Narrow Results

We study the distributions of money in a simple closed economic system for different types of monetary transactions. We know that for arbitrary and random sharing with locally conserving money transactions, the money distribution goes to the Gibb's distribution of statistical mechanics. We then consider the effects of savings, etc. and see how the distribution changes. We also propose a new model where the agents invest equal amounts of money in each transaction. We find that for short time-period, the money distribution obeys a power-law with an exponent very close to unity, and has an exponential tail; after a very long time, this distribution collapses and the entire amount of money goes to a tiny fraction of the population.

Recently, we have extended our "unified" model of evolutionary ecology to incorporate the spatial inhomogeneities of the ecosystem and the migration of individual organisms from one patch to another within the same ecosystem. In this paper, an extension of our recent model is investigated so as to describe the migration and speciation in a more realistic way.

We study the operator-split method Gillespie-MultiParticle method (GMP) for a diffusion-limited reaction in a finite medium by looking at the detailed distribution of reaction times for a broad range of time scales. The diffusion-limited reaction lies at the lower limits of the applicability of the GMP method. We propose an empirical distribution for the inter-reaction times (duration of the dissociated state) that accounts for the power-law and exponential regimes. The knowledge of this distribution can be used to change the reactive method of GMP and enable it to use coarser discretization, with consequent improvement in performance.

We analyzed the rising and relaxation of the cusp-like local peaks superposed with oscillations which were well defined by the Warsaw Stock Exchange index WIG in a daily time horizon. We found that the falling paths of all index peaks were described by a generalized exponential function or the Mittag-Leffler (ML) one superposed with various types of oscillations. However, the rising paths (except the first one of WIG which rises exponentially and the most important last one which rises again according to the ML function) can be better described by bullish anti-bubbles or inverted bubbles.^2–4 The ML function superposed with oscillations is a solution of the nonhomogeneous fractional relaxation equation which defines here our Fractional Market Model (FMM) of index dynamics which can be also called the Rheological Model of Market. This solution is a generalized analog of an exactly solvable fractional version of the Standard or Zener Solid Model of viscoelastic materials commonly used in modern rheology.⁵ For example, we found that the falling paths of the index can be considered to be a system in the intermediate state lying between two complex ones, defined by short and long-time limits of the Mittag-Leffler function; these limits are given by the Kohlrausch-Williams-Watts (KWW) law for the initial times, and the power-law or the Nutting law for asymptotic time. Some rising paths (i.e., the bullish anti-bubbles) are a kind of log-periodic oscillations of the market in the bullish state initiated by a crash. The peaks of the index can be viewed as precritical or precrash ones since:

(i) the financial market changes its state too early from the bullish to bearish one before it reaches a scaling region (defined by the diverging power-law of return per unit time), and

(ii) they are affected by a finite size effect.

These features could be a reminiscence of a significant risk aversion of the investors and their finite number, respectively. However, this means that the scaling region (where the relaxations of indexes are described by the KWW law or stretched exponential decay) was not observed. Hence, neither was the power-law of the instantaneous returns per unit time observed. Nevertheless, criticality or crash is in a natural way contained in our FMM and we found its "finger print".

Correlation-based weighted financial networks are analyzed to present cumulative distribution of strength with a power-law tail, which suggests that a small number of hub-like stocks have greater influence on the whole fluctuation of financial market than others. The relationship between clustering and connectivity of vertices emphasizes hierarchical organization, which has been depicted by minimal span tree in previous work. These results urge us to further study the mixing patter of financial network to understand the tendency for vertices to be connected to vertices that are like (or unlike) them in some way. The measurement of average nearest-neighbor degree running over classes of vertices with degree k shows a descending trend when k increases. This interesting result is first uncovered in our work, and suggests the disassortative mixing of financial network which refers to a bias in favor of connections between dissimilar vertices. All the results in weighted complex network aspect may provide some insights to deeper understand the underlying mechanism of financial market and model the evolution of financial market.

We analyze a spatial susceptible-infected epidemic model using cellular automata and investigate the relations between the power-law distribution of patch sizes and the regime of invasion. The obtained results show that, when the invasion is in the form of coexistence of stable target and spiral wave, power-law will emerge, which may provide a new insight into the control of disease.

The nanofluids are a recent challenging task in a nanotechnology field used in heat transfer enhancement for base fluids. The major purpose of this research is to examine the influences of Hall current on the non-Newtonian power-law nanofluid on an exponentially extending surface. Implementation in the Cattaneo–Christov heat flux and the free stream is performed to analyze the thermal relaxation features. Entropy generation evaluation and Bejan number during the convection flow are investigated. The Runge–Kutta–Fehlberg method is employed to resolve the transformed governing nonlinear equations. The impacts of the key physical factors on the profiles of primary and secondary velocities, temperature and entropy generation are discussed across the graphs. The local skin-friction coefficients, Nusselt and Sherwood numbers are demonstrated in a tabular form under the impacts of key physical parameters. Two different types of power-law indicators including pseudoplastic fluid $(n=0.7)$ and dilatant fluid $(n=1.2)$ are conducted. The results indicated that the flow speed decreases at dilatant fluid compared to pseudoplastic fluid due to higher viscosity. Increasing Hall current parameter powers the axial and secondary velocity profiles. Thermophoresis parameter powers the profiles of the temperature, nanoparticle volume fraction and local entropy generation. The dilatant fluid $(n=1.2)$ gives higher values of $C_{fx},C_{fz},{\mathrm{\text{Nu}}}_x$ and ${\mathrm{\text{Sh}}}_x$ compared to the pseudoplastic fluid $(n=0.7)$.

We have examined a homogeneous massive scalar field minimally coupled to the spatially flat Friedmann–Robertson–Walker metric, in squeezed and coherent state formalisms, in semiclassical theory of gravity. In the oscillatory phase of inflaton, the approximate leading solution to the semiclassical Einstein equation has the same power-law expansion as that of the classical Einstein equation for both coherent and squeezed state formalisms. Quantum fluctuations of the inflaton in coherent and squeezed vacuum states are studied, by using dispersion relations. The uncertainty relation for coherent state does not depend on the coherent state parameter while the uncertainty relation for squeezed vacuum state depends on the associated squeezing parameter and squeezing angle.

The effects of power-law plasticity (yield strength and strain hardening exponent) on the plastic strain distribution underneath a Vickers indenter was systematically investigated by recourse to three-dimensional finite element analysis, motivated by the experimental macro- and micro-indentation on heat-treated Al-Zn-Mg alloy. For meaningful comparison between simulated and experimental results, the experimental heat treatment was carefully designed such that Al alloy achieve similar yield strength with different strain hardening exponent, and vice versa. On the other hand, full 3D simulation of Vickers indentation was conducted to capture subsurface strain distribution. Subtle differences and similarities were discussed based on the strain field shape, size and magnitude for the isolated effect of yield strength and strain hardening exponent.

Recently, methods from the statistical physics of complex systems have been applied successfully to identify universal features in the long-range correlations (LRCs) of written texts. However, in real texts, these universal features are being intermingled with language-specific influences. This paper aims at the characterization and further understanding of the interplay between universal and language-specific effects on the LRCs in texts. To this end, we apply the language-sensitive mapping of written texts to word-length series (wls) and analyse large parallel (of same content) corpora from 10 languages classified to four families (Romanic, Germanic, Greek and Uralic). The autocorrelation functions of the wls reveal tiny but persistent LRCs decaying at large scales following a power-law with a language-independent exponent $\sim$0.60–0.65. The impact of language is displayed in the amplitude of correlations where a relative standard deviation $>$40% among the analyzed languages is observed. The classification to language families seems to play a significant role since, the Finnish and Germanic languages exhibit more correlations than the Greek and Roman families. To reveal the origins of the LRCs, we focus on the long words and perform burst and correlation analysis in their positions along the corpora. We find that the universal features are linked more to the correlations of the inter-long word distances while the language-specific aspects are related more to their distributions.

Effects of scale-free avalanche walks on anomalous diffusions have been studied by introducing simple non-Markovian walk models. The scale-free avalanche walk is realized as a walker goes to one direction consistently in a time interval, the distribution of which follows a power-law. And it is applied to the memory models, in which the entire history of a walk process is memorized or the memory for the latest step is enhanced with time. The power-law avalanche walk with memory effects strengthens the persistence between steps and thus makes the Hurst exponent be larger than the cases without avalanche walks, while does not affect the anti-persistent nature.

The dynamics of the convergence to the critical attractor for the logistic map is investigated. At the border of chaos, when the Lyapunov exponent is zero, the use of the nonextensive statistical mechanics formalism allows to define a weak sensitivity or insensitivity to initial conditions. Using this formalism we analyze how a set of initial conditions spread all over the phase space converges to the critical attractor in the case of infinite bifurcation and tangent bifurcation points. We show that the phenomena is governed in both cases by a power-law regime but the critical exponents depend on the type of bifurcation and may also depend on the numerical experiment set-up. Differences and similarities between the two cases are also discussed.

We consider a scalar-tensor model of dark energy with kinetic and Gauss–Bonnet (GB) couplings. We study the conditions for the existence of quintessential and phantom power-law expansion, and also analyze these conditions in the absence of a potential (closely related to string theory). A mechanism to avoid the Big Rip singularity in various asymptotic limits of the model has been studied. It was found that the kinetic and GB couplings might prevent the Big Rip singularity in a phantom scenario. The autonomous system for the model has been used to study the stability properties of the power-law solution, and the center manifold analysis was used to treat zero eigenvalues.

In this paper, we study the semiclassical evolution of a “coherently oscillating massive scalar field”. This field is nominally linked to the spatially levelled Friedmann–Robertson–Walker (FRW) metric. By use of two-mode quantum optical states formalism, we evaluated the power-law cosmology with scale factor as a power function of cosmic time. In the oscillatory phase of scalar field, it was found that, in all the four different combinations of quantum states, quantum scalar field obeying the time-dependent Schrödinger equation leads to the power-law expansion, in good agreement with that of the matter-dominated era. However, in the semiclassical theory of gravity in contrary to the classical description of gravity, we obtained two behavioral solutions corresponding to two different pattern evolution of the scale factor. This behavioral pattern evolution of scale factor of the universe has an important quantum implication associated with it.

In this paper, by the use of entangled and nonentangled coherent and squeezed state formalism of two-mode nonclassical states, we studied the chaotic inflationary model of a massive scalar field with quadratic potential in the semiclassical gravity, derived from canonical quantum gravity. It was found that the semiclassical quantum gravity leads to the same power-law expansion of the universe as that of the matter-dominated era ${\tau }^{\frac{2}{3}}$ in an oscillatory phase of the scalar field in all the nonclassical quantum states considered. The coherently oscillating scalar field in the expanding universe suffers from the phenomenon of particle creation which restricts the duration of stable coherent oscillations of the scalar field dependent on the parameters of the states considered and affect in a certain way the abundant particle production owing to the parametric resonance of bosonic fields coupled to this coherently oscillating scalar field.

An empirical study accomplished on various samples of marks shows that their distribution on mark levels has some universal features. Using (classical) examination items based on commentaries and debate, the distribution displays a Gaussian form with a variance that does not depend on the profile of classes and age level of pupils. In the case of examination based on items with multiple choices, the deviation from the Gaussian form becomes relevant and the distribution gets the shape of a stable Lévy distribution with a fat tail that can be fitted to a power-law. The difficulties related to the infinite variance, typical for this class of models, can be overcome using the Truncated Lévy Distribution, frequently met in physics of socio-economic systems.

An agent-based model is applied for studying the group-forming behavior numerically. A fixed number of homogeneous agents are distributed on a two-dimensional square lattice system with L × L unit cells. The dynamics of the agents are described in terms of a potential and a friction factor, just like that of a weight on a slope. The potential is defined in an attempt to model the concept of the "biosocial attraction." Depending on the density of the population and the friction factor, four regimes of group-size distributions are identified. They are the fixed, the exponential, the power-law and the black-hole regimes. Especially, beyond the well-known exponential and power-law distributions, a black-hole regime is identified for the lowest values of the friction factor. In the black-hole regime, all the agents in the system are attracted to the only group remaining in the system. An example membership values for each of the four classes are suggested in terms of the two key parameters of the system, the population density and the friction factor. And a phase transition is observed between the exponential and the black-hole regimes. Also the results are discussed in relation to the city-size distributions in a country.

An agent-based model is employed for the study of the group size distributions. A fixed number of homogeneous agents are distributed on a two-dimensional lattice system. The dynamics of the agents is described in terms of the inverse distance potential and the friction factor. From a random initial distribution, the agents move forming groups until all the agents come to a stationary position. For a squared system with L × L cells, the group size distribution showed a well defined power-law behavior up to the cut-off size. But when the system changed to an L × H non-squared one, a "geometric aging effect" emerged. Together with the phase transition, the geometric aging effect is considered to be a generic mechanism of the deviated power-law distributions, such as the "fall-off" and the three-bent-line distributions. The results are discussed in relation to the well-known physical or social phenomena such as the King Effect in the city size distributions, the fall-off distribution of the fish schools, the three-bent-line distributions of the Earth-crossing asteroids and 2D percolation problem.

The cello suites of Johann Sebastian Bach exhibit several types of power-law scaling, the best examples of which can be considered fractal in nature. This article examines scaling with respect to the characteristics of melodic interval and its derivative, melodic moment. A new and effective method for pitch-related analysis is described and then applied to a selection of the 36 pieces that comprise the six cello suites.

Study of heat and mass transfers is carried out for MHD Oldroyd-B fluid (OBF) over an infinite vertical plate having time-dependent velocity and with ramped wall temperature and constant concentration. It is proven in many already published articles that the heat and mass transfers do not really or always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such dynamical process very accurately, thus, a different concept of differentiation is needed to capture such process. Very recently, a new class of differential operators were introduced and have been recognized to be efficient in capturing processes following the power-law, the decay law and the crossover behaviors. For the study of heat and mass transfers, we applied the newly introduced differential operators to model such flow and compare the results with integer-order derivative. Laplace transform and inversion algorithms are used for all the cases to find analytical solutions and to predict the influences of different parameters. The obtained analytical solutions were plotted for different values of fractional orders $\alpha$ and $\beta$, $\lambda$, ${\lambda }_r$, $M$, $G_r$ and $Pr$ on the velocity field. In comparison, Atangana–Baleanu (ABC) fractional derivatives are found to be the best to explain the memory effects than the classical, Caputo (C) and Caputo–Fabrizio (CF) fractional derivatives. Some calculated values for Nusselt number and Sherwood number are presented in tables. Moreover, from the present solutions, the already published results were found as limiting cases.