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Direct Interaction Approximation for generalized stochastic models in the turbulence problem

Bhimsen Shivamoggi, Nicole Tuovila

TL;DR

This work generalizes the direct interaction approximation (DIA) to non-ergodic stochastic systems by incorporating Tsallis non-extensive entropy and a Tsallis-type autocorrelation with gamma-distributed mixing parameters. Using a linear damped stochastic oscillator as a testbed, the authors compare Keller's perturbative approach with DIA across Markovian and non-Markovian regimes, deriving analytic expressions for the relevant Green-function-like quantities and their Laplace transforms. They show that in the white-noise (large-$\lambda$) limit, non-perturbative features are minimized and Tsallis/Uhlenbeck-Ornstein models converge, while in other limits the models exhibit quantitatively similar results with gamma-distribution and Tsallis entropy underpinning the stochastic structure. The findings advance a framework for applying DIA to non-ergodic turbulence problems, highlighting the roles of non-extensive statistics and compound-Gamma autocorrelations in shaping temporal correlations and response functions.

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.

Direct Interaction Approximation for generalized stochastic models in the turbulence problem

TL;DR

This work generalizes the direct interaction approximation (DIA) to non-ergodic stochastic systems by incorporating Tsallis non-extensive entropy and a Tsallis-type autocorrelation with gamma-distributed mixing parameters. Using a linear damped stochastic oscillator as a testbed, the authors compare Keller's perturbative approach with DIA across Markovian and non-Markovian regimes, deriving analytic expressions for the relevant Green-function-like quantities and their Laplace transforms. They show that in the white-noise (large-) limit, non-perturbative features are minimized and Tsallis/Uhlenbeck-Ornstein models converge, while in other limits the models exhibit quantitatively similar results with gamma-distribution and Tsallis entropy underpinning the stochastic structure. The findings advance a framework for applying DIA to non-ergodic turbulence problems, highlighting the roles of non-extensive statistics and compound-Gamma autocorrelations in shaping temporal correlations and response functions.

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.

Paper Structure

This paper contains 25 sections, 3 theorems, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tsallis entropy model
  3. Tsallis autocorrelation model via compound statistics of the Uhlenbeck-Ornstein model for a random process
  4. Linear damped stochastic oscillator
  5. The Markovian Model
  6. The long-range autocorrelation case
  7. Perturbative procedure
  8. DIA Procedure
  9. The short-range autocorrelation case
  10. Perturbative procedure
  11. DIA procedure
  12. The large time limit
  13. The Non-Markovian Model
  14. The long-range autocorrelation case
  15. Perturbative procedure
  16. ...and 10 more sections

Key Result

Lemma 1

In the zero dispersion limit $c \to \infty$, the gamma distribution reduces to $\delta(\lambda - \lambda_0)$,

Figures (4)

  • Figure 1: Plot showing $1- (\lambda/\lambda_0)+\ln(\lambda /\lambda_0) < 0$.
  • Figure 2: The Gamma distribution as a function of the dispersion parameter $c$.
  • Figure 3: Plot showing $I_q(t)$ vs $q$. Here the parameter $\lambda =0.1$, but the behavior is unchanged when varying $\lambda$.
  • Figure 4: Plots of $I_q(t)$ vs $\lambda$ for $q \approx 1$.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof