Gibbsian theory of power-law distributions

It is shown that power-law phase space distributions describe marginally stable Gibbsian equilibria far from thermal equilibrium, which are expected to occur in collisionless plasmas containing fully developed quasistationary turbulence. Gibbsian theory is extended on the fundamental level to statistically dependent subsystems introducing an "ordering parameter" kappa. Particular forms for the entropy and partition functions are derived with superadditive (nonextensive) entropy, and a redefinition of temperature in such systems is given.     

Variability in the power-law distributions of rupture events

Rupture events, as the propagation of cracks or the sliding along faults, associated with the deformation of brittle materials are observed to obey power-law distributions. This is verified at scales ranging from laboratory samples to the Earth’s crust, for various materials and under various loading modes. Besides the claim that this is a universal characteristic of the deformation of heterogeneous media, spatial and temporal variations are observed in the exponent and tail-shape. These have considerable implications for the ability and the reliability of forecasting large events from smaller ones. There is a growing interest in identifying the factors responsible for these variations. In this work, we first present observations at various scales (laboratory tests, field experiments, landslides, mining induced seismicity, crustal Earthquakes) showing that substantial variations exist in both the slope and the tail-shape of the rupture event size distribution. This review allows us to identify potential explanations for these variations (incorrect statistical methods, heterogeneity, stress, brittle/ductile transition, finite size effects, proximity to the failure). A possible link with the critical point theory is also drawn showing that it is able to explain a part of the observed variations considering the distance to the critical point. Using numerical simulations of progressive failure we investigate the role of mechanical properties on the power-law distributions. The results of simulations agree with the critical point theory for various macroscopic behaviors ranging from ductility to brittleness providing a unified framework for the understanding of power-law variability observed in rupture phenomena.     

Fractional-order formulation of power-law and exponential distributions

We present a novel approach to data analysis using fractional order calculus. In principle the approach can be applied to any distribution and shows remarkable improvement even if the parameters of a particular distribution have been optimised to achieve the best fit to data. The method is demonstrated for two important distributions that are used in data analysis, namely, power-law and exponential distributions. We show that the approach can allow composite distributions to be constructed for improved accuracy and robustness in the characterisation of data sets.     

The power-law tail exponent of income distributions

In this paper we tackle the problem of estimating the power-law tail exponent of income distributions by using the Hill's estimator. A subsample semi-parametric bootstrap procedure minimizing the mean squared error is used to choose the power-law cutoff value optimally. This technique is applied to personal income data for Australia and Italy.     

Maximum entropy principle and power-law tailed distributions

In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution p_i by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. Indeed, the Maxwell-Boltzmann distribution is obtained by maximizing the Boltzmann-Shannon entropy under proper constraints. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same function in direct and inverse form. Indeed, the Maxwell-Boltzmann distribution p_i is expressed in terms of the exponential function, while the Boltzmann-Shannon entropy is defined as the mean value of -ln (p_i). In generalized statistical mechanics the second link is customarily relaxed. Of course, the generalized exponential function defining the probability distribution function after inversion, produces a generalized logarithm Λ(p_i). But, in general, the mean value of -Λ(p_i) is not the entropy of the system. Here we reconsider the question first posed in [Phys. Rev. E 66, 056125 (2002) and 72, 036108 (2005)], if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics. Within this scenario, apart from the standard logarithmic-exponential functions that define ordinary statistical mechanics, there emerge other new couples of direct-inverse functions, i.e. generalized logarithms Λ(x) and generalized exponentials Λ^-1(x), defining coherent and self-consistent generalized statistical theories. Interestingly, all these theories preserve the main features of ordinary statistical mechanics, and predict distribution functions presenting power-law tails. Furthermore, the obtained generalized entropies are both thermodynamically and Lesche stable.     

金融市场中幂律分布的经验和理论研究进展--经济物理学研究的一个前沿

对金融市场波动性的研究是经济物理(econophysics)的一个重要内容．物理学家们借鉴物理学研究方法对金融市场中的主要变量进行的经验研究，揭示了金融资产价格涨落及相关变量概率分布尾部的幂律渐近行为、这一性质明显有悖于传统金融学中的正态分布和试图取代它的列维分布，引起人们广泛的兴趣．文章集中介绍了关于金融资产收益率分布尾部幂律性质经验研究的主要方法和结果，以及几种相关的理论解释．     

Four contemporary interpretations of the nature of science

Instrumentalism is an approach to science that treats a theory as a tool and only as a tool for computation; it dispenses with the concept of truth.Conventionalism treats a theory as true by convention if it forms a pattern of observations from which correct predictions can be made.Operationalism denies meaning to the concepts of a theory unless they can be defined operationally. It is argued in this paper that truth-value is indispensable to science, because a theory can be rejected only if an empirical consequence is false and if falsity of a conclusion entails falsity of a premise. This undermines the above positions. The fourth interpretation isinduction. Induction, by contrast, uses the notion of truth-value. What is focused on here is its reliance on the ultimacy ofobservation. The present thesis is that instrumentalism, conventionalism, and induction are different attempts to handle observations. The common problem is the gap between data and theory.All these interpretations share a philosophy of observationalism. The aim of this paper is to show that the several orthodox interpretations of science all fail to solve the problem of the data-theory gap, and to show that they all presuppose a philosophy of observation.     

Equilibration of two power-law tailed distributions in a parton cascade model

We study the process of equilibration between two non-extensive subsystems in the framework of a particular non-extensive Boltzmann equation. We have found that even subsystems with different non-extensive properties achieve a common equilibrium distribution. We extract the thermodynamic temperature from final energy distributions in a particular case.     

Parameter estimation for power-law distributions by maximum likelihood methods

Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods.     

Formation of power-law size distributions of defects during fracture of materials

The experimental data on the surface relief of loaded ribbons of an amorphous alloy have been obtained. The distributions of surface defects formed under loading have been analyzed using the wavelet transform and box counting method. Moreover, the data on the time accumulation of microcracks in the volume of a loaded granite specimen have been examined. It has been shown that power-law size distributions of defects (scaling) appear on the surface and in the bulk before fracture. It has been revealed that the appearance of the power-law distributions is one of the indications of the formation of the self-organized critical state. The formation of the self-organized critical state in the bulk and on the surface of the material has been considered. It has been established that the formation of the self-organized critical state precedes the fracture of a solid.     

A note on possible interpretations of negative binomial multiplicity distributions

Different interpretations of negative binomial multiplicity distributions in high-energy hadronic collisions are briefly discussed. It is shown that the interpretation in terms of independent emission of clusters does describe the data well but does not imply early asymptotic behaviour in the average number of clusters, contrary to previous suggestions. In âddition, an alternative interpretation in terms of a continuous superposition of independent emission processes is shown to also describe the data well. Finally, some consequences of extrapolating the presently successful parametrization are discussed.     

Survivor statistics and damage spreading on social network with power-law degree distributions

Damage spreading(DS) of the random graph networks with power-law degree distributions is investigated using Glauber dynamics. Various subgraphs defined by the probability of acquaintance show distinct features in DS as measured by Hamming distance. A heuristic understanding of the long-time value of damage is achieved through an analysis of the survivor statistics. All survivors are dynamical, flipping in unison for the controlled sample and the damaged sample. Verification of these dynamic survivors is achieved through the introduction of a new measure of self-damage.     

Transition state theory: A generalization to nonequilibrium systems with power-law distributions

Transition state theory (TST) is generalized to nonequilibrium systems with power-law distributions. The stochastic dynamics that gives rise to the power-law distributions for the reaction coordinate and momentum is modeled by Langevin equations and corresponding Fokker-Planck equations. It is considered that a system far away from equilibrium does not have to relax to a thermal equilibrium state with Boltzmann-Gibbs distribution, but asymptotically approaches a nonequilibrium stationary state with a power-law distribution. Thus, we obtain a possible generalization of TST rates to nonequilibrium systems with power-law distributions. Furthermore, we derive the generalized TST rate constants for one-dimensional and n-dimensional Hamiltonian systems away from equilibrium, and obtain a generalized Arrhenius rate for systems with power-law distributions.     

Temporal generation of power-law distributions: A universal ‘oligarchy mechanism’

We present a universal mechanism for the temporal generation of power-law distributions with arbitrary integer-valued exponents.     

Sensitivity of the resilience of congested random networks to rolloff and offset in truncated power-law degree distributions

Random networks were generated with the random configuration model with prescribed truncated power-law degree distributions, parameterized by an exponent, an offset, and an exponential rolloff. As a model of an attack, each network had exactly one of its highest degree nodes removed, with the result that in some cases, one or more remaining nodes became congested with the reassignment of the load. The congested nodes were then removed, and the “cascade failure” process continued until all nodes were uncongested. The ratio of the number of nodes of the largest remaining cluster to the number of nodes in the original network was taken to be a measure of the network's resiliency to highest-degree node removal. We found that the resiliency is sensitive to both rolloff and offset (but not to cutoff) in the degree distribution, and that rolloff tends to decrease resiliency while offset tends to increase it.     

On the derivation of power-law distributions within classical statistical mechanics far from the thermodynamic limit

We show that within classical statistical mechanics, without taking the thermodynamic limit, the most general Boltzmann factor for the canonical ensemble is a q-exponential function. The only assumption here is that microcanonical distributions have to be separated from the total system energy, which is the prerequisite for any sensible measurement. We derive that all separable distributions are parametrized by a mathematical separation constant Q, which can be related to the non-extensivity q-parameter in Tsallis distributions. We further demonstrate that nature fixes the separation constant Q to 1 for large dimensionality of Gibbs $\Gamma$-phase space. Our results will be relevant for systems with a low-dimensional $\Gamma$-space, for example nanosystems, comprised of a small number of particles, or for systems with a dimensionally collapsed phase space, which might be the case for a large class of complex systems.