Non-Gaussian Statistics, Classical Field Theory, and Realizable Langevin Models
Authors: John A. Krommes
The direct-interaction approximation (DIA) to the fourth-order statistic Z ~ <(lj2(2>, where l is a specifed operator and j is a random field, is discussed from several points of view distinct from that of Chen et al. [Phys. Fluids A 1 , 1844 (1989)]. It is shown that the formula for ZDIA already appeared in the seminal work of Martin, Siggia, and Rose [Phys. Rev. A 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. 28, 145 (1971)] and Kraichnan [J. Fluid Mech. 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 ("spurious vertices") is described. It is shown how to derive an improved representation, that realizes cumu- lants through O(j4), by adding to the GLE a particular non-Gaussian correction. A Markovian approximation ZDIA to ZDIA is derived. Both ZDIA and ZDIA incorrectly predict a Gaussian kurtosis for the steady state of a solvable three-mode example.