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Title: Dynamics of Nonlinear Stochastic Systems

Abstract

A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum-mechanical many-body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive-definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse-response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrodinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine themore » average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.« less

Authors:
Publication Date:
Research Org.:
New York Univ., New York
Sponsoring Org.:
USDOE
OSTI Identifier:
4098774
NSA Number:
NSA-15-009324
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 2; Journal Issue: 1; Other Information: Orig. Receipt Date: 31-DEC-61; Journal ID: ISSN 0022-2488
Publisher:
American Institute of Physics (AIP)
Country of Publication:
Country unknown/Code not available
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; EQUATIONS; FREQUENCY; GREEN FUNCTION; MANY BODY PROBLEM; OSCILLATIONS; PERTURBATION THEORY; PULSES; QUANTUM MECHANICS; SCHROEDINGER EQUATION; SPECTRA; TURBULENCE

Citation Formats

Kraichnan, Robert H. Dynamics of Nonlinear Stochastic Systems. Country unknown/Code not available: N. p., 1961. Web. doi:10.1063/1.1724206.
Kraichnan, Robert H. Dynamics of Nonlinear Stochastic Systems. Country unknown/Code not available. https://doi.org/10.1063/1.1724206
Kraichnan, Robert H. 1961. "Dynamics of Nonlinear Stochastic Systems". Country unknown/Code not available. https://doi.org/10.1063/1.1724206.
@article{osti_4098774,
title = {Dynamics of Nonlinear Stochastic Systems},
author = {Kraichnan, Robert H.},
abstractNote = {A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum-mechanical many-body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive-definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse-response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrodinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine the average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.},
doi = {10.1063/1.1724206},
url = {https://www.osti.gov/biblio/4098774}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 1,
volume = 2,
place = {Country unknown/Code not available},
year = {Sun Jan 01 00:00:00 EST 1961},
month = {Sun Jan 01 00:00:00 EST 1961}
}