Abstract
The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the “self-consistent field” approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.
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References
R. H. Kraichnan, Dynamics of nonlinear stochastic systems,J. Math. Phys. 2:124–148 (1961).
Y. M. Chen, On the scattering of waves by random distributions,J. Math. Phys. 5 (11): 1541–1546 (Nov. 1964).
V. Twersky, On the scattering of waves by random distributions,J. Math. Phys. 3:700–715, 724–734 (1962).
G. Adomian, Linear stochastic operators, Ph.D. Dissertation (Physics), UCLA (1961), Chapter 4.
J. B. Keller, Stochastic equations and wave propagation in random media, inAnnual Symposia in Applied Mathematics, Vol. XVI (Amer. Math. Soc., 1964).
J. C. Samuels and A. C. Eringen, On stochastic linear systems,J. Math. Phys. 38:83–103 (1959).
A. T. Bharucha-Reid, On the theory of random equations, inAnnual Symposia in Applied Mathematics, Vol. XVI (Amer. Math. Soc., 1964), pp. 40–69.
A. T. Bharucha-Reid,Random Equations (Academic Press, New York, in publication).
R. Syski, Stochastic differential equations, Chapter 8 inModern Nonlinear Equations, T. L. Saaty, Ed. (McGraw-Hill, New York, 1967).
J. M. Richardson, The application of truncated hierarchy techniques in the solution of a stochastic linear differential equation,Annual Symposia in Applied Mathematics, Vol. XVI (Amer. Math. Soc., 1964).
G. Adomian, Linear stochastic operators,Rev. Mod. Phys. 35 (1): 185–207 (Jan. 1963).
G. Adomian, Linear random operator equations in mathematical physics, Part I,J. Math. Phys. 11 (3): 1069–1084 (March 1970); Parts II and III to appear in Sept. 1971 and Oct. 1971.
G. Adomian, Theory of random systems, inProc. Fourth Internat. Conf. on Information Theory, Statistical Decision Functions, and Stochastic Processes (Czechoslovak Academy of Sciences, Prague, 1967).
G. Adomian, Stochastic Green's functions, inProc. of Annual Symposia in Applied Mathematics, Vol. XVI, Invited Paper (Amer. Math. Soc., 1964).
L. H. Sibul, Stochastic Green's functions and their relation to the resolvent kernels of integral equations, inProc. of the Fifth Allerton Conference on Circuit and Systems Theory (University of Illinois, 1967).
G. Adomian and L. H. Sibul, Propagation in a stochastic medium, forthcoming paper.
Klimontovich,The Statistical Theory of Nonequilibrium Processes in a Plasma (MIT Press, 1967).
L. H. Sibul, Application of linear stochastic operator theory, Ph.D. Dissertation, Pennsylvania State University (1968).
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Supported by the National Aeronautics and Space Administration (Grant NGR 11-003-020) and partially supported by the Office of Naval Research (Contract N 00014-69-A-0423 Themis).
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Adomian, G. The closure approximation in the hierarchy equations. J Stat Phys 3, 127–133 (1971). https://doi.org/10.1007/BF01019846
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DOI: https://doi.org/10.1007/BF01019846