Time scale separation in superstatistical complex systems

复杂系统往往表现出一种动力学，可以看作是几个动力学在不同时间尺度上的叠加。一个简单的例子是在非均匀环境中运动的布朗粒子，它在相对大的尺度上表现出空间和时间的温度波动。有两个相关的随机过程的叠加，一个是由粒子的速度给出的快速过程，一个是描述环境变化的慢得多的随机过程。人们通常称这种类型的系统为“超统计”，因为它们由两种统计的叠加组成，一种是由普通统计力学描述的快速统计，另一种是描述环境变化的慢得多的统计。超统计量是非常普遍的，最近已被应用于各种复杂系统，包括流体动力湍流、模式形成非平衡系统、太阳耀斑、宇宙射线、风速波动、水文气候波动、股价演变、随机网络和随机矩阵理论。这项研究计划的目的是双重的。在理论方面，目标是发展一种广义的统计力学形式，以一种有效的方式描述上述类型的大量复杂系统。与其考虑复杂系统的每一个细节，不如用很少的相关变量寻求有效的描述。为此，热力学的方法是一般化的：人们从更一般的熵函数开始，考虑到环境的变化（或者，一般来说，相关系统参数的大规模波动）。扩展理论还考虑了局部系统松弛到平衡状态的速度，从而描述了有限时间尺度的分离效应。在应用方面，目的是将上述理论应用于由不同复杂系统（模式形成的颗粒气体、癫痫发作期间的大脑活动、日本和加利福尼亚的地震活动、股价指数的演变、湍流中的速度差异）产生的大量时间序列。我们将研究哪些超统计现象是普遍存在的。独立于所研究的复杂系统的细节)，并且是特定于特定系统的。可能的普适性分类将直接从数据中提取出来。将开发特定应用的模型来解释观察到的缓慢变化的系统参数的概率分布。

Complex system often exhibit a dynamics that can be regarded as superpositionof several dynamics on different time scales.A simple example is a Brownian partice that moves in an inhomogeneousenvironment which exhibits temperature fluctuations in space and time on a relatively large scale. There is a superposition of two relevant stochastic processes,a fast one given by the velocity of the particle and a much slower onedescribing changes in the environment. It has become common to call thesetypes of systems 'superstatistical' since they consist of a superposition of twostatistics, a fast one as described by ordinary statistical mechanicsand a much slower one describing changes of the environment. The superstatistics is very general and has been recently applied to a variety of complex systems, including hydrodynamicturbulence, pattern forming nonequilibrium systems, solar flares, cosmic rays,wind velocity fluctuations, hydro-climatic fluctuations, share price evolution,random networks and random matrix theory.The aim of the research proposal is twofold.On the theoretical side, the aim is to develop a generalisedstatistical mechanics formalism that describes a large variety of complexsystems of the above type in an effective way. Rather thantaking into account every detail of the complex system, one seeksfor an effective description with few relevant variables. For thisthe methods of thermodynamics are generalised:One starts with more general entropy functionsthat take into account changes of the environment(or, in general, large-scale fluctuations of a relevant system parameter) as well. An extended theory also takes into account how fast the local system relaxes to equilibrium,thus describing finite time scale separation effects.On the applied side, the aim is to apply the above theory to a large variety of time series generated by different complexsystems (pattern forming granular gases, brain activityduring epileptic seizures, earthquake activity in Japan and California, evolutionof share price indices, velocity differences in turbulent flows).It will be investigated which superstatistical phenomena are universal(i.e. independent of details of the complex system studied) and whichare specific to a particular system. Possible universality classeswill be extracted directly from the data. Application-specific modelswill be developed to explain the observed probability distributionsof the slowly varying system parameters.