The h-theory: universality classes of hierarchical complex systems

Abstract

Complex dynamical systems can be characterized through the time series associated with dynamical variables, which yield important information on the underlying stochastic process. The probability density function, the temporal correlation function, the power spectrum, and the memory function are examples of statistical properties that can be extracted from the time series. In this thesis, we are particularly interested in describing complex phenomena in which the stationary distribution of the time series of the main dynamical variable (the signal) exhibits large deviations from Gaussian statistics, possibly showing long and heavy tails. This kind of phenomena is present in many areas of physics, biology, and economics. However, our interest is focused on spectral fluctuations in non-integrable ballistic cavities, intensity fluctuations in random lasers, turbulence in fluids, stock prices fluctuations in financial markets. We shall attempt to describe these phenomena as a composition of distributions with distinct space/time scales which arise from a hierarchical dynamics with a coupling between contiguous scales. The model to be used, denominated H-Theory, was recently proposed by our research group and consists of a set of coupled stochastic differential equations, whose stationary solution leads to a parametric family of distributions represented by Fox H-function. This result unifies and generalizes the universality classes of superstatistics, which is a formalism that has been successfully used to describe systems with two separated time scales.