Abstract
We consider a system composed of many subsystems which are coupled to individual reservoirs at different temperatures. We show how the solution of a many-dimensional Fokker-Planck equation may be reduced to a Fokker-Planck equation of dimensionn, wheren is the number of relevant constants of motion. We treat also a Fokker-Planck equation with continuously many variables and the time-dependent one. The usefulness of the present procedure to determine explicitly distribution functions is exhibited by several examples. If all temperatures are equal the Boltzman distribution function is obtained as a special case. Using the method of quantum-classical correspondence, the distribution function for quantum systems may be found.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
See e.g. the articles by Fröhlich, H., Graham, R., Großmann, S., Haken, H., Kawasaki, K., Kubo, M., Landauer, R., Woo, J. W. F., Mori, H., Prigogine, I., Lefever, R., Thomas, H., Wagner, M.: In: Synergetics, ed. Haken, H. Stuttgart: Teubner 1973, where many further references may be found
Graham, R., Haken, H.: Z. Physik213, 240 (1968);273, 31 (1970)
Haken, H.: In: Festkörperprobleme X, ed. O. Madelung. Braunschweig 1970
De Giorgio, V., Scully, M. O.: Phys. Rev.A2, 1170 (1970)
Großmann, S., Richter, P. H.: Z. Physik242, 458 (1970)
Dohm, V.: Solid State Comm.11, 1273 (1972), and to be published
Its usefulness in treating mathematical models of sociology was recently demonstrated by Weidlich, W., see l.c. [1]
for references see [9]
Ming, Chen Wang, Uhlenbeck, G. E.: Rev. Mod. Phys.17, 23 (1945)
Graham, R., Haken, H.: Z. Physik243, 289 (1971).
See also Graham, R.: In: Springer tracts on physics, in press. For important special cases see Stratonovich, R. L.: Topics in the theory of random noise. New York: Gordon and Breach 1963.
Haken, H.: Z. Physik219 246 (1969).
Wöhrstein, H.: (part of a thesis, Stuttgart) has found, that perturbation theory is applicable, ifL1 fulfills the conditions of detailed balance.
See e.g. Graham, R., Haken, H.: Z. Physik234, 193 (1970);235, 166 (1970)
For a review see Haken, H.: Encyclopedia of physics, vol. XXV/2c. Berlin-Heidelberg-New York: Springer 1970
Weidlich, W., Haake, F.: Z. Physik185, 30 (1965);186, 203 (1965); and in particular Lax, M., Louisell, W.: IEEE J. Quantum Electronics QE3, 47 (1965)
Two-level system: Haken, H., Risken, H., Weidlich, W.: Z. Physik206, 573 (1967)
Haken, H.: Z. Physik219, 411 (1969)
l.c. [9] and Haken, H., Vollmer, H. D.: Z. Physik242, 416 (1971)
This problem including quantum mechanical fluctuations has been treated by Haken, H.: Z. Physik181, 96 (1964). For a detailed account for the theory initiated by that paper see l.c. [9]. For more recent publications see Hepp, K., Lieb, E., to be published in Helv. Phys. Acta, and Dohm, V. l.c. [2]. Dohm treats this problem using a method equivalent to degenerate perturbation theory applied to thedensity matrix equation
Haken, H., Wöhrstein, H.: Optics Comm. in press
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haken, H. Distribution function for classical and quantum systems far from thermal equilibrium. Z. Physik 263, 267–282 (1973). https://doi.org/10.1007/BF01391586
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01391586