Randomness, evenness, and Rényi’s index

This paper links together the notion of entropy and the notion of inequality indices—the former is applied in Statistical Physics to measure randomness, and the latter is applied in Economics to measure evenness. We explore the profound similarities between these diametric notions, construct a mathematical transformation between them, and show how randomness can be used to measure evenness-and vice versa. In particular, we devise and study Rényi’s index—a randomness-based measure of evenness with special properties. Rényi’s index is established as an effectual gauge of statistical heterogeneity in the context of general probability laws defined on the positive half-line.     

How random is a random vector?

Over 80 years ago Samuel Wilks proposed that the “generalized variance” of a random vector is the determinant of its covariance matrix. To date, the notion and use of the generalized variance is confined only to very specific niches in statistics. In this paper we establish that the “Wilks standard deviation” –the square root of the generalized variance–is indeed the standard deviation of a random vector. We further establish that the “uncorrelation index” –a derivative of the Wilks standard deviation–is a measure of the overall correlation between the components of a random vector. Both the Wilks standard deviation and the uncorrelation index are, respectively, special cases of two general notions that we introduce: “randomness measures” and “independence indices” of random vectors. In turn, these general notions give rise to “randomness diagrams”—tangible planar visualizations that answer the question: How random is a random vector? The notion of “independence indices” yields a novel measure of correlation for Lévy laws. In general, the concepts and results presented in this paper are applicable to any field of science and engineering with random-vectors empirical data.     

Econophysical visualization of Adam Smith’s invisible hand

Consider a complex system whose macrostate is statistically observable, but yet whose operating mechanism is an unknown black-box. In this paper we address the problem of inferring, from the system’s macrostate statistics, the system’s intrinsic force yielding the observed statistics. The inference is established via two diametrically opposite approaches which result in the very same intrinsic force: a top-down approach based on the notion of entropy, and a bottom-up approach based on the notion of Langevin dynamics. The general results established are applied to the problem of visualizing the intrinsic socioeconomic force–Adam Smith’s invisible hand–shaping the distribution of wealth in human societies. Our analysis yields quantitative econophysical representations of figurative socioeconomic forces, quantitative definitions of “poor” and “rich”, and a quantitative characterization of the “poor-get-poorer” and the “rich-get-richer” phenomena.     

Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

The growth statistics of Zipfian ensembles: Beyond Heaps’ law

We consider an evolving ensemble assembled from a set of n different elements via a stochastic growth process in which independent and identically distributed copies of the elements arrive randomly in time, and their statistics are governed by Zipf’s law. The associated “Heaps process” is the stochastic process tracking the fraction of different element copies present in the evolving ensemble at any given time point. For example, the evolving ensemble is a text assembled from a stream of words, and the Heaps process keeps count of the number of different words in the evolving text. A detailed asymptotic statistical analysis of the Heaps process, in the limit n→∞, is conducted. This paper establishes a comprehensive “Heapsian analysis” of the growth statistics of Zipfian ensembles. The analysis presented far extends and generalizes Heaps’ law, which asserts that the number of different words in a text of length l follows a power law in the variable l.     

A probabilistic walk up power laws

We establish a path leading from Pareto’s law to anomalous diffusion, and present along the way a panoramic overview of power-law statistics. Pareto’s law is shown to universally emerge from “Central Limit Theorems” for rank distributions and exceedances, and is further shown to be a finite-dimensional projection of an infinite-dimensional underlying object — Pareto’s Poisson process  . The fundamental importance and centrality of Pareto’s Poisson process is described, and we demonstrate how this process universally generates an array of anomalous diffusion statistics characterized by intrinsic power-law structures: sub-diffusion and super-diffusion, Lévy laws and the “Noah effect”, long-range dependence and the “Joseph effect”, 1/f$1/f$ noises, and anomalous relaxation.     

Quantum-like microeconomics: Statistical model of distribution of investments and production

In this paper we demonstrate that the probabilistic quantum-like (QL) behavior-the Born’s rule, interference of probabilities, violation of Bell’s inequality, representation of variables by in general noncommutative self-adjoint operators, Schrödinger’s dynamics-can be exhibited not only by processes in the micro world, but also in economics. In our approach the QL-behavior is induced not by properties of systems. Here systems (commodities) are macroscopic. They could not be superpositions of two different states. In our approach the QL-behavior of economical statistics is a consequence of the organization of the process of production as well as investments. In particular, Hamiltonian (“financial energy”) is determined by rate of return.     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

The multi-agent Parrondo’s model based on the network evolution

A multi-agent Parrondo’s model is proposed in the paper. The model includes link A based on the rewiring mechanism (the network evolution) + game B (dependent on the spatial neighbors). Moreover, to produce the paradoxical effect and analyze the “agitating” effect of the network evolution, the dynamic processes of the network evolution + game B are studied. The simulation results and the theoretical analysis both show that the network evolution can make game B which is losing produce the winning paradoxical effect. Furthermore, we obtain the parameter space where the strong or weak Parrondo’s paradox occurs. Each size of the region of the parameter space is larger than the one in the available multi-agent Parrondo’s model of game A + game B. This result shows that the “agitating” effect of rewiring based on the network evolution is better than that of the zero-sum game between individuals.     

Formulation of the Hellmann–Feynman theorem for the “second choice” version of Tsallis’ thermostatistics

An approach to formulating the Hellmann–Feynman theorem within the “second choice” formalism of non-extensive statistical mechanics is considered. For the state of thermal equilibrium, we derive a relation of Hellmann–Feynman type between the derivative of the non-extensive free energy with respect to the external parameter and the quantum statistical q$q$-average of the derivative of the Hamilton operator. We also give a proper extension for an arbitrary observable commuting with the Hamiltonian. Some reasons for the usefulness of new formulas are discussed.     

Spin-charge gauge compositeness of electron in HTS: Hints from metal-insulator crossover and entropy behavior

We show that the coexistence of Fermi arcs and metal-insulator crossover of the in-plane resistivity can give a hint of a peculiar “gauge compositeness” of the electron in hole-doped high T_c cuprates and a similar hint also comes from the negative intercept at T=0 of the electronic entropy extrapolated from moderate temperatures in the “pseudogap phase”. An implementation of this “compositeness” within the spin-charge gauge approach is outlined and is employed to discuss the above phenomena.     

Quantum dice

In a letter to Born, Einstein wrote  [42]: “Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the ‘old one.’ I, at any rate, am convinced that He does not throw dice.” In this paper we take seriously Einstein’s famous metaphor, and show that we can gain considerable insight into quantum mechanics by doing something as simple as rolling dice. More precisely, we show how to perform measurements on a single die, to create typical quantum interference effects, and how to connect (entangle) two identical dice, to maximally violate Bell’s inequality.     

The Pietra term structures of financial assets

This paper explores an elemental connection between call options-the most commonly tradable financial derivatives, implied volatility term structures-critical “market information” emanating from call-option prices, and the Pietra index-a quantitative economic measure of societal egalitarianism. Our study: (i) unveils an intrinsic “Pietra structure” of call-option prices; (ii) introduces the notion of the “Pietra term structures” of financial assets; (iii) describes the probabilistic meaning of the Pietra term structures; (iv) establishes an explicit nonlinear one-to-one mapping between the Pietra term structures and the implied volatility term structures of financial assets. The results presented in this paper provide a deep insight into the econophysics of call options and implied volatility term structures.     

Lorenzian analysis of infinite Poissonian populations and the phenomena of Paretian ubiquity

The Lorenz curve is a universally calibrated statistical tool measuring quantitatively the distribution of wealth within human populations. We consider infinite random populations modeled by inhomogeneous Poisson processes defined on the positive half-line—the randomly scattered process-points representing the wealth of the population-members (or any other positive-valued measure of interest such as size, mass, energy, etc.). For these populations the notion of “macroscopic Lorenz curve” is defined and analyzed, and the notion of “Lorenzian fractality” is defined and characterized. We show that the only non-degenerate macroscopically observable Lorenz curves are power-laws manifesting Paretian statistics—thus providing a universal “Lorenzian explanation” to the ubiquitous appearance of Paretian probability laws in nature.     

Where do we stand on econophysics?

This issue contains papers selected from the contributions presented at the 5th International Conference on “Applications of Physics in Financial Analysis” (APFA5) held in Torino from June 29th to July 1st, 2006 (http://www.polito.it/apfa5). The issue collects recent applications of models and methods of statistical physics to economic problems. This interdisciplinary field of research, known as Econophysics, has seen intensive growth over the last decade. The challenge for econophysicists will be to go beyond the traditional views of economics and physics unifying the separate lines of development followed by the two disciplines over great part of the 20th century.“The conventional view serves to protect us from the painful job of thinking”, John Kenneth Galbraith (1908-2006).     

Quantum-like dynamics of decision-making

In cognitive psychology, some experiments for games were reported, and they demonstrated that real players did not use the “rational strategy” provided by classical game theory and based on the notion of the Nasch equilibrium. This psychological phenomenon was called the disjunction effect. Recently, we proposed a model of decision making which can explain this effect (“irrationality” of players) Asano et al. (2010, 2011) [23] and [24]. Our model is based on the mathematical formalism of quantum mechanics, because psychological fluctuations inducing the irrationality are formally represented as quantum fluctuations Asano et al. (2011) [55]. In this paper, we reconsider the process of quantum-like decision-making more closely and redefine it as a well-defined quantum dynamics by using the concept of lifting channel, which is an important concept in quantum information theory. We also present numerical simulation for this quantum-like mental dynamics. It is non-Markovian by its nature. Stabilization to the steady state solution (determining subjective probabilities for decision making) is based on the collective effect of mental fluctuations collected in the working memory of a decision maker.     

Ultrasonic measurements of particle retention by a porous medium

Ultrasound transmission and reflection are utilized to characterize the particle retention in depth of fluid-saturated porous samples under a flow of silt solution. The effect of the concentration of particles in the fluid is investigated via measurements of “comparison coefficients” which are the ratio of the Fourier transforms of the reflected (transmitted) signals before and after the particle flow. Numerical computations of the latter coefficients using Biot-Stoll’s theory are compared to the experimental data. The frequential evolution of physical parameters such as bulk and shear moduli are sought. To this end, a gradient descent algorithm is utilized to minimize the differences between the experimental and calculated comparison coefficients. Several concentrations of suspended particles are investigated to check the validity of this inversion method and a good agreement between theory and experiments is observed.     

Correlated randomness and switching phenomena

One challenge of biology, medicine, and economics is that the systems treated by these serious scientific disciplines have no perfect metronome in time and no perfect spatial architecture—crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. Further, many of these processes and structures have the remarkable feature of “switching” from one behavior to another as if by magic. The past century has, philosophically, been concerned with placing aside the human tendency to see the universe as a fine-tuned machine. Here we will address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at some of the many spatial and temporal patterns in biology, medicine, and economics and even begin to characterize the switching phenomena that enables a system to pass from one state to another. Inspired by principles developed by A. Nihat Berker and scores of other statistical physicists in recent years, we discuss some applications of correlated randomness to understand switching phenomena in various fields. Specifically, we present evidence from experiments and from computer simulations supporting the hypothesis that water’s anomalies are related to a switching point (which is not unlike the “tipping point” immortalized by Malcolm Gladwell), and that the bubbles in economic phenomena that occur on all scales are not “outliers” (another Gladwell immortalization). Though more speculative, we support the idea of disease as arising from some kind of yet-to-be-understood complex switching phenomenon, by discussing data on selected examples, including heart disease and Alzheimer disease.     

Income distribution patterns from a complete social security database

We analyze the income distribution of employees for 9 consecutive years (2001–2009) using a complete social security database for an economically important district of Romania. The database contains detailed information on more than half million taxpayers, including their monthly salaries from all employers where they worked. Besides studying the characteristic distribution functions in the high and low/medium income limits, the database allows us a detailed dynamical study by following the time-evolution of the taxpayers income. To our knowledge, this is the first extensive study of this kind (a previous Japanese taxpayers survey was limited to two years). In the high income limit we prove once again the validity of Pareto’s law, obtaining a perfect scaling on four orders of magnitude in the rank for all the studied years. The obtained Pareto exponents are quite stable with values around α≈2.5$\alpha \approx 2.5$, in spite of the fact that during this period the economy developed rapidly and also a financial-economic crisis hit Romania in 2007–2008. For the low and medium income category we confirmed the exponential-type income distribution. Following the income of employees in time, we have found that the top limit of the income distribution is a highly dynamical region with strong fluctuations in the rank. In this region, the observed dynamics is consistent with a multiplicative random growth hypothesis. Contrarily with previous results obtained for the Japanese employees, we find that the logarithmic growth-rate is not independent of the income.