Non-Gaussian statistics, classical field theory, and realizable Langevin models

Journal Article · Wed May 01 04:00:00 EDT 1996 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PhysRevE.53.4865· OSTI ID:283825

Krommes, J A ^[1]

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1. Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543 (United States)

The direct-interaction approximation (DIA) to the fourth-order statistic {ital Z}{approximately}{l_angle}({lambda}{psi}{sup 2}){sup 2}{r_angle}, where {lambda} is a specified operator and {psi} is a random field, is discussed from several points of view distinct from that of Chen {ital et} {ital al}. [Phys. Fluids A {bold 1}, 1844 (1989)]. It is shown that the formula for {ital Z}{sub DIA} already appeared in the seminal work of Martin, Siggia, and Rose [Phys. Rev. A {bold 8}, 423 (1973)] on the functional approach to classical statistical dynamics. It does not follow from the original generalized Langevin equation (GLE) of Leith [J. Atmos. Sci. {bold 28}, 145 (1971)] and Kraichnan [J. Fluid Mech. {bold 41}, 189 (1970)] (frequently described as an amplitude representation for the DIA), in which the random forcing is realized by a particular superposition of products of random variables. The relationship of that GLE to renormalized field theories with non-Gaussian corrections ({open_quote}{open_quote}spurious vertices{close_quote}{close_quote}) is described. It is shown how to derive an improved representation, which realizes cumulants through {ital O}({psi}{sup 4}), by adding to the GLE a particular non-Gaussian correction. A Markovian approximation {ital Z}{sub DIA}{sup {ital M}} to {ital Z}{sub DIA} is derived. Both {ital Z}{sub DIA} and {ital Z}{sub DIA}{sup {ital M}} incorrectly predict a Gaussian kurtosis for the steady state of a solvable three-mode example. {copyright} {ital 1996 The American Physical Society.}

• Research Organization: Princeton Plasma Physics Laboratory
• DOE Contract Number: AC02-76CH03073
• OSTI ID: 283825
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 5 Vol. 53; ISSN PLEEE8; ISSN 1063-651X
• Country of Publication: United States
• Language: English

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