Modeling collective motion in nonconservative systems, such as granular materials, is difficult since a general microscopic-to-macroscopic approach is not available: there is no Hamiltonian, no known stationary densities in phase space, and not a known small set of relevant variables. Phenomenological coarse-grained models are a good alternative, provided that one has identified a few slow observables and collected a sufficient amount of data for their dynamics. Here we study the case of a vibrofluidized dense granular material. The experimental study of a tracer, dispersed into the media, showed evidence of many time scales: Fast ballistic, intermediate caged, slow superdiffusive, and very slow diffusive. A numerical investigation has demonstrated that a tracer's superdiffusion is related to slow rotating drifts of the granular medium. Here we offer a deeper insight into the slow scales of the granular medium, and we propose a phenomenological model for such a “secular” dynamics. Based upon the model for the granular medium, we also introduce a model for the tracer (fast and slow) dynamics, which consists in a stochastic system of equations for three coupled variables, and is therefore more refined and successful than previous models.

中文翻译：

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稠密的振动流化颗粒材料中的时间较慢。

由于无法使用一般的微观到宏观方法，因此很难在非保守系统（例如颗粒材料）中对集体运动进行建模：没有哈密顿量，相空间中没有已知的固定密度，也没有已知的少量相关变量。现象学的粗粒度模型是一个很好的选择，只要已识别出一些缓慢的可观测对象并为它们的动力学收集了足够的数据即可。在这里，我们研究了一种纤维化致密颗粒材料的情况。对示踪剂进行分散在介质中的实验研究显示出许多时间尺度的证据：快速弹道，中间笼状，缓慢超扩散和非常缓慢扩散。数值研究表明，示踪剂的超扩散与颗粒介质的缓慢旋转漂移有关。在这里，我们对粒状介质的慢尺度提供了更深入的了解，并为这种“长期”动力学提出了一种现象学模型。在粒状介质模型的基础上，我们还引入了示踪剂（快速和慢速）动力学模型，该模型包含三个耦合变量的随机方程组，因此比以前的模型更加完善和成功。