Paolo Grigolini

Collective behavior and evolutionary games – An introduction

This article is an introduction to a special issue in Chaos, Solitons & Fractals with the goal of attracting submissions that identify unifying principles that describe the essential aspects of collective behavior, and which thus allow for a better interpretation and foster the understanding of the complexity arising in such systems.

Scaling Detection in Time Series: Diffusion Entropy Analysis

The methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance. This means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved. We illustrate a method of statistical analysis based on the Shannon entropy of the diffusion process generated by the time series, called diffusion entropy analysis (DEA). We adopt artificial Gauss and Lévy time series, as prototypes of ordinary and anomalous statistics, respectively, and we analyze them with the DEA and four ordinary methods of analysis, some of which are very popular. We show that the DEA determines the correct scaling exponent even when the statistical properties, as well as the dynamic properties, are anomalous. The other four methods produce correct results in the Gauss case but fail to detect the correct scaling in the case of Lévy statistics.

Bistable flow driven by coloured gaussian noise: A critical study

AbstractA one-dimensional bistable flow driven by additive, exponentially correlated Gaussian noise is considered. The small relaxation time Fokker-Planck approximations, widely used in the recent literature, are derived and possible shortcomings of those approximation schemes are discussed. In particular, it is pointed out that higher order non-Fokker-Planck type contributions are generally of the same order as the Fokker-Planck terms. In principle, those contributions cannot be neglected if the global behavior of the probability solutions is to be described accurately. The result for the activation rate (Arrhenius factor), as evaluated from the approximative Fokker-Planck schemes, does not coincide in leading order in the correlation time τ of the noise with a computer simulation of the rate at low noise level. This result indicates that the wings of the stationary probability

Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

\bar p(x)

ASYMMETRIC ANOMALOUS DIFFUSION: AN EFFICIENT WAY TO DETECT MEMORY IN TIME SERIES

are in leading order in τ not recovered correctly from the approximative Fokker-Planck schemes. Some implications of our study for adiabatic elimination procedures are also discussed.

The non‐Markovian relaxation process as a ‘‘contraction’’ of a multidimensional one of Markovian type

We study the statistical properties of time distribution of seismicity in California by means of a new method of analysis, the diffusion entropy. We find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. We prove that this distribution is an inverse power law with an exponent mu=2.06+/-0.01. We propose the long-range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes.

Fractional calculus as a macroscopic manifestation of randomness

We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity<gamma</=2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q=(gamma+3)/(gamma+1)(0<gamma</=2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for Levy-like superdiffusion (i.e., nu=1 and 0<gamma</=2). Finally, for (gamma,nu)=(2,0) we obtain q=5/3, which differs from the value q=2 corresponding to the gamma=2 solutions available in the literature (nu<1 porous medium equation), thus exhibiting nonuniform convergence.

THE THERMODYNAMICS OF SOCIAL PROCESSES: THE TEEN BIRTH PHENOMENON

The first step of our approach consists of relating the generalized Brownian motion in a double‐well potential to a suitable time‐independent Fokker–Planck operator implying that an arbitrary large number of ‘‘virtual’’ variables be used. Then, to simplify the solution of this multidimensional Fokker–Planck equation, we develop a procedure of adiabatic elimination of the fastly relaxing variables. As a significant feature of this reduction scheme, we point out that no limitation on the number of the virtual variables is implied. The explicit form of the first correction term to the Smoluchowski equation is also shown to depend on whether or not the stochastic force is white. Via a comparison with the analytical results of Grote and Hynes’ theory [J. Chem. Phys. 73, 2715 (1980)] it is argued that the ‘‘exact’’ approach and the ‘‘reduction’’ procedure can be regarded as being complementary to one another.