Abstract:
A solution is found for the mean value of the Green's function of a stochastic linear system of general form with Gaussian fluctuating parameters. The method is based on constructing higher approximations of the Dyson equation by closing the chains of equations for the mean values of the variational derivatives of the solution at a certain step. It is shown that for the case of exponentially correlated fluctuations of the parameters of the system, the exact solution of the Dyson equation can be represented as an infinite continued fraction. The results are illustrated by the finding of the dynamical characteristics of an harmonic oscillator which has fluctuations of the eigenfrequency and the losses.
Citation:
O. V. Muzychuk, “Construction of an exact solution of the Dyson equation for the mean value of the Green's function”, TMF, 28:3 (1976), 371–380; Theoret. and Math. Phys., 28:3 (1976), 851–857
\Bibitem{Muz76}
\by O.~V.~Muzychuk
\paper Construction of an exact solution of the Dyson equation for the mean value of the Green's function
\jour TMF
\yr 1976
\vol 28
\issue 3
\pages 371--380
\mathnet{http://mi.mathnet.ru/tmf4273}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=449269}
\transl
\jour Theoret. and Math. Phys.
\yr 1976
\vol 28
\issue 3
\pages 851--857
\crossref{https://doi.org/10.1007/BF01029178}
Linking options:
https://www.mathnet.ru/eng/tmf4273
https://www.mathnet.ru/eng/tmf/v28/i3/p371
This publication is cited in the following 15 articles:
Alexander A. Dubkov, “Statistical time-reversal symmetry and its physical applications”, Chemical Physics, 375:2-3 (2010), 364
O. V. Muzychuk, “Effective frequency characteristic of a linear system with intensive nonwhite fluctuations of parameters”, Radiophys Quantum Electron, 33:3 (1990), 237
R. V. Bobrik, “Hierarchies of moment equations for the solution of the Schrödinger equation with random potential and their closure”, Theoret. and Math. Phys., 68:2 (1986), 841–847
U. Behn, “Tight Binding Quasiparticle Motion in a Poisson cOrrelated Discrete Stochastic Potential. The Density of States”, Physica Status Solidi (b), 113:1 (1982), 219
Yu. N. Barabanenkov, M. I. Kalinin, “Application of an undecoupled Dyson-type equation for estimating the error of the one-group approximation in the theory of conservative stochastic linear dynamical systems”, Radiophys Quantum Electron, 25:6 (1982), 459
G. N. Bochkov, A. A. Dubkov, “Functional statistical analysis of non-Gaussian parametric systems. I”, Radiophys Quantum Electron, 24:12 (1981), 996
G. I. Babkin, V. I. Klyatskin, “Analysis of the Dyson equation for stochastic integral equations”, Theoret. and Math. Phys., 41:3 (1979), 1080–1086
L. A. Apresyan, “Pade approximants (review)”, Radiophys Quantum Electron, 22:6 (1979), 449
O. V. Muzychuk, “Statistical description of linear systems with parameter fluctuations which are not delta-correlated”, Radiophys Quantum Electron, 22:10 (1979), 863
V. I. Klyatskin, “Markov processes, correlations of functionals, and stochastic equations”, Radiophys Quantum Electron, 22:6 (1979), 495
S. E. Pitovranov, V. N. Chetverikov, “Corrections to the diffusion approximation in stochastic differential equations”, Theoret. and Math. Phys., 35:2 (1978), 415–422
O. V. Muzychuk, “Statistical means in dynamic systems with a certain type of non-Gaussian parameter fluctuations”, Radiophys Quantum Electron, 21:2 (1978), 148
A. N. Malakhov, O. V. Muzychuk, I. E. Pozumentov, “Differential description of stochastic linear systems with nonwhite parameter fluctuations”, Radiophys Quantum Electron, 21:9 (1978), 887
A. N. Malakhov, O. V. Muzychuk, “Moment and cumulant functions of stochastic linear systems”, Radiophys Quantum Electron, 21:1 (1978), 47
A. A. Dubkov, O. V. Muzychuk, “Analysis of higher approximations of Dyson's equation for the mean value of the Green function”, Radiophys Quantum Electron, 20:6 (1977), 623
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