Direct Interaction Approximation for generalized stochastic models in the turbulence problem

TL;DR

This paper extends the direct interaction approximation (DIA) to non-ergodic turbulence models using Tsallis entropy, analyzing a stochastic oscillator and revealing new mathematical properties of related models.

Contribution

It introduces a non-extensive entropy-based DIA framework for turbulence, addressing non-ergodic systems with influence bias and comparing it to traditional models.

Findings

Non-perturbative effects are minimized in the white-noise limit.

Tsallis and Uhlenbeck-Ornstein models yield similar results in certain limits.

New mathematical properties of gamma distribution and Tsallis entropy are identified.

Abstract

The purpose of this paper is to consider the application of the direct interaction approximation (DIA) developed by Kraichnan to generalized stochastic models in the turbulence problem. Previous developments 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 consider the introduction of an influence bias discriminating rare and frequent events explicitly, as it behooves non-ergodic systems, which is dealt with by a using a Tsallis type autocorrelation model with an underlying non-extensive entropy measure. As an example, we consider a linear damped stochastic oscillator system, and describe the resulting stochastic process. The non-perturbative aspects excluded by Keller's perturbative procedure are found to be minimized in the white-noise limit. In the opposite limit, the physical variances between the random process models don't seem to materialize, and the Uhlenbeck-Ornstein and Tsallis type models are found to yield the same result. In the process, we also deduce some apparently novel mathematical properties of the stochastic models associated with the present investigation -- the gamma distribution and the Tsallis non-extensive entropy.