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Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the master equation

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Zeitschrift für Physik B Condensed Matter

Abstract

We determine the path integral solution of a stochastic process described by a generalized Langevin equation with coordinate-dependent fluctuating forces and white spectrum. Since such equations do not permit a unique determination of the distribution function but require the Ito or Stratonovich prescription, we first pass over to the corresponding Fokker-Planck equation adopting such a prescription. By means of the one-dimensional case we show that the path integral solutions are not uniquely determined in form but allow for a class of equivalent representations. Adopting an especially suitable representation we then present the path integral solution of the multi-dimensional Fokker-Planck equation. The exponent of the exponential function occurring in that solution can be interpreted as generalization of the Onsager-Machlup function. Finally we give a path-integral solution of the master equation. In the “Fokker-Planck limit” again a generalized Onsager-Machlup function is obtained.

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Authors and Affiliations

  1. Institut für theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, D-7000, Stuttgart 80, Vaihingen, Federal Republic of Germany

    H. Haken

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  1. H. Haken
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Haken, H. Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the master equation. Z Physik B 24, 321–326 (1976). https://doi.org/10.1007/BF01360904

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  • DOI: https://doi.org/10.1007/BF01360904

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