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

The direct-interaction approximation (DIA) to the fourth-order statistic $Z\sim 〈{\left(\lambda {\psi }^2\right)}^2〉$, where $\lambda$ is a specified operator and $\psi$ 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 $Z_{\mathrm{DIA}}$ 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, which realizes cumulants through $O\left({\psi }^4\right)$, by adding to the GLE a particular non-Gaussian correction. A Markovian approximation $Z_{\mathrm{DIA}}^M$ to $Z_{\mathrm{DIA}}$ is derived. Both $Z_{\mathrm{DIA}}$ and $Z_{\mathrm{DIA}}^M$ incorrectly predict a Gaussian kurtosis for the steady state of a solvable three-mode example.