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.
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References
Langevin Equations: An Example
For general approaches see R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon & Breach, New York-London 1963)
For general approaches see M. Lax: Rev. Mod. Phys. 32, 25 (1960);
For general approaches see M. Lax: Rev. Mod. Phys. 38, 358, 541 (1966);
For general approaches see M. Lax: 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)
Reservoirs and Random Forces
Here we present a simple example. For general approaches see R. Zwanzig: J. Stat. Phys. 9, 3, 215 (1973)
Here we present a simple example. For general approaches see H. Haken: Rev. Mod. Phys. 47, 67 (1975)
The Fokker-Planck Equation
For general approaches see R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon & Breach, New York-London 1963)
For general approaches see M. Lax: Rev. Mod. Phys. 32, 25 (1960);
For general approaches see M. Lax: Rev. Mod. Phys. 38, 358, 541 (1966);
For general approaches see M. Lax: Phys. Rev. 145, 110 (1966).
For general approaches see H. Haken: Rev. Mod. Phys. 47, 67 (1975)
with further references P. Hänggi, H. Thomas: Phys. Rep. 88, 208 (1982)
Some Properties and Stationary Solution of the Fokker-Planck Equation
The “potential case” is treated by 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)
The more general case for systems in detailed balance is treated by R. Graham: Z. Phys. B40, 149 (1981)
H. Risken: Z. Phys. 251, 231 (1972);
see also H. Haken: Rev. Mod. Phys. 47, 67 (1975)
Time-Dependent Solutions of the Fokker-Planck Equation
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)
Solution of the Fokker-Planck Equation by Path Integrals
L. Onsager, S. Machlup: Phys. Rev. 91, 1505, 1512 (1953)
I. M. Gelfand, A. M. Yaglome: J. Math. Phys. I, 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 16C, 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.
Phase Transition Analogy
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)
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)
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)
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)
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)
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)
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)
The modern and powerful renormalization group technique of Wilson is reviewed by K. G. Wilson, J. Kogut: Phys. Rep. 12C, 75 (1974)
The modern and powerful renormalization group technique of Wilson is reviewed by 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),
Phase Transition Analogy in Continuous Media: Space Dependent Order Parameter
a) References to Systems in Thermal Equilibrium
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
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)
Further papers on this evaluation are: M. Nauenberg, F. Kuttner, M. Fusman: Phys. Rev. A13, 1185 (1976)
b) References to Systems Far from Thermal Equilibrium (and Nonphysical Systems)
R. Graham, H. Haken: Z. Phys. 237, 31 (1970)
For related topics see H. Haken: Rev. Mod. Phys. 47, 67 (1975)
and the articles by various authors in H. Haken, ed.: Synergetics (Teubner, Stuttgart 1973)
and the articles by various authors in H. Haken, M. Wagner, eds.: Cooperative Phenomena (Springer, Berlin-Heidelberg-New York 1973)
and the articles by various authors in H. Haken, ed.: Cooperative Effects (North Holland, Amsterdam 1974)
and the articles by various authors in H. Haken (ed.): Springer Series in Synergetics Vols. 2–20 (Springer, Berlin-Heidelberg-New York)
Cooperative Effects in the Laser. Self-Organization and Phase Transition
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)
The Laser Equations in the Mode Picture
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)
The Order Parameter Concept
Compare especially H. Haken: Rev. Mod. Phys. 47, 67 (1975)
The Single Mode Laser
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-Heidelberg-New 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)
The laser distribution function was derived by R. D. Hempstead, M. Lax: J. Phys. Rev. 161, 350 (1967)
For a fully quantum mechanical distribution function cf. W. Weidlich, H. Risken, H. Haken: Z. Phys. 201, 396 (1967)
For a fully quantum mechanical distribution function cf. M. Scully, W. E. Lamb: Phys. Rev. 159, 208 (1967):
For a fully quantum mechanical distribution function cf. M. Scully, W. E. Lamb: Phys. Rev. 166, 246 (1968)
The Multimode Laser
H. Haken: Z. Phys. 219, 246 (1969)
Laser with Continuously Many Modes. Analogy with Superconductivity
For a somewhat different treatment see R. Graham, H. Haken: Z. Phys. 237, 31 (1970)
First-Order Phase Transitions of the Single Mode Laser
J. F. Scott, M. Sargent III, C. D. Cantrell: Opt. Commun. 15, 13 (1975)
W. W. Chow, M. O. Scully, E. W. van Stryland: Opt. Commun. 15, 6 (1975)
Hierarchy of Laser Instabilities and Ultrashort Laser Pulses
We follow essentially H. Haken, H. Ohno: Opt. Commun. 16, 205 (1976)
We follow essentially 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);
For a machine calculation see H. Risken, K. Nummedal: 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. 51A, 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)
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)
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)
Some monographs in hydrodynamics: Chia-Shun-Yih: Fluid Mechanics (McGraw Hill, New York 1969)
Some monographs in hydrodynamics: G. K. Batchelor: An Introduction to Fluid Dynamics (University Press, Cambridge 1970)
Some monographs in hydrodynamics: S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford 1961)
Stability problems are treated particularly by Chandrasekhar l.c. and by C. C. Lin: Hydrodynamic Stability (University Press, Cambridge 1967)
Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations. 8.14 The Fokker-Planck Equation and Its Stationary Solution
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):
For related work see R. Graham: 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)
Higher instabilities are discussed by D. Ruelle, F. Takens: Comm. Math. Phys. 20, 167 (1971)
Higher instabilities are discussed by J. B. McLaughlin, P. C. Martin: Phys. Rev. A12, 186 (1975)
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)
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 Appl. 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.
A Model for the Statistical Dynamics of the Gunn Instability Near Threshold
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)
Elastic Stability: Outline of Some Basic Ideas
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)
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
This work was extended by Zhabotinsky and his coworkers in a series of papers A. N. Zaikin, A. M. Zhabotinsky: Nature 225, 535 (1970)
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)
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. 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.
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)
A detailed review over the whole field gives the Y. Schiffmann: Phys. Rep. 64, 88 (1980)
For chemical oscillations see especially G. Nicolis, J. Portnow: Chem. Rev. 73, 365 (1973)
Deterministic Processes, Without Diffusion, One Variable. 9.3 Reaction and Diffusion Equations
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)
React ion-Diffusion Model with Two or Three Variables; the Brusselator and the Oregonator
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)
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)
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)
The “Oregonator” model reaction was formulated and treated by R. J. Field, E. Korös, R. M. Noyes: J. Am. Chem. Soc. 49, 8649 (1972)
The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: Nature 237, 390 (1972)
The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: J. Chem Phys. 60, 1877 (1974)
The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: J. Am. Chem. Soc. 96, 2001 (1974)
Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable
A first treatment of this model is due to V. J. McNeil, D. F. Walls: J. Stat. Phys. 10, 439 (1974)
Stochastic Model for a Chemical Reaction with Diffusion. One Variable
The master equation with diffusion is derived by H. Haken: Z. Phys. B20, 413 (1975)
We essentially follow 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)
Related to this chapter are the papers by M. Malek-Mansour, G. Nicolis: preprint Febr. 1975
Stochastic Treatment of the Brusselator Close to Its Soft Mode Instability
We essentially follow H. Haken: Z. Phys. B20, 413 (1975)
Chemical Networks
Related to this chapter are G. F. Oster, A. S. Perelson: Chem. Reaction Dynamics. Arch. Rat. Mech. Anal. 55, 230 (1974)
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)
with further references. 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. 31a, 255 (1976)
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Haken, H. (1983). Chance and Necessity. In: Synergetics. Springer Series in Synergetics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88338-5_6
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