Advertisement

Synergetics pp 147-189 | Cite as

Chance and Necessity

Reality Needs Both
  • Hermann Haken
Part of the Springer Series in Synergetics book series (SSSYN, volume 1)

Abstract

Consider a football dribbled ahead over the grass by a football (soccer) player. Its velocity v changes due to two causes. The grass continuously slows the ball down by a friction force whereas the football player randomly increases the velocity of the ball by his kicks.

Keywords

Correlation Function Stationary Solution Friction Force Langevin Equation Planck Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

Langevin Equations: An Example

  1. For general approaches see R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon & Breach, New York-London 1963)Google Scholar
  2. For general approaches see M. Lax: Rev. Mod. Phys. 32, 25 (1960);ADSzbMATHCrossRefGoogle Scholar
  3. For general approaches see M. Lax: Rev. Mod. Phys. 38, 358, 541 (1966);MathSciNetADSGoogle Scholar
  4. For general approaches see M. Lax: Phys. Rev. 145, 110 (1966).ADSCrossRefGoogle Scholar
  5. H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar
  6. with further references P. Hänggi, H. Thomas: Phys. Rep. 88, 208 (1982)CrossRefGoogle Scholar

Reservoirs and Random Forces

  1. Here we present a simple example. For general approaches see R. Zwanzig: J. Stat. Phys. 9, 3, 215 (1973)ADSCrossRefGoogle Scholar
  2. Here we present a simple example. For general approaches see H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar

The Fokker-Planck Equation

  1. For general approaches see R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon & Breach, New York-London 1963)Google Scholar
  2. For general approaches see M. Lax: Rev. Mod. Phys. 32, 25 (1960);ADSzbMATHCrossRefGoogle Scholar
  3. For general approaches see M. Lax: Rev. Mod. Phys. 38, 358, 541 (1966);MathSciNetADSGoogle Scholar
  4. For general approaches see M. Lax: Phys. Rev. 145, 110 (1966).ADSCrossRefGoogle Scholar
  5. For general approaches see H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar
  6. with further references P. Hänggi, H. Thomas: Phys. Rep. 88, 208 (1982)CrossRefGoogle Scholar

Some Properties and Stationary Solution of the Fokker-Planck Equation

  1. The “potential case” is treated by R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon & Breach, New York-London 1963)Google Scholar
  2. The more general case for systems in detailed balance is treated by R. Graham, H. Haken: Z. Phys. 248, 289 (1971)MathSciNetADSGoogle Scholar
  3. The more general case for systems in detailed balance is treated by R. Graham: Z. Phys. B40, 149 (1981)ADSGoogle Scholar
  4. H. Risken: Z. Phys. 251, 231 (1972);ADSCrossRefGoogle Scholar
  5. see also H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar

Time-Dependent Solutions of the Fokker-Planck Equation

  1. The solution of the n-dimensional Fokker-Planck equation with linear drift and constant diffusion coefficients was given by M. C. Wang, G. E. Uhlenbeck: Rev. Mod. Phys. 17, 2 and 3 (1945)MathSciNetCrossRefGoogle Scholar
  2. For a short representation of the results see H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar

Solution of the Fokker-Planck Equation by Path Integrals

  1. L. Onsager, S. Machlup: Phys. Rev. 91, 1505, 1512 (1953)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. I. M. Gelfand, A. M. Yaglome: J. Math. Phys. I, 48 (1960)ADSCrossRefGoogle Scholar
  3. R. P. Feynman, A. R. Hibbs: Quantum Mechanics and Path Integrals (McGraw-Hill, New York 1965)zbMATHGoogle Scholar
  4. F. W. Wiegel: Path Integral Methods in Statistical Mechanics, Physics Reports 16C, No. 2 (North Holland, Amsterdam 1975)Google Scholar
  5. R. Graham: In Springer Tracts in Modern Physics, Vol. 66 (Springer, Berlin-Heidelberg-New York 1973) p. 1Google Scholar
  6. A critical discussion of that paper gives W. Horsthemke, A. Bach: Z. Phys. B22, 189 (1975)ADSGoogle Scholar
  7. We follow essentially H. Haken: Z. Phys. B24, 321 (1976) where also classes of solutions of Fokker-Planck equations are discussed.MathSciNetADSGoogle Scholar

Phase Transition Analogy

  1. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles L. D. Landau, I. M. Lifshitz: In Course of Theoretical Physics, Vol. 5: Statistical Physics (Pergamon Press, London-Paris 1959)Google Scholar
  2. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles R. Brout: Phase Transitions (Benjamin, New York 1965)Google Scholar
  3. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles L. P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palcanskas, M. Rayl, J. Swift, D. Aspnes, J. Kane: Rev. Mod. Phys. 39, 395 (1967)ADSCrossRefGoogle Scholar
  4. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles M. E. Fischer: Repts. Progr. Phys. 30, 731 (1967)CrossRefGoogle Scholar
  5. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles H. E. Stanley: Introduction to Phase Transitions and Critical Phenomena. Internat. Series of Monographs in Physics (Oxford University, New York 1971)Google Scholar
  6. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles A. Münster: Statistical Thermodynamics, Vol. 2 (Springer, Berlin-Heidelberg-New York and Academic Press, New York-London 1974)Google Scholar
  7. The theory of phase transitions of systems in thermal equlibrium is presented, for example, in the following books and articles C. Domb, M. S. Green, eds.: Phase Transitions and Critical Phenomena, Vols. 1–5 (Academic Press, London 1972–76)Google Scholar
  8. The modern and powerful renormalization group technique of Wilson is reviewed by K. G. Wilson, J. Kogut: Phys. Rep. 12C, 75 (1974)ADSCrossRefGoogle Scholar
  9. The modern and powerful renormalization group technique of Wilson is reviewed by S.-K. Ma: Modern Theory of Critical Phenomena (Benjamin, London 1976)Google Scholar
  10. The profound and detailed analogies between a second order phase transition of a system in thermal equilibrium (for instance a superconductor) and transitions of a non-equilibrium system were first derived in the laser-case in independent papers by R. Graham, H. Haken: Z. Phys. 213, 420 (1968) and in particular Z. Phys. 237, 31 (1970),ADSCrossRefGoogle Scholar
  11. who treated the continuum mode laser, and by V. DeGiorgio, M. O. Scully: Phys. Rev. A2, 1170 (1970),ADSGoogle Scholar

Phase Transition Analogy in Continuous Media: Space Dependent Order Parameter

  1. a) References to Systems in Thermal EquilibriumGoogle Scholar
  2. The Ginzburg-Landau theory is presented, for instance, by N. R. Werthamer: In Superconductivity, Vol. 1, ed. by R. D. Parks (Marcel Dekker Inc., New York 1969) p. 321Google Scholar
  3. The exact evaluation of correlation functions is due to D. J. Scalapino, M. Sears, R. A. Ferrell: Phys. Rev. B6, 3409 (1972)ADSGoogle Scholar
  4. Further papers on this evaluation are: L. W. Gruenberg, L. Gunther: Phys. Lett. 38A, 463 (1972)ADSGoogle Scholar
  5. Further papers on this evaluation are: M. Nauenberg, F. Kuttner, M. Fusman: Phys. Rev. A13, 1185 (1976)ADSGoogle Scholar
  6. b) References to Systems Far from Thermal Equilibrium (and Nonphysical Systems)Google Scholar
  7. R. Graham, H. Haken: Z. Phys. 237, 31 (1970)MathSciNetADSCrossRefGoogle Scholar
  8. For related topics see H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar
  9. and the articles by various authors in H. Haken, ed.: Synergetics (Teubner, Stuttgart 1973)zbMATHGoogle Scholar
  10. and the articles by various authors in H. Haken, M. Wagner, eds.: Cooperative Phenomena (Springer, Berlin-Heidelberg-New York 1973)zbMATHGoogle Scholar
  11. and the articles by various authors in H. Haken, ed.: Cooperative Effects (North Holland, Amsterdam 1974)Google Scholar
  12. and the articles by various authors in H. Haken (ed.): Springer Series in Synergetics Vols. 2–20 (Springer, Berlin-Heidelberg-New York)Google Scholar

Cooperative Effects in the Laser. Self-Organization and Phase Transition

  1. The dramatic change of the statistical properties of laser light at laser threshold was first derived and predicted by H. Haken: Z. Phys. 181, 96 (1964)ADSCrossRefGoogle Scholar

The Laser Equations in the Mode Picture

  1. For a detailed review on laser theory see H. Haken: In Encyclopedia of Physics, Vol. XXV/c: Laser Theory (Springer, Berlin-Heidelberg-New York 1970)Google Scholar

The Order Parameter Concept

  1. Compare especially H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar

The Single Mode Laser

  1. The dramatic change of the statistical properties of laser light at laser threshold was first derived and predicted by H. Haken: Z. Phys. 181, 96 (1964)ADSCrossRefGoogle Scholar
  2. For a detailed review on laser theory see H. Haken: In Encyclopedia of Physics, Vol. XXV/c: Laser Theory (Springer, Berlin-Heidelberg-New York 1970)Google Scholar
  3. Compare especially H. Haken: Rev. Mod. Phys. 47, 67 (1975)MathSciNetADSCrossRefGoogle Scholar
  4. The laser distribution function was derived by H. Risken: Z. Phys. 186, 85 (1965)ADSCrossRefGoogle Scholar
  5. The laser distribution function was derived by R. D. Hempstead, M. Lax: J. Phys. Rev. 161, 350 (1967)ADSCrossRefGoogle Scholar
  6. For a fully quantum mechanical distribution function cf. W. Weidlich, H. Risken, H. Haken: Z. Phys. 201, 396 (1967)ADSCrossRefGoogle Scholar
  7. For a fully quantum mechanical distribution function cf. M. Scully, W. E. Lamb: Phys. Rev. 159, 208 (1967):ADSCrossRefGoogle Scholar
  8. For a fully quantum mechanical distribution function cf. M. Scully, W. E. Lamb: Phys. Rev. 166, 246 (1968)ADSCrossRefGoogle Scholar

The Multimode Laser

  1. H. Haken: Z. Phys. 219, 246 (1969)ADSCrossRefGoogle Scholar

Laser with Continuously Many Modes. Analogy with Superconductivity

  1. For a somewhat different treatment see R. Graham, H. Haken: Z. Phys. 237, 31 (1970)MathSciNetADSCrossRefGoogle Scholar

First-Order Phase Transitions of the Single Mode Laser

  1. J. F. Scott, M. Sargent III, C. D. Cantrell: Opt. Commun. 15, 13 (1975)ADSCrossRefGoogle Scholar
  2. W. W. Chow, M. O. Scully, E. W. van Stryland: Opt. Commun. 15, 6 (1975)ADSCrossRefGoogle Scholar

Hierarchy of Laser Instabilities and Ultrashort Laser Pulses

  1. We follow essentially H. Haken, H. Ohno: Opt. Commun. 16, 205 (1976)ADSCrossRefGoogle Scholar
  2. We follow essentially H. Ohno, H. Haken: Phys. Lett. 59A, 261 (1976), and unpublished workADSGoogle Scholar
  3. For a machine calculation see H. Risken, K. Nummedal: Phys. Lett. 26A, 275 (1968);ADSGoogle Scholar
  4. For a machine calculation see H. Risken, K. Nummedal: J. appl. Phys. 39, 4662 (1968)ADSCrossRefGoogle Scholar
  5. For a discussion of that instability see also R. Graham, H. Haken: Z. Phys. 213, 420 (1968)ADSCrossRefGoogle Scholar
  6. For temporal oscillations of a single mode laser cf. K. Tomita, T. Todani, H. Kidachi: Phys. Lett. 51A, 483 (1975)ADSGoogle Scholar
  7. For further synergetic effects see R. Bonifacio (ed.): Dissipative Systems in Quantum Optics, Topics Current Phys., Vol. 27 (Springer, Berlin-Heidelberg-New York 1982)Google Scholar

Instabilities in Fluid Dynamics: The Bénard and Taylor Problems. 8.10 The Basic Equations. 8.11 Introduction of new variables 8.12 Damped and Neutral Solutions (R ≤ Rc)

  1. Some monographs in hydrodynamics: L. D. Landau, E. M. Lifshitz: In Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Pergamon Press, London-New York-Paris-Los Angeles 1959)Google Scholar
  2. Some monographs in hydrodynamics: Chia-Shun-Yih: Fluid Mechanics (McGraw Hill, New York 1969)Google Scholar
  3. Some monographs in hydrodynamics: G. K. Batchelor: An Introduction to Fluid Dynamics (University Press, Cambridge 1970)Google Scholar
  4. Some monographs in hydrodynamics: S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford 1961)zbMATHGoogle Scholar
  5. Stability problems are treated particularly by Chandrasekhar l.c. and by C. C. Lin: Hydrodynamic Stability (University Press, Cambridge 1967)Google Scholar

Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations. 8.14 The Fokker-Planck Equation and Its Stationary Solution

  1. We follow essentially H. Haken: Phys. Lett. 46A, 193 (1973)ADSGoogle Scholar
  2. and in particular Rev. Mod. Phys. 47, 67 (1976)Google Scholar
  3. For related work see R. Graham: Phys. Rev. Lett. 31, 1479 (1973):ADSCrossRefGoogle Scholar
  4. For related work see R. Graham: Phys. Rev. 10, 1762 (1974)ADSGoogle Scholar
  5. A. Wunderlin: Thesis, Stuttgart University (1975)Google Scholar
  6. J. Swift, P. C. Hohenberg: Phys. Rev. A15, 319 (1977)ADSGoogle Scholar
  7. For the analysis of mode-configurations, but without fluctuations, cf. A. Schlüter, D. Lortz, F. Busse: J. Fluid Mech. 23, 129 (1965)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. F. H. Busse: J. Fluid Mech. 30, 625 (1967)ADSzbMATHCrossRefGoogle Scholar
  9. A. C. Newell, J. A. Whitehead: J. Fluid Mech. 38, 279 (1969)ADSzbMATHCrossRefGoogle Scholar
  10. R. C. Diprima, H. Eckhaus, L. A. Segel: J. Fluid Mech. 49, 705 (1971)ADSzbMATHCrossRefGoogle Scholar
  11. Higher instabilities are discussed by F. H. Busse: J. Fluid Mech. 52, 1, 97 (1972)ADSzbMATHCrossRefGoogle Scholar
  12. Higher instabilities are discussed by D. Ruelle, F. Takens: Comm. Math. Phys. 20, 167 (1971)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. Higher instabilities are discussed by J. B. McLaughlin, P. C. Martin: Phys. Rev. A12, 186 (1975)ADSGoogle Scholar
  14. Higher instabilities are discussed by J. Gollup, S. V. Benson: In Pattern Formation by Dynamic Systems and Pattern Recognition, (ed. by H. Haken), Springer Series in Synergetic Vol. 5 (Springer, Berlin-Heidelberg-New York 1979)Google Scholar
  15. where further references may be found. A review on the present status of experiments and theory give the books Fluctuations, Instabilities and Phase Transitions, ed. by T. Riste (Plenum Press, New York 1975)Google Scholar
  16. H. L. Swinney, J. P. Gollub (eds.): Hydrodynamic Instabilities and the Transitions to Turbulence, Topics Appl. Phys., Vol. 45 (Springer, Berlin-Heidelberg-New York 1981)Google Scholar
  17. For a detailed treatment of analogies between fluid and laser instabilities c.f. M. G. Velarde: In Evolution of Order and Chaos, ed. by H. Haken, Springer Series in Synergetics, Vol.17 (Springer, Berlin-Heidelberg-New York 1982) where further references may be found.Google Scholar

A Model for the Statistical Dynamics of the Gunn Instability Near Threshold

  1. J. B. Gunn: Solid State Commun. 1, 88 (1963)ADSCrossRefGoogle Scholar
  2. J. B. Gunn: IBM J. Res. Develop. 8, (1964)Google Scholar
  3. For a theoretical discussion of this and related effects see for instance H. Thomas: In Synergetics, ed. by H. Haken (Teubner, Stuttgart 1973)Google Scholar
  4. Here, we follow essentially K. Nakamura: J. Phys. Soc. Jap. 38, 46 (1975)ADSCrossRefGoogle Scholar

Elastic Stability: Outline of Some Basic Ideas

  1. Introductions to this field give J. M. T. Thompson, G. W. Hunt: A General Theory of Elastic Stability (Wiley, London 1973)zbMATHGoogle Scholar
  2. K. Huseyin: Nonlinear Theory of Elastic Stability (Nordhoff, Leyden 1975)zbMATHGoogle Scholar
  3. Concentration oscillations were reported as early as 1921 by C. H. Bray: J. Am. Chem. Soc. 43, 1262 (1921)CrossRefGoogle Scholar
  4. A different reaction showing oscillations was studied by B. P. Belousov: Sb. ref. radats. med. Moscow (1959)Google Scholar
  5. This work was extended by Zhabotinsky and his coworkers in a series of papers V. A. Vavilin, A. M. Zhabotinsky, L. S. Yaguzhinsky: Oscillatory Processes in Biological and Chemical Systems (Moscow Science Publ. 1967) p. 181Google Scholar
  6. This work was extended by Zhabotinsky and his coworkers in a series of papers A. N. Zaikin, A. M. Zhabotinsky: Nature 225, 535 (1970)ADSCrossRefGoogle Scholar
  7. This work was extended by Zhabotinsky and his coworkers in a series of papers A. M. Zhabotinsky, A. N. Zaikin: J. Theor. Biol. 40, 45 (1973)CrossRefGoogle Scholar
  8. A theoretical model accounting for the occurrence of spatial structures was first given by A. M. Turing: Phil. Trans. Roy. Soc. B 237, 37 (1952)ADSCrossRefGoogle Scholar
  9. Models of chemical reactions showing spatial and temporal structures were treated in numerous publications by Prigogine and his coworkers. P. Glansdorff, I. Prigogine: Thermodynamik Theory of Structures, Stability and Fluctuations (Wiley, New York 1971)Google Scholar
  10. with many references, and G. Nicolis, I. Prigogine: Self-organization in Non-equilibrium Systems (Wiley, New York 1977)Google Scholar
  11. Prigogine has coined the word “dissipative structures”. Glansdorff and Prigogine base their work on entropy production principles and use the excess entropy production as means to search for the onset of an instability. The validity of such criteria has been critically investigated by R. Landauer: Phys. Rev. A12, 636 (1975). The Glansdorff-Prigogine approach does not give an answer to what happens at the instability point and how to determine or classify the new evolving structures. An important line of research by the Brussels school, namely chemical reaction models, comes closer to the spirit of Synergetics.ADSGoogle Scholar
  12. A review of the statistical aspects of chemical reactions can be found in D. McQuarry: Supplementary Review Series in Appl. Probability (Methuen, London 1967)Google Scholar
  13. A detailed review over the whole field gives the Faraday Symposium 9: Phys. Chemistry of Oscillatory Phenomena, London (1974)Google Scholar
  14. A detailed review over the whole field gives the Y. Schiffmann: Phys. Rep. 64, 88 (1980)MathSciNetADSCrossRefGoogle Scholar
  15. For chemical oscillations see especially G. Nicolis, J. Portnow: Chem. Rev. 73, 365 (1973)CrossRefGoogle Scholar

Deterministic Processes, Without Diffusion, One Variable. 9.3 Reaction and Diffusion Equations

  1. We essentially follow F. Schlögl: Z. Phys. 253, 147 (1972),ADSCrossRefGoogle Scholar
  2. who gave the steady state solution. The transient solution was determined by H. Ohno: Stuttgart (unpublished)Google Scholar

React ion-Diffusion Model with Two or Three Variables; the Brusselator and the Oregonator

  1. We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reaction-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, l.c. For related treatments see J. F. G. Auchmuchty, G. Nicolis: Bull. Math. Biol. 37, 1 (1974)Google Scholar
  2. We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reaction-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, l.c. For related treatments see Y. Kuramoto, T. Tsusuki: Progr. Theor. Phys. 52, 1399 (1974)ADSCrossRefGoogle Scholar
  3. We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reaction-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, l.c. For related treatments see M. Herschkowitz-Kaufmann: Bull. Math. Biol. 37, 589 (1975)CrossRefGoogle Scholar
  4. The “Oregonator” model reaction was formulated and treated by R. J. Field, E. Korös, R. M. Noyes: J. Am. Chem. Soc. 49, 8649 (1972)CrossRefGoogle Scholar
  5. The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: Nature 237, 390 (1972)ADSCrossRefGoogle Scholar
  6. The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: J. Chem Phys. 60, 1877 (1974)ADSCrossRefGoogle Scholar
  7. The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: J. Am. Chem. Soc. 96, 2001 (1974)CrossRefGoogle Scholar

Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable

  1. A first treatment of this model is due to V. J. McNeil, D. F. Walls: J. Stat. Phys. 10, 439 (1974)ADSCrossRefGoogle Scholar

Stochastic Model for a Chemical Reaction with Diffusion. One Variable

  1. The master equation with diffusion is derived by H. Haken: Z. Phys. B20, 413 (1975)ADSGoogle Scholar
  2. We essentially follow C. H. Gardiner, K. J. McNeil, D. F. Walls, I. S. Matheson: J. Stat. Phys. 14, 4, 307 (1976)ADSCrossRefGoogle Scholar
  3. Related to this chapter are the papers by G. Nicolis, P. Aden, A. van Nypelseer: Progr. Theor. Phys. 52, 1481 (1974)ADSCrossRefGoogle Scholar
  4. Related to this chapter are the papers by M. Malek-Mansour, G. Nicolis: preprint Febr. 1975Google Scholar

Stochastic Treatment of the Brusselator Close to Its Soft Mode Instability

  1. We essentially follow H. Haken: Z. Phys. B20, 413 (1975)ADSGoogle Scholar

Chemical Networks

  1. Related to this chapter are G. F. Oster, A. S. Perelson: Chem. Reaction Dynamics. Arch. Rat. Mech. Anal. 55, 230 (1974)MathSciNetCrossRefGoogle Scholar
  2. Related to this chapter are A. S. Perelson, G. F. Oster. Chem. Reaction Dynamics, Part II; Reaction Networks. Arch Rat. Mech. Anal. 57, 31 (1974/75)MathSciNetCrossRefGoogle Scholar
  3. with further references. G. F. Oster, A. S. Perelson, A. Katchalsky: Quart. Rev. Biophys. 6, 1 (1973)CrossRefGoogle Scholar
  4. O. E. Rössler: In Lecture Notes in Biomathematics, Vol. 4 (Springer, Berlin-Heidelberg-New York 1974) p. 419Google Scholar
  5. O. E. Rössler: Z. Naturforsch. 31a, 255 (1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Hermann Haken
    • 1
  1. 1.Institut für Theoretische PhysikUniversität StuttgartStuttgart 80Fed. Rep. of Germany

Personalised recommendations