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)


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. The equation of motion of the football is precisely given by Newton’s law: Mass · acceleration = force, i.e.,
$$ m \cdot \mathop{v}\limits^{ \bullet } = F $$


Correlation Function Stationary Solution Friction Force Order Phase Transition Langevin Equation 
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Langevin Equations: An Example For general approaches see

  1. R. L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1 (Gordon & Breach, New York-London 1963)Google Scholar
  2. M. Lax: Rev. Mod. Phys. 32, 25 (1960);ADSzbMATHCrossRefGoogle Scholar
  3. M. Lax: Rev. Mod. Phys. 38, 358, (1966);MathSciNetADSGoogle Scholar
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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
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The Fokker-Planck Equation Same references as for Section 6.1

  1. 6.4 Some Properties and Stationary Solution of the Fokker-Planck EquationGoogle Scholar
  2. 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
  3. The more general case for systems in detailed balance is treated by R. Graham, H. Haken: Z. Phys. 248, 289 (1971)MathSciNetGoogle Scholar
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  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 (1945)MathSciNetCrossRefGoogle Scholar
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Solution of the Fokker-Planck Equation by Path Integrals

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  8. 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 equilibrium 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
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  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 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 byADSCrossRefGoogle Scholar
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Phase Transition Analogy in Continuous Media: Space Dependent Order Parameter a) References to Systems in Thermal Equilibrium

  1. 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
  2. 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)ADSGoogle Scholar
  3. Further papers on this evaluation are: L. W. Gruenberg, L. Gunther: Phys. Lett. 38A, 463 (1972)ADSGoogle Scholar
  4. M. Nauenberg, F. Kuttner, M. Fusman: Phys. Rev. A 13, 1185 (1976)ADSCrossRefGoogle Scholar

References to Systems Far from Thermal Equilibrium (and Nonphysical Systems)

  1. R. Graham, H. Haken: Z. Phys. 237, 31 (1970) Furthermore the Chapters 8 and 9MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

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

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