Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. This work presents the first detailed information-theoretic analysis of turbulence in such strongly interacting systems. The analysis involves both energy and entropy and elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models with triplet interactions of neighboring modes and show that it has quadratic conservation laws defined by the Fibonacci numbers. Depending on how the interaction time changes with the mode number, three types of turbulence were found: single direct cascade, double cascade, and the first-ever case of a single inverse cascade. We describe quantitatively how deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. We reveal where the information (entropy deficit) is encoded and disentangle the communication channels between modes, as quantified by the mutual information in pairs and the interaction information inside triplets.

中文翻译：

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斐波那契湍流

热平衡和湍流之间的区别从来没有如此显着，因为当二次不变量使平衡统计具有独立波动的模式时恰好是高斯的。这发生在两种截然不同但联系紧密的系统中：不可压缩流体动力学和共振相互作用波。这项工作首次对这种强相互作用系统中的湍流进行了详细的信息理论分析。该分析涉及能量和熵，并阐明了空间和时间在设置级联方向和沿其统计的变化方面的基本作用。我们引入了一个非常简单但丰富的离散模型系列，这些模型具有相邻模式的三重相互作用，并表明它具有由斐波那契数定义的二次守恒定律。根据相互作用时间如何随模式数变化，发现了三种类型的湍流：单直接级联、双级联和有史以来第一个单逆级联的情况。我们定量地描述了从热平衡一直到湍流级联的偏差如何使统计越来越非高斯，并找到单模概率分布的自相似形式。我们揭示了信息（熵赤字）的编码位置，并解开了模式之间的通信渠道，正如通过成对的互信息和三元组内的交互信息所量化的那样。我们定量地描述了从热平衡一直到湍流级联的偏差如何使统计越来越非高斯，并找到单模概率分布的自相似形式。我们揭示了信息（熵赤字）的编码位置，并解开了模式之间的通信渠道，正如通过成对的互信息和三元组内的交互信息所量化的那样。我们定量地描述了从热平衡一直到湍流级联的偏差如何使统计越来越非高斯，并找到单模概率分布的自相似形式。我们揭示了信息（熵赤字）的编码位置，并解开了模式之间的通信渠道，正如通过成对的互信息和三元组内的交互信息所量化的那样。