Fokker-Planck Equation

  • Hannes Risken
Part of the Springer Series in Synergetics book series (SSSYN, volume 18)


As shown in Sects. 3.1, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (3.1, 31). For nonlinear Lange vin equations (3.67, 110) expectation values are much more difficult to obtain, so here we first try to derive an equation for the distribution function. As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [1.1] and Planck [1.2]: many review articles and books on the Fokker-Planck equation now exist [1.5–15].


Langevin Equation Diffusion Matrix Kolmogorov Equation Covariant Vector Riemann Curvature Tensor 
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  1. 4.1
    P. Hänggi: Helv. Phys. Acta 51, 183 (1978)MathSciNetGoogle Scholar
  2. 4.2
    F. J. Dyson: Phys. Rev. 75, 486 (1949)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 4.3
    R. F. Pawula: Phys. Rev. 162, 186 (1967)ADSCrossRefGoogle Scholar
  4. 4.4
    I. S. Gradshteyn, I. M. Ryzhik: Tables of Integrals, Series and Products (Academic, New York 1965) p. 338Google Scholar
  5. 4.5
    H. Haken: Z. Physik B24, 321 (1976)MathSciNetADSGoogle Scholar
  6. 4.6
    C. Wissel: Z. Physik B35, 185 (1979)MathSciNetADSGoogle Scholar
  7. 4.7
    L. Onsager, S. Machlup: Phys. Rev. 91, 1505, 1512 (1953)MathSciNetADSMATHGoogle Scholar
  8. 4.8
    R. Graham: Springer Tracts in Mod. Phys. 66, 1, (Springer, Berlin, Heidelberg, New York 1973)Google Scholar
  9. 4.9
    R. Graham: Z. Physik B26, 281 (1977)ADSGoogle Scholar
  10. 4.10
    R. Graham: In Stochastic Processes in Nonequilibrium Systems, Proc., Sitges (1978), Lecture Notes Phys. Vol. 84 (Springer, Berlin, Heidelberg, New York 1978) p. 83Google Scholar
  11. 4.11
    H. Leschke, M. Schmutz: Z. Physik B27, 85 (1977)MathSciNetADSGoogle Scholar
  12. 4.12
    U. Weiss: Z. Physik B30, 429 (1978)ADSGoogle Scholar
  13. 4.13
    H. Risken, H. D. Vollmer: Z. Physik B35, 313 (1979)MathSciNetADSGoogle Scholar
  14. 4.14
    H. D. Vollmer: Z. Physik B33, 103 (1979)ADSGoogle Scholar
  15. 4.15
    H. Margenau, G. M. Murphy: The Mathematics of Physics and Chemistry (Van Nostrand, Princeton, NJ 1964)MATHGoogle Scholar
  16. 4.16
    J. Mathews, R. Walker: Mathematical Methods of Physics (Benjamin, Menlo Park, CA 1973) Chap. 15Google Scholar
  17. 4.17
    A. Duschek, A. Hochrainer: Grundzüge der Tensorrechnung in analytischer Darstellung, Teil II Tensoranalysis (Springer, Wien 1950)Google Scholar
  18. 4.18
    R. Graham: Z. Physik B26, 397 (1977)ADSGoogle Scholar
  19. 4.19
    4.19 H. Grabert, R. Graham, M. S. Green: Phys. Rev. A21, 2136 (1980)MathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Hannes Risken
    • 1
  1. 1.Abteilung für Theoretische PhysikUniversität UlmUlmFed. Rep. of Germany

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