Universality classes of fluctuation dynamics in hierarchical complex systems

Abstract

A unified approach is proposed to describe the statistics of the short time dynamics of multiscale complex systems. The probability density function of the relevant time series (signal) is represented as a statistical superposition of a large time-scale distribution weighted by the distribution of certain internal variables that characterize the slowly changing background. The dynamics of the background is formulated as a hierarchical stochastic model whose form is derived from simple physical constraints, which in turn restrict the dynamics to only two possible classes. The probability distributions of both the signal and the background have simple representations in terms of Meijer G-functions. The two universality classes for the background dynamics manifest themselves in the signal distribution as two types of tails: power law and stretched exponential, respectively. A detailed analysis of empirical data from classical turbulence and financial markets shows excellent agreement with the theory.

Emergence of skewed non-Gaussian distributions of velocity increments in isotropic turbulence

“…Physically, the deterministic term in Eq. (5) represents the coupling between adjacent scales, whereas the stochastic term emerges from the complex interactions among all scales and is necessary for intermittency [14]. We further observe that a rescaling of variables i → ζ i properly leaves the model dynamics unchanged, which is a required property for a multiplicative cascade model [30] in the sense that it implies f ( i | i−1 )d i = g(x)dx, for…”

“…We work under the formalism of a unified hierarchical approach to describe the statistics of fluctuations in multiscale complex systems [13][14][15]. This framework, called H-theory, is an extension to multiscale systems of the compounding [5,7] or superstatistics [21] approaches to describe complex fluctuating phenomena.…”

“…Experimental and theoretical studies on homogeneous and isotropic turbulent flows indicate [5,7,8,10,22,23,26] that the conditional distribution P (δv r | ) is given by a Gaussian with variance . For the sake of simplicity, a Gaussian with zero mean is often considered in theoretical turbulence models [5,13,14,21,25], leading to symmetric (i.e., non-skewed) distributions P (δv r ).…”

“…, N , where i ≥ 0 represents the energy transfer rate from the hierarchy level i to smaller scales, γ i > 0 is a relaxation rate, κ i > 0 characterizes the strength of the multiplicative noise (and hence of the intermittency) in the hierarchical level i, and W i denotes a Wiener process. The intermittency model (5) with α = 0 has been introduced in [14]. The generalization above (with α = 0) is important to consider because the parameter α > 0 can be associated with a residual dissipation in the inertial range (see below), which is usually neglected in phenomenological cascade models.…”

“…Our intermittency model is built upon a hierarchy of multiple coupled scales of energy transfer rates [13][14][15]. The marginal distribution of short-scale velocity increments P (δv r ) is related to the energy transfer rate ε at a larger scale [see Eq.…”