930, Oscillations sinusoidales et de relaxation., L’Onde Electrique, 9: 245, 293.Google Scholar
  107. Van Kampen, N.G., 1977, A soluble model for diffusion in a bistable potential., Jour, of Stat. Phys., 17: 71.ADSCrossRefGoogle Scholar
  108. Velarde, M., and Normand, C., 1980, Convection, Scientific American, July.Google Scholar
  109. Vidal, C., and Roux, J., 1980, La turbulence chimique existetelle?, La Recherche, 107: 66.Google Scholar
  110. Walgraef, D., Dewel, G., and Borckmans, P., 1980, Fluctuations near nonequilibrium phase transitions to nonuniform states, Phys. Rev., A 21: 397.ADSCrossRefGoogle Scholar
  111. Walgraef, D., Dewel, G., and Borckmans, P., 1981 (in press), Non- equilibrium phase transitions and chemical instabilities, Advanc. in Chem. Phys.Google Scholar
  112. Zhabotinski, A., 1974, “Self-Oscillating Concentrations,” Nauka, Moscow.Google Scholar
  113. Zwanzig, R., 1972, Collective modes in classical liquids, in: “Statistical Mechanics: New Concepts, New Problems, New Applications,” S. Rice, K. Freed and J. Light, ed., U. of Chicago Press.Google Scholar
  114. Zwanzig, R., 1972, Nonlinear dynamics of collective modes, in: “Statistical Mechanics: New Concepts, New Problems, New Applications,” S. Rice, K. Freed and J. Light, ed., U. of Chicago Press.Google Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • G. Nicolis
    • 1
  1. 1.Department of PhysicsFree University of BrusselsBrusselsBelgium

Personalised recommendations