Appendix: The Master Equation

  • Günter Haag
Part of the Studies in Operational Regional Science book series (SORS, volume 6)


The master equation provides a fairly general mathematical method for describing the time development of any complex system (see Weidlich and Haag1). Before going into details of its structure, some examples will be given that illustrate the scope of its applications which ranges from physics, chemistry and biology to economics (Haken2,3) sociology and psychology. We shall also remark on the relation between deterministic and probabilistic descriptions of systems.


Transition Rate Master Equation Detailed Balance Multinomial Logit Model Decision Behaviour 
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  1. 1.
    W. Weidlich, G. Haag, Concepts and Mode/s of a Quantitative Sociology, The Dynamics of Interacting Populations, Springer Ser. Synergetics, Vol. 14 (Springer, Berlin, Heidelberg, New York, 1983 ).Google Scholar
  2. 2.
    H. Haken, Synergetics, an Introduction, 2nd. ed., Springer Series Synergetics, Vol. 1 ( Springer, Berlin, Heidelberg, New York, 1977 ).Google Scholar
  3. 3.
    H. Haken, Advanced Synergetics, Instability Hierarchies on Self-Organizing Systems and Devices, Springer Series in Synergetics, Vol. 20 ( Springer, Berlin, Heidelberg, New York, 1983 ).Google Scholar
  4. 4.
    R. L. Stratonovich, Topics in the Theory of Random Noice, Vol. 1 and 2 (Gordon and Breach, New York 1963 and 1967).Google Scholar
  5. 5.
    A.T. Bharucha-Reid, Elements of the Theory of Markov Processes and their Applications (Mc-Graw-Hill, New York 1960).Google Scholar
  6. 6.
    N. Wax (ed.), Selected Papers on Noise and Stochastic Processes ( Dover, New York 1954 ).Google Scholar
  7. 7.
    C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Ser. Synergetics, Vol. 13 ( Springer, Berlin, Heidelberg, New York 1983 ).Google Scholar
  8. 8.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry ( North-Holland, Amsterdam 1981).Google Scholar
  9. 9.
    G. Haag, W. Weidlich, P. Alber, Approximation Methods for Stationary Master Equations, Z. Physik B 26, 207–215 (1977).CrossRefGoogle Scholar
  10. 10.
    G. Haag, Transition Factor Method for Discrete Master Equations and Application to Chemical Reactions, Z. Physik B 29, 153–159 (1978).CrossRefGoogle Scholar
  11. 11.
    G. Haag, P. Hänggi, Exact Solution of Discrete Master Equations in Terms of Continued Fractions, Z. Physik B 34, 411–417 (1979).CrossRefGoogle Scholar
  12. 12.
    P. Hänggi, G. Haag, Continued Fraction Solutions of Discrete Master Equations not Obeying Detailed Balance, Z. Physik B 39, 269–279 (1980).CrossRefGoogle Scholar
  13. 13.
    W. Weidlich, On the Structure of Exact Solutions of Discrete Master Equations, Z. Physik B 30, 345 (1978).CrossRefGoogle Scholar
  14. 14.
    H. Haken, The Stationary Solution of the Master Equation for Detailed Balance, Phys. Letters 46A, 7 (1974).Google Scholar
  15. 15.
    A. de Palma, Individual Decision-Making in Dynamic Collective Systems, Part 1 and Part 2,Journal of Mathematical Sociology, forthcoming (1987).Google Scholar
  16. 16.
    A. de Palma, Cl. Lefevre, Simplification Procedures for a Probabilistic Choice Model, Journal of Mathematical Sociology, 8, 43–60 (1981).CrossRefGoogle Scholar
  17. 17.
    T. Domencich, D. McFadden, Urban Travel Demand: A Behavioural Analysis, (North Holland, Amsterdam, 1975 ).Google Scholar
  18. 18.
    G. Leonardi, Trainsient and Asymptotic Behaviour of a Random Utility Based Stochasitic Search Process in Continuous Space and Time, Collaborative Paper, WP-83-, Laxenburg (1983).Google Scholar
  19. 19.
    W. Weidlich, G. Haag (eds.), Interregional Migration, Dynamic Theory and Comparative Analysis (Springer, Berlin, Heidelberg, New York, 1988).Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Günter Haag
    • 1
  1. 1.Deutsche Physikalische GesellschaftGermany

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