Complexity as a Form of Transition From Dynamics to Thermodynamics: Application to Sociological and Biological Processes.

This dissertation addresses the delicate problem of establishing the statistical mechanical foundation of complex processes. These processes are characterized by a delicate balance of randomness and order, and a correct paradigm for them seems to be the concept of sporadic randomness. First of all, we have studied if it is possible to establish a foundation of these processes on the basis of a generalized version of thermodynamics, of non-extensive nature. A detailed account of this attempt is reported in Ignaccolo and Grigolini (2001), which shows that this approach leads to inconsistencies. It is shown that there is no need to generalize the Kolmogorov-Sinai entropy by means of a non-extensive indicator, and that the anomaly of these processes does not rest on their non-extensive nature, but rather in the fact that the process of transition from dynamics to thermodynamics, this being still extensive, occurs in an exceptionally extended time scale. Even, when the invariant distribution exists, the time necessary to reach the thermodynamic scaling regime is infinite. In the case where no invariant distribution exists, the complex system lives forever in a condition intermediate between dynamics and thermodynamics. This discovery has made it possible to create a new method of analysis of non-stationary time series which is currently applied to problems of sociological and physiological interest

One, No One, and One Hundred Thousand: The Paradigm of the Z–R Relationship

AbstractThe Z–R relationship is a scaling-law formulation, Z = ARb, connecting the radar reflectivity Z to the rain rate R. However, more than 100 Z–R relationships, with different values of the parameters, have been reported in literature. This abundance of relationships is in itself a strong indication that no one "physical" relationship exists, a state of affairs that we find similar to that of the protagonist of Luigi Pirandello's novel One, No One and One Hundred Thousand. Nevertheless the "elevation" of a simple linear fit in the (logR, logZ) space to the role of "scaling law" is such a widespread tenet in literature that it eclipses the simple realization that the abundance of different intercepts and slopes reflects the inhomogeneous nature of rain, and, in ultimate analysis, the statistical variability existing between the number of drops and drop size distribution. Here, we "eliminate" the contribution of the number of drops by rescaling both reflectivity and rainfall rate to per unit drop variables, (Z, R) → (z, r), so that the remaining variability is due only to the variability of the drop size distribution. We use a worldwide database of disdrometer data to show that for the rescaled variables (z, r) only "one," albeit approximate, scaling law exists

Renewal and memory properties in the random growth of surfaces

Article discussing the renewal and memory properties in the random growth of surfaces

Dynamics of Electroencephalogram Entropy and Pitfalls of Scaling Detection

This article discusses dynamics of electroencephalogram entropy and pitfalls of scaling detection. Herein the authors study the time evolution of diffusion entropy to elucidate the scaling of EGG time series

Brain, Music, and Non-Poisson Renewal Processes

Article discussing research that shows both music composition and brain function, as revealed by the electroencephalogram (EEG) analysis, are renewal non-Poisson processes living in the nonergodic dominion