Weak turbulence has been investigated in nonlinear-nonequilibrium physics to understand universal characteristics near the transition point of order to disorder states. Here the one-dimensional Nikolaevskii turbulence, which is a mathematical model of weak turbulence, was studied theoretically. We calculated the velocity field of the Nikolaevskii turbulence assuming convective structure and carried out tagged-particle simulations in the flow to clarify the Nikolaevskii turbulence from the Lagrangian description. The tagged-particle diffuses in the disturbed flow, and the diffusion was superdiffusive in an intermediate timescale between ballistic and normal-diffusive scale. The diffusion of the slow structure is characterized by the power-law for the control parameter near the transition point of the Nikolaevskii turbulence, suggesting that the diffusive characteristics of the slow structure hold scale-invariance. We proposed a simplified model named two-scale Brownian motion, which reveals a hierarchy in the Nikolaevskii turbulence.

中文翻译：

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尼古拉耶夫斯基湍流中的慢扩散结构

在非线性非平衡物理学中已经研究了微弱的湍流，以了解在从无序到有序状态转变点附近的普遍特征。在理论上研究了一维尼古拉耶夫斯基湍流，它是弱湍流的数学模型。我们计算了假定为对流结构的尼古拉耶夫斯基湍流的速度场，并在流中进行了标记粒子模拟，以根据拉格朗日描述来阐明尼古拉耶夫斯基湍流。带标记的粒子在扰动流中扩散，并且扩散在弹道和正扩散尺度之间的中间时间尺度上是超扩散的。慢速结构的扩散特征在于尼古拉耶夫斯基湍流过渡点附近的控制参数的幂律，这表明慢速结构的扩散特性具有尺度不变性。我们提出了一个简化的模型，称为两尺度布朗运动，揭示了尼古拉耶夫斯基湍流中的层次结构。