标题： 广义随机模型在湍流问题中的直接相互作用近似 显示英文标题

标题： Direct Interaction Approximation for generalized stochastic models in the turbulence problem

摘要： 本文的目的是考虑Kraichnan开发的直接相互作用近似（DIA）在湍流问题中的广义随机模型的应用。 先前的发展基于Boltzmann-Gibbs对底层熵度量的规定，该度量表现出可扩展性特性，适用于遍历系统。 在这里，我们考虑引入一种影响偏差，明确区分罕见事件和频繁事件，这适用于非遍历系统，通过使用具有底层非可扩展熵度量的Tsallis类型自相关模型来处理。 作为例子，我们考虑一个线性阻尼随机振子系统，并描述由此产生的随机过程。 Keller的微扰程序排除的非微扰方面在白噪声极限中被发现被最小化。 在相反的极限情况下，随机过程模型之间的物理方差似乎没有显现出来，Uhlenbeck-Ornstein和Tsallis类型模型被发现产生相同的结果。 在此过程中，我们还推导了一些与当前研究相关的随机模型的明显新颖的数学性质——伽马分布和Tsallis非可扩展熵。 显示英文摘要

摘要： 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.