Abstract
Over the past years the field of synergetics has been mushrooming. An ever-increasing number of scientific papers are published on the subject, and numerous conferences all over the world are devoted to it. Depending on the particular aspects of synergetics being treated, these conferences can have such varied titles as “Nonequilibrium Nonlinear Statistical Physics,” “Self-Organization,” “Chaos and Order,” and others. Many professors and students have expressed the view that the present book provides a good introduction to this new field. This is also reflected by the fact that it has been translated into Russian, Japanese, Chinese, German, and other languages, and that the second edition has also sold out. I am taking the third edition as an opportunity to cover some important recent developments and to make the book still more readable.
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References, Further Reading, and Comments
1. Goal
H. Haken, R. Graham: Synergetik-Die Lehre vom Zusammenwirken. Umschau 6. 191 (1971)
H. Haken (ed.): Synergetics (Proceedings of a Symposium on Synergetics. Elmau 1972 ) ( B. G. Teubner, Stuttgart 1973 )
H. Haken (ed.): Cooperative Effects, Progress in Synergetics ( North Holland, Amsterdam 1974 )
H. Haken: Cooperative effects in systems far from thermal equilibrium and in nonphysical systems. Rev. Mod. Phys. 47, 67 (1975)
H. Haken (ed.): Springer Series in Synergetics,Vols. 2–20 (Springer, Berlin, Heidelberg, New York)
For a popularisation see H. Haken: Erfolgsgeheimnisse der Natur (Deutsche Verlagsanstalt, Stuttgart 1981 ) English edition in preparation.
J. T. Bonner, D. S. Barkley, E. M. Hall, T. M. Konijn, J. W. Mason, G. O’Keefe, P. B. Wolfe: Develop. Biol. 20, 72 (1969)
T. M. Konijn: Advanc. Cycl. Nucl. Res. 1, 17 (1972)
A. Robertson, D. J. Drage, M. H. Cohen: Science 175, 333 (1972)
G. Gerisch, B. Hess: Proc. nat. Acad. Sci (Wash.) 71, 2118 (1974)
G. Gerisch: Naturwissenschaften 58, 430 (1971)
2. Probability
Kai Lai Chung: Elementary Probability Theory with Stochastic Processes ( Springer, Berlin-Heidelberg-New York 1974 )
W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 1 (Wiley, New York 1968 ), Vol. 2 ( Wiley, New York 1971 )
R.C. Dubes: The Theory of Applied Probability (Prentice Hall, Englewood Cliffs, N.J. 1968 )
Yu. V. Prokhorov, Yu. A. Rozanov: Probability Theory. In Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 157 (Springer, Berlin-Heidelberg-New York 1968 )
J. L. Doob: Stochastic Processes ( Wiley, New York-London 1953 )
M. Loève: Probability Theory (D. van Nostrand, Princeton, N.J.-Toronto-New York-London 1963 )
R. von Mises: Mathematical Theory of Probability and Statistics ( Academic Press, New York-London 1964 )
3. Information
Monographs on this subject are: L. Brillouin: Science and Information Theory ( Academic Press, New York-London 1962 )
L. Brillouin: Scientific Uncertainty and Information ( Academic Press, New York-London 1964 )
Information theory was founded by C. E. Shannon: A mathematical theory of communication. Bell System Techn. J. 27, 370–423, 623–656 (1948)
C. E. Shannon: Bell System Techn. J. 30, 50 (1951)
C. E. Shannon, W. Weaver: The Mathematical Theory of Communication (Univ. of Illin. Press, Urbana 1949 )
Some conceptions, related to information and information gain (H-theorem!) were introduced by L. Boltzmann: Vorlesungen über Gastheorie, 2 Vols. (Leipzig 1896, 1898 )
For a detailed treatment and definition see S. Kullback: Ann. Math. Statist. 22, 79 (1951)
S. Kullback: Information Theory and Statistics (Wiley, New York 1951 ) Here we follow our lecture notes.
We follow in this chapter essentially E. T. Jaynes: Phys. Rev. 106, 4, 620 (1957); Phys. Rev. 108, 171 (1957)
E. T. Jaynes: In Delaware Seminar in the Foundations of Physics (Springer, Berlin-Heidelberg-New York 1967)
Early ideas on this subject are presented in W. Elsasser: Phys. Rev. 52, 987 (1937); Z. Phys. 171, 66 (1968)
The approach of this chapter is conceptually based on Jaynes’ papers, I.c. Section 3.3. For textbooks giving other approaches to thermodynamics see
Landau-Lifshitz: In Course of Theoretical Physics,Vol. 5: Statistical Physics (Pergamon Press. London-Paris 1952)
R. Becker: Theory of Heat ( Springer, Berlin-Heidelberg-New York 1967 )
A. Münster: Statistical Thermodynamics, Vol. 1 (Springer, Berlin-Heidelberg-New York 1969 ) H. B. Callen: Thermodynamics ( Wiley, New York 1960 )
P. T. Landsberg: Thermodynamics ( Wiley, New York 1961 )
R. Kubo: Thermodynamics ( North Holland, Amsterdam 1968 )
W. Brenig: Statistische Theorie der Wärme ( Springer, Berlin-Heidelberg-New York 1975 )
W. Weidlich: Thermodynamik and statistische Mechanik ( Akademische Verlagsgesellschaft, Wiesbaden 1976 )
An interesting and promising link between irreversible thermodynamics and network theory has been established by
A. Katchalsky, P. F. Curran: Nonequilibrium Thermodynamics in Biophysics ( Harvard University Press, Cambridge Mass. 1967 )
For a recent representation including also more current results see
J. Schnakenberg: Thermodynamic Network Analysis of Biological Systems, Universitext ( Springer, Berlin-Heidelberg-New York 1977 )
For detailed texts on irreversible thermodynamics see
I. Prigogine: Introduction to Thermodynamics of Irreversible Processes (Thomas, New York 1955 ) I. Prigogine: Non-equilibrium Statistical Mechanics ( Interscience, New York 1962 )
S. R. De Groot, P. Mazur: Non-equilibrium Thermodynamics (North Holland, Amsterdam 1962 ) R. Haase: Thermodynamics of Irreversible Processes (Addison-Wesley, Reading, Mass. 1969 )
D. N. Zubarev: Non-equilibrium Statistical Thermodynamics ( Consultants Bureau, New York-London 1974 )
Here, we present a hitherto unpublished treatment by the present author.
For the problem subjectivistic-objectivistic see for example
E. T. Jaynes: Information Theory. In Statistical Physics, Brandeis Lectures, Vol. 3 ( W. A. Benjamin, New York 1962 )
Coarse graining is discussed by A. Münster: In Encyclopedia of Physics, ed. by S. Flügge, Vol. 1I1/2: Principles of Thermodynamics and Statistics ( Springer, Berlin-Göttingen-Heidelberg 1959 )
The concept of entropy is discussed in all textbooks on thermodynamics, cf. references to Section 3.4.
4. Chance
For detailed treatments of Brownian motion see for example
N. Wax, ed.: Selected Papers on Noise and Statistical Processes (Dover Publ. Inc., New York 1954) with articles by S. Chandrasekhar, G. E. Uhlenbeck and L. S. Ornstein, Ming Chen Wang and G. E. Uhlenbeck, M. Kac
T. T. Soong: Random Differential Equations in Science and Engineering ( Academic Press, New York 1973 )
See for instance M. Kac: Am. Math. Month. 54, 295 (1946)
M. S. Bartlett: Stochastic Processes (Univ. Press, Cambridge 1960 )
See references on stochastic processes, Chapter 2. Furthermore
R. L. Stratonovich: Topics in the Theory of Random Noise (Gordon Breach, New York-London, Vol. 1 1963, Vol. II 1967 )
M. Lax: Rev. Mod. Phys. 32, 25 (1960); 38, 358 (1965); 38, 541 (1966)
Path integrals will be treated later in our book (Section 6.6), where the corresponding references may be found.
How to Use Joint Probabilities. Momrats. Characteristic Function. Gaussian Processes Same references as on Section 4. 3.
The master equation does not only play an important role in (classical) stochastic processes, but also in quantum statistics. Here are some references with respect to quantum statistics:
H. Pauli: Probleme der Modernen Physik. Festschrift zum 60. Geburtstage A. Sommerfelds, cd. by P. Debye ( Hirzel, Leipzig 1928 )
L. van Hove: Physica 23, 441 (1957)
S. Nakajiama: Progr. Theor. Phys. 20, 948 (1958)
R. Zwanzig: J. Chem. Phys. 33, 1338 (1960)
E. W. Montroll: Fundamental Problems in Statistical Mechanics, compiled by E. D. G. Cohen (North Holland, Amsterdam 1962 )
P. N. Argyres, P. L. Kelley: Phys. Rev. 134, A98 (1964)
For a recent review see F. Haake: In Springer Tracts in Modern Physics, Vol. 66 ( Springer, Berlin-Heidelberg-New York 1973 ) p. 98.
H. Haken: Phys. Lett. 46A, 443 (1974); Rev. Mod. Phys. 47, 67 (1975), where further discussions are given.
R. Landauer: J. Appl. Phys. 33, 2209 (1962)
G. Kirchhoff: Ann. Phys. Chem., Bd. LXXII 1847, Bd. 12, S. 32
G. Kirchhoff: Poggendorffs Ann. Phys. 72, 495 (1844)
R. Bott, J. P. Mayberry: Matrices and Trees, Economic Activity Analysis ( Wiley. New York 1954 )
E. L. King, C. Altmann: J. Phys. Chem. 60, 1375 (1956)
T. L. Hill: J. Theor. Biol. 10, 442 (1966)
A very elegant derivation of Kirchhoff’s solution was recently given by W. Weidlich; Stuttgart (unpublished)
I. Schnakenberg: Rev. Mod. Phys. 48, 571 (1976)
J. Keizer: On the Solutions and the Steady States of a Master Equation ( Plenum Press, New York 1972 )
P. and T. Ehrenfest: Phys. Z. 8, 311 (1907) and also A. Münster: In Encyclopedia of Physics, ed. by S. Flügge, Vol. III/2; Principles of Thermodynamics and Statistics ( Springer, Berlin-Göttingen-Heidelberg 1959 )
5. Necessity
Monographs on dynamical systems and related topics are N. N. Bogoliubov, Y. A. Mitropolsky: Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan Publ. Corp., Delhi 1961 )
N. Minorski: Nonlinear Oscillations ( Van Nostrand, Toronto 1962 )
A. Andronov, A. Vitt, S. E. Khaikin: Theory of Oscillators ( Pergamon Press, London-Paris 1966 )
D. H. Sattinger In Lecture Notes in Mathematics, Vol. 309: Topics in Stability and Bifurcation Theory, ed. by A. Dold, B. Eckmann ( Springer, Berlin-Heidelberg-New York 1973 )
M. W. Hirsch, S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra ( Academic Press, New York-London 1974 )
V. V. Nemytskn, V. V. Stepanov: Qualitative Theory of Differential Equations (Princeton Univ. Press, Princeton, N.J. 1960 )
Many of the basic ideas are duc to
H. Poincaré: Oeuvres, Vol. 1 ( Gauthiers-Villars, Paris 1928 )
H. Poincaré: Sur l’equilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7 (1885)
H. Poincaré: Figures d’equilibre d’une masse fluide (Paris 1903)
H. Poincaré: Sur le problème de trois corps et les équations de la dynamique. Acta Math. 13 (1890) H. Poincaré: Les méthodes nouvelles de la méchanique céleste ( Gauthier-Villars, Paris 1892–1899 )
J. La Salle, S. Lcfshctz: Stability by Ljapunov’s Direct Method with Applications ( Academic Press, New York-London 1961 )
W. Hahn: Stability of Motion. In Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Bd. 138 ( Springer, Berlin-Heidelberg-New York 1967 )
D. D. Joseph: Stability of Fluid Motions I and 17, Springer Tracts in Natural Philosophy, Vols. 27, 28 ( Springer, Berlin-Heidelberg-New York 1976 )
Exercises 5.3: F. Schlögl: Z. Phys. 243, 303 (1973)
A. Lotka: Proc. Nat. Acad. Sci. (Wash.) 6, 410 (1920)
V. Volterra: Leçons sur la théorie mathematiques de la lutte pour la vie (Paris 1931)
N. S. God, S. C. Maitra, E. W. Montroll: Rev. Mod. Phys. 43, 231 (1971)
B. van der Pol: Phil. Mag. 43, 6, 700 (1922); 2, 7, 978 (1926); 3, 7, 65 (1927)
H. T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover Publ. Inc., New York 1962 )
G. Ioss, D. D. Joseph: Elementary Stability and Bifurcation Theory (Springer, Berlin, Heidelberg, New York 1980 )
R. Thom: Structural Stability and Morphogenesis (W. A. Benjamin, Reading, Mass. 1975 ) Thom’s book requires a good deal of mathematical background. Our “pedestrian’s” approach provides a simple access to Thom’s classification of catastrophes. Our interpretation of how to apply these results to natural sciences, for instance biology is, however, entirely different from Thom’s.
T. Poston, I. Steward: Catastrophe Theory and its Applications ( Pitman, London 1978 )
E. C. Zeeman: Catastrophe Theory ( Addison-Wesley, New York 1977 )
P. T. Sounders: An Introduction to Catastrophe Theory ( Cambridge University Press, Cambridge 1980 )
6. Chance and Necessity
For general approaches see R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 ( Gordon * Breach, New York-London 1963 )
M. Lax: Rev. Mod. Phys. 32, 25 (1960); 38, 358, 541 (1966); Phys. Rev. 145, 110 (1966)
H. Haken: Rev. Mod. Phys. 47, 67 (1975)
with further references P. Hänggi, H. Thomas: Phys. Rep. 88, 208 (1982)
Here we present a simple example. For general approaches see R. Zwanzig: J. Stat. Phys. 9, 3, 215 (1973)
H. Haken: Rev. Mod. Phys. 47, 67 (1975)
R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon * Breach, New York-London 1963 )
The more general case for systems in detailed balance is treated by
R. Graham, H. Haken: Z. Phys. 248, 289 (1971)
R. Graham: Z. Phys. B40, 149 (1981)
H. Risken: Z. Phys. 251, 231 (1972);
H. Haken: Rev. Mod. Phys. 47, 67 (1975)
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)
For a short representation of the results see
H. Haken: Rev. Mod. Phys. 47, 67 (1975)
L. Onsager, S. Machlup: Phys. Rev. 91, 1505, 1512 (1953)
I. M. Gelfand, A. M. Yaglome: J. Math. Phys. 1, 48 (1960)
R. P. Feynman, A. R. Hibbs: Quantum Mechanics and Path Integrals ( McGraw-Hill, New York 1965 )
F. W. Wiegel: Path Integral Methods in Statistical Mechanics, Physics Reports I6C, No. 2 ( North Holland, Amsterdam 1975 )
R. Graham: In Springer Tracts in Modern Physics, Vol. 66 ( Springer, Berlin-Heidelberg-New York 1973 ) p. 1
A critical discussion of that paper gives W. Horsthemke, A. Bach: Z. Phys. B22, 189 (1975)
We follow essentially H. Haken: Z. Phys. B24,321 (1976) where also classes of solutions of Fokker-Planck equations are discussed.
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 (Perga-mon Press, London-Paris 1959)
R. Brout: Phase Transitions ( Benjamin, New York 1965 )
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)
M. E. Fischer: Repts. Progr. Phys. 30, 731 (1967)
H. E. Stanley: Introduction to Phase Transitions and Critical Phenomena. Internat. Series of Monographs in Physics (Oxford University, New York 1971 )
A. Münster: Statistical Thermodynamics, Vol. 2 ( Springer, Berlin-Heidelberg-New York and Academic Press, New York-London 1974 )
C. Domb, M. S. Green, eds.: Phase Transitions and Critical Phenomena,Vols. 1–5 (Academic Press, London 1972–76)
The modern and powerful renormalization group technique of Wilson is reviewed by
K. G. Wilson, J. Kogut: Phys. Rep. 12 C, 75 (1974)
S.-K. Ma: Modern Theory of Critical Phenomena (Benjamin, London 1976 )
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), who treated the continuum mode laser, and by
V. DeGiorgio, M. O. Scully: Phys. Rev. A2, 1170 (1970)
For further references elucidating the historical development see Section 13.
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. 321
with further references The exact evaluation of correlation functions is due to D. J. Scalapino, M. Sears, R. A. Ferrell: Phys. Rev. B6, 3409 (1972)
Further papers on this evaluation are:
L. W. Gruenberg, L. Gunther: Phys. Lett. 38A, 463 (1972)
M. Nauenberg, F. Kuttner, M. Fusman: Phys. Rev. A13, 1185 (1976)
R. Graham, H. Haken: Z. Phys. 237, 31 (1970) Furthermore the Chapter 8 and 9
7. Self-Organization
H. Haken: unpublished material
A different approach to the problem of self-organization has been developed by
J. v. Neuman: Theory of Self-reproducing Automata. ed. and completed by Arthur W. Burks (University of Illinois Press, 1966 )
For a detailed discussion of reliability as well as switching, especially of computer elements, see R. Landauer: IBM Journal 183 (July 1961)
R. Landauer: J. Appl. Phys. 33, 2209 (1962)
R. Landauer, J. W. F. Woo: In Synergetics, ed. by H. Haken ( Teubner, Stuttgart 1973 )
H. Haken: Z. Phys. B 20, 413 (1975)
Haken: unpublished References, Further Reading, and Comments
H. Haken: Z. Phys. B21, 105 (1975)
H. Haken: Z. Phys. B22, 69 (1975); B23, 388 (1975)
For another treatment of the slaving principle see A. Wunderlin, H. Haken: Z. Phys. B44, 135 (1981) H. Haken, A. Wundcrlin: Z. Phys. B47, 179 (1982)
A. Wunderlin, H. Haken: Z. Phys. B21, 393 (1975)
For related work see E. Hopf: Berichte der Math.-Phys. Klasse der Sächsischen Akademie der Wissenschaften, Leipzig XCIV, 1 (1942)
A. Schlüter, D. Lortz, F. Busse. J. Fluid Mech. 23, 129 (1965)
A. C. Newell, J. A. Whitehead: J. Fluid Mech. 38, 279 (1969)
R. C. Diprima, W. Eckhaus, L. A. Segel: J. Fluid Mech. 49, 705 (1971)
8. Physical Systems
H. Haken: Rev. Mod. Phys. 47, 67 (1975)
H. Haken, ed.: Synergetics ( Teubner, Stuttgart 1973 )
H. Haken, M. Wagner, eds.: Cooperative Phenomena ( Springer, Berlin-Heidelberg-New York 1973 )
H. Haken, cd.: Cooperative Effects ( North Holland, Amsterdam 1974 )
H. Haken (ed.): Springer Series in Synergetics Vols. 2–20 (Springer, Berlin-Heidelberg-New York)
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)
For a detailed review on laser theory see
H. Haken: In Encyclopedia of Physics,Vol. XXV!c: Laser Theory (Springer, Berlin-HeidelbergNew York 1970)
Compare especially H. Haken: Rev. Mod. Phys. 47, 67 (1975)
The laser distribution function was derived by H. Risken: Z. Phys. 186, 85 (1965) and
R. D. Hempstead, M. Lax: Phys. Rev. 161, 350 (1967)
For a fully quantum mechanical distribution function cf. W. Weidlich, H. Risken, H. Haken: Z. Phys. 201, 396 (1967)
M. Scully, W. E. Lamb: Phys. Rev. 159, 208 (1967): 166, 246 (1968)
H. Haken: Z. Phys. 219, 246 (1969)
For a somewhat different treatment see R. Graham, H. Haken: Z. Phys. 237, 31 (1970)
J. F. Scott, M. Sargentlll, C. D. Cantrell: Opt. Commun. 15, 13 (1975)
W. W. Chow, M. O. Scully, E. W. van Stryland: Opt. Commun. 15, 6 (1975)
We follow essentially H. Haken, H. Ohno: Opt. Commun. 16, 205 (1976)
H. Ohno, H. Haken: Phys. Lett. 59A, 261 (1976), and unpublished work
For a machine calculation see H. Risken, K. Nummedal: Phys. Lett. 26A, 275 (1968); J. appl. Phys. 39, 4662 (1968)
For a discussion of that instability see also
R. Graham, H. Haken: Z. Phys. 213, 420 (1968)
For temporal oscillations of a single mode laser cf.
K. Tomita, T. Todani, H. Kidachi: Phys. Lett. 51 A, 483 (1975)
For further synergetic effects see R. Bonifacio (ed.): Dissipative Systems in Quantum Optics, Topics Current Phys., Vol. 27 ( Springer, Berlin-Heidelberg-New York 1982 )
Some monographs in hydrodynamics: L. D. Landau, E. M. Lifshitz: In Course of Theoretical Physics,Vol. 6: Fluid Mechanics (Perga-mon Press, London-New York-Paris-Los Angeles 1959)
Chia-Shun-Yih: Fluid Mechanics ( McGraw Hill, New York 1969 )
G. K. Batchelor: An Introduction to Fluid Dynamics ( University Press, Cambridge 1970 )
S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford 1961, Stability problems are treated particularly by Chandrasekhar I.c. and by C. C. Lin: Hydrodynamic Stability (University Press, Cambridge 1967 ) We follow essentially
H. Haken: Phys. Lett. 46A, 193 (1973) and in particular Rev. Mod. Phys. 47, 67 (1976)
For related work see R. Graham: Phys. Rev. Lett. 31, 1479 (1973): Phys. Rev. 10, 1762 (1974)
A. Wunderlin: Thesis, Stuttgart University (1975)
J. Swift, P. C. Hohenberg: Phys. Rev. A15, 319 (1977)
For the analysis of mode-configurations, but without fluctuations, cf. A. Schlüter, D. Lortz, F. Busse: J. Fluid Mech. 23, 129 (1965)
F. H. Busse: J. Fluid Mech. 30. 625 (1967)
A. C. Newell, J. A. Whitehead: J. Fluid Mech. 38, 279 (1969)
R. C. Diprima, H. Eckhaus, L. A. Segel: J. Fluid Mech. 49, 705 (1971)
Higher instabilities are discussed by F. H. Busse: J. Fluid Mech. 52, 1, 97 (1972)
D. Ruelle, F. Takens: Comm. Math. Phys. 20, 167 (1971)
J. B. McLaughlin, P. C. Martin: Phys. Rev. Al2, 186 (1975)
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) 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)
H. L. Swinney, J. P. Gollub (eds.): Hydrodynamic Instabilities and the Transitions to Turbulence. Topics App1. Phys., Vol. 45 ( Springer, Berlin-Heidelberg-New York 1981 )
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.
J. B. Gunn: Solid State Commun. 1, 88 (1963)
J. B. Gunn: IBM J. Res. Develop. 8, (1964)
For a theoretical discussion of this and related effects see for instance H. Thomas: In Synergetics, ed. by H. Haken ( Teubner, Stuttgart 1973 )
Here, we follow essentially K. Nakamura: J. Phys. Soc. Jap. 38, 46 (1975)
Introductions to this field give J. M. T. Thompson, G. W. Hunt: A General Theory of Elastic Stability ( Wiley. London 1973 )
K. Huseyin: Nonlinear Theory of Elastic Stability ( Nordhoff, Leyden 1975 )
9. Chemical and Biochemical Systems
In this chapter we particularly consider the occurrence of spatial or temporal structures in chemical reactions.
Concentration oscillations were reported as early as 1921 by
C. H. Bray: J. Am. Chem. Soc. 43, 1262 (1921)
A different reaction showing oscillations was studied by
B. P. Belousov: Sb. ref. radats. med. Moscow (1959)
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. 181
A. N. Zaikin, A. M. Zhabotinsky: Nature 225, 535 (1970)
A. M. Zhabotinsky, A. N. Zaikin: J. Theor. Biol. 40, 45 (1973)
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)
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) with many references, and
G. Nicolis, I. Prigogine: Self-organization in Non-equilibrium Systems (Wiley, New York 1977) 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. A 12, 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.
A review of the statistical aspects of chemical reactions can be found in
D. McQuarry: Supplementary Review Series in Appl. Probability (Methuen, London 1967 ) A detailed review over the whole field gives the
Faraday Symposium 9: Phys. Chemistry of Oscillatory Phenomena, London (1974) Y. Schiffmann: Phys. Rep. 64, 88 (1980)
For chemical oscillations see especially
G. Nicolis, J. Portnow: Chem. Rev. 73, 365 (1973)
We essentially follow F. Schlögl: Z. Phys. 253, 147 (1972)
who gave the steady state solution. The transient solution was determined by
H. Ohno: Stuttgart (unpublished)
We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reac- tion-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, 1.c. For related treatments see
J. F. G. Auchmuchty, G. Nicolis: Bull. Math. Biol. 37, 1 (1974)
Y. Kuramoto, T. Tsusuki: Progr. Theor. Phys. 52, 1399 (1974)
M. Herschkowitz-Kaufmann: Bull. Math. Biol. 37, 589 (1975)
The Belousov-Zhabotinsky reaction is described in the already cited articles by Belousov and Zhabotinsky.
The “Oregonator” model reaction was formulated and treated by
R. J. Field, E. Korös, R. M. Noyes: J. Am. Chem. Soc. 49, 8649 (1972) R. J. Field, R. M. Noyes: Nature 237, 390 (1972)
R. J. Field, R. M. Noyes: J. Chem Phys. 60, 1877 (1974)
R. J. Field, R. M. Noyes: J. Am. Chem. Soc. 96, 2001 (1974)
A first treatment of this model is due to
V. J. McNeil, D. F. Walls: J. Stat. Phys. 10, 439 (1974)
The master equation with diffusion is derived by
H. Haken: Z. Phys. B20, 413 (1975)
C. H. Gardiner, K. J. McNeil, D. F. Walls, I. S. Matheson: J. Stat. Phys. 14. 4. 307 (1976)
Related to this chapter are the papers by
G. Nicolis, P. Aden, A. van Nypelseer: Progr. Theor. Phys. 52, 1481 (1974)
M. Malek-Mansour, G. Nicolis: preprint Febr. 1975 We essentially follow
H. Haken: Z. Phys. B 20. 413 (1975)
Related to this chapter are G. F. Oster, A. S. Perelson: Chem. Reaction Dynamics. Arch. Rat. Mech. Anal. 55. 230 (1974)
A. S. Perelson, G. F. Oster: Chem. Reaction Dynamics, Part IL Reaction Networks. Arch Rat. Mech. Anal. 57, 31 (1974/75)
G. F. Oster, A. S. Perelson, A. Katchalsky: Quart. Rev. Biophys. 6, 1 (1973)
O. E. Rössler: In Lecture Notes in Biomathematics, Vol. 4 ( Springer, Berlin-Heidelberg-New York 1974 ) p. 419
O. E. Rössler: Z. Naturforsch. Ma, 255 (1976)
10. Applications to Biology
For general treatments see N. S. Goel, N. Richter-Dyn: Stochastic Models in Biology ( Academic Press. New York 1974 )
D. Ludwig: In Lecture Notes in Biomathematics, Vol. 3: Stochastic Population Theories, ed. by S. Levin (Springer, Berlin-Heidelberg-New York 1974 )
For a different treatment of the problem of this section see V. T. N. Reddy: J. Statist. Phys. 13, 1 (1975)
The equations discussed here seem to have first occurred in the realm of laser physics, where they explained mode-selection in lasers (H. Haken, H. Sauermann: Z. Phys. 173, 261 (1963)). The application of laser-type equations to biological processes was suggested by
H. Haken: Talk at the Internat. Conference From Theoretical Physics to Biology, ed. by M. Marois, Versailles 1969
H. Haken: In From Theoretical Physics to Biology, ed. by M. Marois ( Karger, Basel 1973 )
A comprehensive and detailed theory of evolutionary processes has been developed by M. Eigen: Die Naturwissenschaften 58, 465 (1971).
With respect to the analogies emphasized in our book it is interesting to note that Eigen’s “Bewertungsfunktion” is identical with the saturated gain function (8.35) of multimode lasers.
An approach to interpret evolutionary and other processes as games is outlined by M. Eigen, R. Winkler-Oswatitsch: Das Spiel ( Piper, München 1975 )
An important new concept is that of hypercycles and, connected with it, of “quasi-species” M. Eigen, P. Schuster: Naturwissensch. 64, 541 (1977); 65, 7 (1978): 65, 341 (1978)
We present here a model due to Gicrer and Meinhardt cf.
A. Gierer, M. Meinhardt: Biological pattern formation involving lateral inhibition. Lectures on Mathematics in the Life Sciences 7, 163 (1974)
H. Meinhardt: The Formation of Morphogenetic Gradients and Fields. Ber. Deutsch. Bot. Ges. 87, 101 (1974)
H. Meinhardt, A. Gicrer: Applications of a theory of biological pattern formation based on lateral inhibition. J Cell. Sci. 15, 321 (1974)
H. Meinhardt: preprint 1976
H. Meinhardt: Models of Biological Pattern Formation ( Academic, London 1982 )
We present here results by H. Haken and H. Olbrich. J. Math. Biol. 6, 317 (1978)
11. Sociology and Economics
We present here Weidlich’s model.
W. Weidlich: Collective Phenomena 1, 51 (1972)
W. Weidlich: Brit. J. math. stat. Psycho!. 24, 251 (1971)
W. Weidlich: In Svnergetics, ed. by H. Haken ( Teubner, Stuttgart 1973 )
The following monographs deal with a mathematization of sociology:
J. S. Coleman: Introduction to Mathematical Sociology ( The Free Press, New York 1964 )
D. J. Bartholomew: Stochastic Models for Social processes ( Wiley, London 1967 )
W. Weidlich, G. Haag: Concepts and Models of a Quantitative Sociology, Springer Ser. Synergetics, Vol. 14 ( Springer, Berlin-Heidelberg-New York 1983 )
I present here my “translation” [first published in the German version: Synergetik. Eine Einführung (Springer, Berlin Heidelberg, New York 1982)] of an economic model by G. Mensch et al. into the formalism and language of synergetics.
G. Mensch, K. Kaasch, A. Kleinknecht, R. Schnapp: “Innovation Trends, and Switching Between Full-and Under-Employment Equilibria”, IIMjdp 80–5, Discussion paper series International Institute of Management, Wissenschaftszentrum Berlin (1950–1978)
12. Chaos
For mathematically rigorous treatments of examples of chaos by means of mappings and other topological methods see
S. Smale: Bull. A. M. S. 73, 747 (1967)
T. Y. Li, J. A. Yorke: Am. Math. Monthly 82, 985 (1975)
D. Ruelle, F. Takens: Commun. math. Phys. 20, 167 (1971)
E. N. Lorenz: J. Atmospheric Sci. 20, 130 (1963)
E. N. Lorenz: J. Atmospheric Sci. 20, 448 (1963)
Historically, the first papers showing a “strange attractor”. For further treatments of this model see
J. B. McLaughlin, P. C. Martin: Phys. Rev. Al2, 186 (1975)
M. Lücke: J. Stat. Phys. 15, 455 (1976)
C. T. Sparrow: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors ( Springer, Berlin-Heidelberg-New York 1982 )
For the laser fluid analogy presented in this chapter see H. Haken: Phys. Lett. 53 A, 77 (1975)
H. Haken, A. Wunderlin: Phys. Lett. 62A, 133 (1977)
H. Haken, J. Zorell: Unpublished
M. Lücke: J. Stat. Phys. 15, 455 (1976) Y. Aizawa, I. Shimada: Preprint 1977
H. Poincaré: Les méthodes nouvelles de la méchanique céleste. Gauthier-Villars, Paris (1892/99), Reprint ( Dover Publ., New York 1960 )
For electronic devices especially Khaikin’s “universal circuit” see
A. A. Andronov, A. A. Vitt, S. E. Khaikin: Theory of Oscillators ( Pergamon Press, OxfordLondon-Edinburgh-New York-Toronto-Paris-Frankfurt 1966 )
K. Nakamura: Progr. Theoret. Phys. 57, 1874 (1977)
Numerous chemical reaction models (without diffusion) have been treated by
O. E. Roessler. For a summary and list of reference consult
O. E. Roessler: In Synergetics, A Workshop, ed. by H. Haken ( Springer, Berlin-Heidelberg-New York, 1977 )
For chemical reaction models including diffusion see
Y. Kuramoto, T. Yamada: Progr. Theoret. Phys. 56, 679 (1976)
T. Yamada, Y. Kuramoto: Progr. Theoret. Phys. 56, 681 (1976)
Modulated chemical reactions have been trated by
K. Tomita, T. Kai, F. Hikami: Progr. Theoret. Phys. 57, 1159 (1977)
For experimental evidence of chaos in chemical reactions see
R. A. Schmitz, K. R. Graziani, J. L. Hudson: J. Chem. Phys. 67, 3040 (1977);
O. E. Roessler, to be published J. A. Jacobs: Phys. Reports 26, 183 (1976) with further references
R. M. May: Nature 261, 459 (1976)
M. I. Rabinovich: Sov. Phys. Usp. 21, 443 (1978)
A. S. Monin: Soy. Phys. Usp. 21, 429 (1978)
D. Ruelle: La Recherche N° 108, Février (1980)
Some fundamental works on period doubling are
S. Grossmann, S. Thomae: Z. Naturforsch. A32, 1353 (1977)
This paper deals with the logistic map given in the text. The universal behavior of period doubling was discovered by
M. J. Feigenbaum: J. Stat. Phys. 19, 25 (1978): Phys. Lett. A74, 375 (1979)
An extensive presentation of later results, as well as many further references, are given in
P. Collet, J. P. Eckmann: Iterated Maps on the Interval as Dynamical System ( Birk häuser, Boston 1980 )
L. Garrido (ed.): Dynamical Systems and Chaos, Lecture Notes Phys., Vol. 179 ( Springer, Berlin, Heidelberg, New York 1983 )
H. Haken (ed.): Chaos and Order in Nature, Springer Ser. Synergetics, Vol. 11 ( Springer, Berlin Heidelberg, New York 1981 )
H. Haken (ed.): Evolution of Order and Chaos, Springer Ser. Synergetics, Vol. 17 ( Springer, Berlin, Heidelberg, New York 1982 )
The influence of fluctuations on period doubling has been studied by the following authors:
G. Mayer-Kress, H. Haken: J. Stat. Phys. 24, 345 (1981)
J. P. Crutchfield, B. A. Huberman: Phys. Lett. A77, 407 (1980)
A. Zippelius, M. Lücke: J. Stat. Phys. 24, 345 (1981)
J. P. Crutchfield, M. Nauenberg, J. Rudnick: Phys. Rev. Lett. 46, 933 (1981)
B. Shraiman, C. E. Wayne, P. C. Martin: Phys. Rev. Lett. 46, 935 (1981)
The corresponding Kolmogorov equation is established and discussed in
H. Haken, G. Mayer-Kress: Z. Phys. B43. 185 (1981)
H. Haken, A. Wunderlin: Z. Phys. B46, 181 (1982)
13. Some Historical Remarks and Outlook
J. F. G. Auchmuchty, G. Nicolis: Bull. Math. Biol. 37, 323 (1975)
L. von Bertalanffi: Blätter für Deutsche Philosophie 18. Nr. 3 and 4 (1945); Science 111, 23 (1950); Brit. J. Phil. Sci. 1, 134 (1950): Biophysik des Fließgleichgewichts ( Vieweg, Braunschweig 1953 )
G. Czajkowski: Z. Phys. 270, 25 (1974)
V. DeGiorgio, M. O. Scully: Phys. Rev. A 2, 117a (1970)
P. Glansdorff, I. Prigogine: Thermodynamic Theory of Structure, Stability and Fluctuations ( Wiley, New York 1971 )
R. Graham, H. Haken: Z. Phys. 213, 420 (1968); 237, 31 (1970)
H. Haken: Z. Phys. 181, 96 (1964)
M. Herschkowitz-Kaufman: Bull. Math. Biol. 37, 589 (1975)
K. H. Janssen: Z. Phys. 270, 67 (1974)
G. J. Klir: The Approach to General Systems Theory ( Van Nostrand Reinhold Comp., New York 1969 )
G. J. Klir, ed.: Trends in General Systems Theory ( Wiley, New York 1972 )
R. Landauer: IBM J. Res. Dev. 5, 3 (1961); J. Appl. Phys. 33, 2209 (1962); Ferroelectrics 2, 47 (1971)
E. Laszlo (cd.): The Relevance of General Systems Theory ( George Braziller, New York 1972 )
I. Matheson, D. F. Walls, C. W. Gardiner: J. Stat. Phys. 12, 21 (1975)
A. Nitzan, P. Ortoleva, J. Deutch, J. Ross: J. Chem. Phys. 61, 1056 (1974)
Prigogine, G. Nicolis: J. Chem. Phys. 46, 3542 (1967)
I. Prigogine, R. Lefever: J. Chem. Phys. 48, 1695 (1968)
A. M. Turing: Phil. Trans. Roy. Soc. B 234, 37 (1952)
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Haken, H. (2004). An Introduction. In: Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10184-1_1
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