Analysis of return distributions in the coherent noise model

Open Problems within Nonextensive Statistical Mechanics

Nonextensive statistical mechanics has developed into an important framework for modeling the thermodynamics of complex systems and the information of complex signals. To mark the 80th birthday of the field's founder, Constantino Tsallis, a review of open problems that can stimulate future research is provided. Over the thirty-year development of NSM, a variety of criticisms have been published ranging from questions about the justification for generalizing the entropy function to the interpretation of the generalizing parameter q. While these criticisms have been addressed in the past and the breadth of applications has demonstrated the utility of the NSM methodologies, this review provides insights into how the field can continue to improve the understanding and application of complex system models. The review starts by grounding q-statistics within scale-shape distributions and then frames a series of open problems for investigation. The open problems include using the degrees of freedom to quantify the difference between entropy and its generalization, clarifying the physical interpretation of the parameter q, improving the definition of the generalized product using multidimensional analysis, defining a generalized Fourier transform applicable to signal processing applications, and re-examining the normalization of nonextensive entropy. This review concludes with a proposal that the shape parameter is a candidate for defining the statistical complexity of a system.

Enthusiasm and Skepticism: Two Pillars of Science—A Nonextensive Statistics Case

Science and its evolution are based on complex epistemological structures. Two of the pillars of such a construction definitively are enthusiasm and skepticism, both being ingredients without which solid knowledge is hardly achieved and certainly not guaranteed. Our friend and colleague Jean Willy André Cleymans (1944–2021), with his open personality, high and longstanding interest for innovation, and recognized leadership in high-energy physics, constitutes a beautiful example of the former. Recently, Joseph I. Kapusta has generously and laboriously offered an interesting illustration of the latter pillar, in the very same field of physics, concerning the very same theoretical frame, namely, nonextensive statistical mechanics and the nonadditive q-entropies on which it is based. I present here a detailed analysis, point by point, of Kapusta’s 19 May 2021 talk and, placing the discussion in a sensibly wider and updated perspective, I refute his bold conclusion that indices q have no physical foundation.

Exploring the Neighborhood of q-Exponentials

The q-exponential form eqx≡[1+(1−q)x]1/(1−q)(e1x=ex) is obtained by optimizing the nonadditive entropy Sq≡k1−∑ipiqq−1 (with S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from q-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding q-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing Sq as particular case.

Beyond Boltzmann-Gibbs-Shannon in Physics and Elsewhere

The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy SBG started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.

Noisy coupled logistic maps in the vicinity of chaos threshold

We focus on a linear chain of N first-neighbor-coupled logistic maps in the vicinity of their edge of chaos in the presence of a common noise. This model, characterised by the coupling strength ϵ and the noise width σmax, was recently introduced by Pluchino et al. [Phys. Rev. E 87, 022910 (2013)]. They detected, for the time averaged returns with characteristic return time τ, possible connections with q-Gaussians, the distributions which optimise, under appropriate constraints, the nonadditive entropy, Sq, basis of nonextensive statistics mechanics. Here, we take a closer look on this model, and numerically obtain probability distributions which exhibit a slight asymmetry for some parameter values, in variance with simple q-Gaussians. Nevertheless, along many decades, the fitting with q-Gaussians turns out to be numerically very satisfactory for wide regions of the parameter values, and we illustrate how the index q evolves with (N,τ,ϵ,σmax). It is nevertheless instructive on how careful one must be in such numerical analysis. The overall work shows that physical and/or biological systems that are correctly mimicked by this model are thermostatistically related to nonextensive statistical mechanics when time-averaged relevant quantities are studied.

Comment on "Universal relation between skewness and kurtosis in complex dynamics"

In a recent paper [M. Cristelli, A. Zaccaria, and L. Pietronero, Phys. Rev. E 85, 066108 (2012)], the authors analyzed the relation between skewness and kurtosis for complex dynamical systems, and they identified two power-law regimes of non-Gaussianity, one of which scales with an exponent of 2 and the other with 4/3. They concluded that the observed relation is a universal fact in complex dynamical systems. In this Comment, we test the proposed universal relation between skewness and kurtosis with a large number of synthetic data, and we show that in fact it is not a universal relation and originates only due to the small number of data points in the datasets considered. The proposed relation is tested using a family of non-Gaussian distribution known as q-Gaussians. We show that this relation disappears for sufficiently large datasets provided that the fourth moment of the distribution is finite. We find that kurtosis saturates to a single value, which is of course different from the Gaussian case (K=3), as the number of data is increased, and this indicates that the kurtosis will converge to a finite single value if all moments of the distribution up to fourth are finite. The converged kurtosis value for the finite fourth-moment distributions and the number of data points needed to reach this value depend on the deviation of the original distribution from the Gaussian case.

Noise, synchrony, and correlations at the edge of chaos

We study the effect of a weak random additive noise in a linear chain of N locally coupled logistic maps at the edge of chaos. Maps tend to synchronize for a strong enough coupling, but if a weak noise is added, very intermittent fluctuations in the returns time series are observed. This intermittency tends to disappear when noise is increased. Considering the probability distribution functions (pdfs) of the returns, we observe the emergence of fat tails which can be satisfactorily reproduced by q-Gaussians' curves typical of nonextensive statistical mechanics. The interoccurrence times of these extreme events are also studied in detail. Similarities with the recent analysis of financial data are also discussed.

Predictability of the coherent-noise model and its applications

We study the threshold distribution function of the coherent-noise model for the case of infinite number of agents. This function is piecewise constant with a finite number of steps n. The latter exhibits a 1/f behavior as a function of the order of occurrence of an avalanche and hence versus natural time. An analytic expression of the expectation value E(S) for the size S of the next avalanche is obtained and used for the prediction of the next avalanche. Apart from E(S), the number of steps n can also serve for this purpose. This enables the construction of a similar prediction scheme which can be applied to real earthquake aftershock data.