From time series to superstatistics

Abstract

Complex nonequilibrium systems are often effectively described by a 'statistics of a statistics', in short, a 'superstatistics'. We describe how to proceed from a given experimental time series to a superstatistical description. We argue that many experimental data fall into three different universality classes: χ 2 -superstatistics (Tsallis statistics), inverse χ 2 -superstatistics, and log-normal superstatistics. We discuss how to extract the two relevant well separated superstatistical time scales τ and T , the probability density of the superstatistical parameter β, and the correlation function for β from the experimental data. We illustrate our approach by applying it to velocity time series measured in turbulent Taylor-Couette flow, which is well described by log-normal superstatistics and exhibits clear time scale separation.

Superstatistics, thermodynamics, and fluctuations

“…A superstatistical system is characterized additionally by the condition that the local relaxation time of the system is short compared to the typical time scale of changes of β , so that each cell can be formally assumed to be in local equilibrium. Sometimes this property will be satisfied for a given complex system, sometimes not [9]. For our approach to be applicable, we must have sufficiently large separation of these two time scales.…”

“…[1], a lot of efforts have been made for further theoretical elaboration [5][6][7][8][9][10][11][12]. At the same time, it has also been applied successfully to a variety of systems and phenomena, including hydrodynamic turbulence [9,13,14], pattern formation [15], cosmic rays [16], solar flares [17], mathematical finance [18][19][20], random matrices [21], complex networks [22], wind velocity fluctuations [23], and hydro-climatic fluctuations [24].…”

Superstatistics: Theoretical Concepts and Physical Applications

“…Fig. 1 shows the function κ(∆t) for an example of a time series that has been studied in [2], the longitudinal velocity difference u(t) = v(t + δ) − v(t) in a turbulent Taylor-Couette flow on a scale δ. For each scale δ the relevant superstatistical time scale T leading to locally Gaussian behaviour can be extracted as the intersection with the line κ = 3.…”

“…Then there is a relatively fast dynamics given by the velocity of the Brownian particle and a slow one given by the temperature changes of the environment, which is spatiotemporally inhomogeneous. The two effects produce a superposition of two statistics, or in a short, a 'superstatistics' [1,2,3,4,5,6,7,8]. The concept of a superstatistics was introduced in [1], in the mean time many applications for a variety of complex systems have been pointed out [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].…”

“…Recent successful applications include hydrodynamic turbulence [9,10,2,11,12], pattern forming systems [14], cosmic rays [15], solar flares [13], mathematical finance [17,18,19], random matrix theory [20], networks [21], quantum systems at low temperatures [23], wind velocity fluctuations [16,25], hydro-climatic fluctuations [22] and delay statistics in traffic models [24]. The aim in the following is to explain the basic concepts and applications in an easy-going way.…”