Abstract
For the statistical behavior of macrovariables described in terms of Langevin equations with a in general colored random force we deduce useful formulas which simplify the calculation of correlation functions. Utilizing these results and the stochastic properties of the random force we derive an exact time-convolutionless masterequation for the probability hereby showing the mathematical equivalence of the formally different approaches of a Langevin description and a masterequation description. We study in detail the class of time-instantaneous Langevin equations and the important class of retarded (Mori-type) Langevin equations with both, Gaussian and general colored random forces. Using the generalization of the nonlinear Langevin equation for continuous Markov processes with white Gaussian noise and white generalized Poisson noise we show that the resulting masterequation can be recast in the Kramers-Moyal form. Interpreting this Langevin equation in the Stratonovitch sense we deduce the fluctuation induced drift (spurious drift) which can be divided up into two parts, the well known part induced by white Gaussian noise and the one induced by white generalized Poisson noise.
Similar content being viewed by others
References
Mori, H.: Prog. Theor. Phys.33, 423 (1965) Mori, H., Fujisaka, H.: Prog. Theor. Phys.49, 764 (1973)
Nordholm, K.S.J., Zwanzig, R.: J. Stat. Phys.13, 347 (1975)
Kawasaki, K., Gunton, J.D.: Phys. Rev. A8, 2048 (1973)
Grabert, H.: Z. Physik B27, 95 (1977)
Graham, R.: Springer Tracts in Modern Physics66, 1 (1973)
Van Kampen, N.G.: Adv. Chem. Phys.34, 245 (1976)
Kramers, H.A.: Physica7, 284 (1940) Moyal, J.E.: J. Roy. Stat. Soc. B11, 150 (1949)
Alekseev, V.V.: Sov. Phys. Usp.19, 1007 (1976)
Dibrov, B.F., Livshits, M.A., Volkenstein, M.V.: J. Theor. Biol.69, 23 (1977)
Hänggi, P., Thomas, H., Grabert, H., Talkner, P.: J. Stat. Phys.18, 155 (1978)
Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals, paragraph 12. New York: McGraw Hill 1965
Grabert, H., Hänggi, P., Talkner, P.: Z. Physik B26, 398 (1977)
Bedeaux, D.: Phys. Lett.62A, 10 (1977)
Clark, J.M.C.: Representation of Nonlinear Stochastic Systems with Applications to Filtering Theory. Ph. D Thesis, Imperial College, London, 1966
Wong, E., Zakai, M.: Ann. Math. Stat.36, 1560 (1965)
Arnold, L.: Stochastic Differential Equations. New York: J. Wiley 1974
Kubo, R.: Rep. Prog. Phys.29, 255 (1966)
Stratonovitch, R.L.: Topics in the Theory of Random Noise, Vol. I. New York: Gordon & Breach 1963
Gerlich, G.: Physica82A, 477 (1977)
Hänggi, P., Thomas, H.: Z. Physik B26, 85 (1977)
Leibowitz, M.A.: J. Math. Phys.4, 852 (1963)
Author information
Authors and Affiliations
Additional information
This work has been supported by a grant of the Volkswagenstiftung
Rights and permissions
About this article
Cite this article
Hänggi, P. Correlation functions and masterequations of generalized (non-Markovian) Langevin equations. Z Physik B 31, 407–416 (1978). https://doi.org/10.1007/BF01351552
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01351552