Application of the direct interaction approximation in turbulence theory to generalized stochastic models

The purpose of this paper is to seek mathematical insights into Kraichnan's direct-interaction approximation (DIA) in turbulence theory via its application to generalized stochastic models. Previous developments [R. H. Kraichnan, J. Math. Phys. 2, 124-148 (1961); Phys. Fluids 8, 575-598 (1965); B. K. Shivamoggi et al., J. Math. Anal. Appl. 229, 639-658 (1999); B. K. Shivamoggi and N. Tuovila, Chaos 29, 063124 (2019)] were based on the Boltzmann-Gibbs prescription for the underlying entropy measure, which exhibits the extensivity property and is suited for ergodic systems. Here, we proceed further and consider the introduction of an influence bias discriminating rare and frequent events explicitly, as it behooves non-ergodic systems by a using a Tsallis type [C. Tsallis, J. Stat. Phys. 52, 479-487 (1988)] autocorrelation model with an underlying non-extensive entropy measure. As an example of this development, we consider a linear damped stochastic oscillator system and explore the Markovian and non-Markovian regimes separately using Keller's perturbative and the DIA non-perturbative procedures. We find that, in the asymptotic regimes of short-range (white-noise) and long-range (black-noise) autocorrelations, the Tsallis model yields the same result as the Uhlenbeck-Ornstein model. Furthermore, the non-perturbative aspects excluded by Keller's perturbative procedure are found to be negligible in these asymptotic regimes. During the course of this investigation, we also exhibit some apparently novel mathematical properties of the stochastic models in question-the gamma distribution and the Tsallis non-extensive entropy.