Framework Of Nonextensive Statistical Mechanics Research Articles

Temporal variations of the probability distribution of voronoi cells generated by earthquake epicenters

The area of the cells of Voronoi tessellations has been modelled through different probability distributions among which the most promising are the generalized gamma and tapered Pareto distributions. In particular the latter has been used to model times and distances between successive earthquakes besides area and perimeter of cells generated by earthquake epicenters. In the framework of nonextensive statistical mechanics applied in geophysics, variables like seismic moment, inter-event time or Euclidean distance between successive earthquakes or length of faults in a given region have been studied through the so-calledq-exponential distributions obtained by maximizing the Tsallis entropy under suitable conditions. These distributions take also the name of generalized Pareto distributions in the context of the limit distributions of excesses. In this work we consider the spatial distribution of a set of earthquakes and its temporal variations by modelling the area of Voronoi cells generated by the epicenters through a generalized Pareto distribution. Following the Bayesian paradigm we analyze the recent seismicity of the central Italy and we compare the posterior marginal likelihood of the aforementioned distributions in shifting time windows. We point out that the best fitting distribution varies over time and the trend of all three distributions converges to that of the exponential distribution in the preparatory phase for the mainshock.

Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect.

This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the “entropy defect” in the composition of a system, we show how the nonextensive entropy of systems with correlations differs from the sum of the entropies of their constituents of these systems. A system is composed extensively when its elementary subsystems are independent, interacting with no correlations; this leads to an extensive system entropy, which is simply the sum of the subsystem entropies. In contrast, a system is composed nonextensively when its elementary subsystems are connected through long-range interactions that produce correlations. This leads to an entropy defect that quantifies the missing entropy, analogous to the mass defect that quantifies the mass (energy) associated with assembling subatomic particles. We develop thermodynamic definitions of kappa and temperature that connect with the corresponding kinetic definitions originated from kappa distributions. Finally, we show that the entropy of a system, composed by a number of subsystems with correlations, is determined using both discrete and continuous descriptions, and find: (i) the resulted entropic form expressed in terms of thermodynamic parameters; (ii) an optimal relationship between kappa and temperature; and (iii) the correlation coefficient to be inversely proportional to the temperature logarithm.

Non-extensive statistical mechanics for acoustic emission in disordered media: Entropy, size effect, and self-organization

The present work explores acoustic emission phenomenon of concrete like disordered material within the framework of non-extensive statistical mechanics. The aim of the study is threefold. At first, we re-derive the non-extensive distribution function using the power-function and exponential-function ansatzes. We assert that the power-function ansatz based distribution model, compared to exponential-function based model, is superior due to its ability to recover the exponential distribution in the limit when the tail index q→1 and due to its higher sensitivity for long-tail. The ability of recovering exponential distribution in the limit q→1 preserves the essence of Tsallis non-extensive statistical mechanics formulation and retains the pertinent meaning of entropic index q of the distribution. Second, we study the size effect on the various parameters of the distribution models and discover the size-independence of the entropic index. Thirdly, the self-organization phenomenon often observed in the complex dynamic systems is commented. The existence of criticality near failure for quasi-static loading of concrete beams appear speculative and the criticality might exist in midway of damage progress.

Anisotropic Kappa Distributions. I. Formulation Based on Particle Correlations

Abstract We develop the theoretical basis for the connection of the variety of anisotropic distributions with the statistical correlations among particles’ velocity components. By examining the most common anisotropic distribution function, we derive the correlation coefficient among particle energies, show how this correlation is connected to the effective dimensionality of the velocity distribution, and derive the connection between anisotropy and adiabatic polytropic index. Having established the importance of the correlation among particles in the formulation of anisotropic kappa distributions, we generalize these distributions within the framework of nonextensive statistical mechanics and based on the types of homogeneous or heterogeneous correlations among the particles’ velocity components. The formulation of the developed generalized distributions mediates the main two types of anisotropic kappa distributions that consider either (a) equal correlations, or (b) zero correlations, among different velocity components. Finally, the developed anisotropic kappa distributions are expressed in terms of the energy and pitch angle in arbitrary reference frames.

Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities.

The concept of duality of probability distributions constitutes a fundamental “brick” in the solid framework of nonextensive statistical mechanics—the generalization of Boltzmann–Gibbs statistical mechanics under the consideration of the q-entropy. The probability duality is solving old-standing issues of the theory, e.g., it ascertains the additivity for the internal energy given the additivity in the energy of microstates. However, it is a rather complex part of the theory, and certainly, it cannot be trivially explained along the Gibb’s path of entropy maximization. Recently, it was shown that an alternative picture exists, considering a dual entropy, instead of a dual probability. In particular, the framework of nonextensive statistical mechanics can be equivalently developed using q- and 1/q- entropies. The canonical probability distribution coincides again with the known q-exponential distribution, but without the necessity of the duality of ordinary-escort probabilities. Furthermore, it is shown that the dual entropies, q-entropy and 1/q-entropy, as well as, the 1-entropy, are involved in an identity, useful in theoretical development and applications.

A new approach to the generalization of Planck’s law of black-body radiation

In this study, Planck’s law of black-body radiation has been modified within the framework of nonextensive statistical mechanics. The average energy of radiation has been derived by introducing the nonextensive partition function in the statistical relation of internal energy. The spectral energy density and spectral radiance have also been computed. The derived expression has been compared with the earlier developed approximate schemes (i.e. asymptotic approach and factorization approach) and with the recently obtained exact result. We utilize the exact and approximate Stefan–Boltzmann laws in order to compare with the new approach introduced here.

Ambulance Service Resource Planning for Extreme Temperatures: Analysis of Ambulance 999 Calls during Episodes of Extreme Temperature in London, UK

The association between episodes of extreme temperature and ambulance 999 calls has not yet been properly quantified. In this study we propose a statistical physics-based method to estimate the true mean number of ambulance 999 calls during episodes of extreme temperatures. Simple arithmetic mean overestimates the true number of calls during such episodes. Specifically, we apply the physics-based framework of nonextensive statistical mechanics (NESM) for estimating the probability distribution of extreme events to model the positive daily variation of ambulance calls. In addition, we combine NESM with the partitioned multiobjective method (PMRM) to determine the true mean of the positive daily difference of calls during periods of extreme temperature. We show that the use of the standard mean overestimates the true mean number of ambulance calls during episodes of extreme temperature. It is important to correctly estimate the mean value of ambulance 999 calls during such episodes in order for the ambulance service to efficiently manage their resources.

Derivation of the entropic formula for the statistical mechanics of space plasmas

Abstract. Kappa distributions describe velocities and energies of plasma populations in space plasmas. The statistical origin of these distributions is associated with the framework of nonextensive statistical mechanics. Indeed, the kappa distribution is derived by maximizing the q entropy of Tsallis, under the constraints of the canonical ensemble. However, the question remains as to what the physical origin of this entropic formulation is. This paper shows that the q entropy can be derived by adapting the additivity of energy and entropy.

Non-Gaussian Closed Form Solutions for Geometric Average Asian Options in the Framework of Non-Extensive Statistical Mechanics.

In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics of returns. Thus, the description of underlying asset price and the pricing of options are more accurate. Moreover, using the martingale method, we obtain closed form solutions for geometric average Asian options. Furthermore, the numerical analysis shows that the model can avoid underestimating risks relative to the Black-Scholes model.

On the Simplification of Statistical Mechanics for Space Plasmas

Space plasmas are frequently described by kappa distributions. Non-extensive statistical mechanics involves the maximization of the Tsallis entropic form under the constraints of canonical ensemble, considering also a dyadic formalism between the ordinary and escort probability distributions. This paper addresses the statistical origin of kappa distributions, and shows that they can be connected with non-extensive statistical mechanics without considering the dyadic formalism of ordinary/escort distributions. While this concept does significantly simplify the usage of the theory, it costs the definition of a dyadic entropic formulation, in order to preserve the consistency between statistical mechanics and thermodynamics. Therefore, the simplification of the theory by means of avoiding dyadic formalism is impossible within the framework of non-extensive statistical mechanics.

Canonical ensemble in non-extensive statistical mechanics

The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, using the maximization of entropy. In this paper we investigate the canonical ensemble in the non-extensive statistical mechanics using a more traditional way, by considering a small system interacting with a large reservoir via short-range forces. The reservoir is characterized by generalized entropy instead of the Boltzmann–Gibbs entropy. Assuming equal probabilities for all available microstates we derive the equations of the non-extensive statistical mechanics. Such a procedure can provide deeper insight into applicability of the non-extensive statistics.

The Generalized Reaction Rate Theories in the Systems with Power-Law Distributions

The calculations of reaction rate are important in studying the different processes in physics, chemistry, biology, engineering etc. There exist many theories of reaction rate, such as transition state theory, collision theory, and unimolecular reaction theory. One key assumption in these theories is that thermodynamic equilibrium prevails throughout the entire system studied for all degrees of freedom. According to the Boltzmann-Gibbs statistical mechanics, a Maxwell-Boltzmann distribution whose form is exponential law holds in the whole time. This exponential law form of reaction rate is usually called Arrhenius behavior in chemistry. However, plenty of experimental results show non-Arrhenius behavior, such as power-law behavior. Besides, in reacting systems, the processes of evolution from one meta-stable state to another neighboring metastable state happen all the time, therefore the equilibrium assumption would be quite far-fetched. In these cases, current theories are no longer valid, and they can not even serve as a conceptual guide for understanding the critical factors that determine rates. Therefore, providing corresponding theoretical description for non-Arrhenius behavior or power-law behavior becomes urgent. In this general paper, generalized reaction rate theories with power-law distributions in the framework of nonextensive statistical mechanics were discussed based on experimental and observing facts.

Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas

AbstractEmpirical kappa distributions provide a straightforward replacement of the Maxwell distribution for systems out of thermal equilibrium such as space plasmas. Kappa distributions have become increasingly widespread across space physics with the number of relevant publications following, remarkably, an exponential growth rate. However, a breakthrough in the field came with the connection of kappa distributions with the framework of nonextensive statistical mechanics. This introductory paper clarifies fundamental physical concepts and provides mathematical formulations of the theory of kappa distributions, which are a consequence of the connection of kappa distributions with a solid statistical background. Among others, the paper presents the existence of a consistent definition of temperature in systems out of thermal equilibrium described by kappa distributions, the physical meaning of the kappa index, and the formulation of the kappa distribution of a Hamiltonian. In addition, the paper examines the most frequent values of kappa indices in space plasmas. Statistical analysis reveals trends between the characteristic values of density, temperature, and kappa index of space plasmas. Finally, understanding the kinetic interpretation of the temperature as the mean kinetic energy, and of the kappa index as the correlation of kinetic energies, helps to develop all the possible formulations of isotropic/anisotropic kappa distributions.

Probabilistic model of N correlated binary random variables and non-extensive statistical mechanics

Abstract The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, thus simple models exhibiting some important properties described by the non-extensive statistical mechanics are useful to provide deeper physical insights. In this article we present a simple model, consisting of a one-dimensional chain of particles characterized by binary random variables, that exhibits both the extensivity of the generalized entropy with q 1 and a q -Gaussian distribution in the limit of the large number of particles.

Continuous transition from the extensive to the non-extensive statistics in an agent-based herding model

Systems with long-range interactions often exhibit power-law distributions and can by described by the non-extensive statistical mechanics framework proposed by Tsallis. In this contribution we consider a simple model reproducing continuous transition from the extensive to the non-extensive statistics. The considered model is composed of agents interacting among themselves on a certain network topology. To generate the underlying network we propose a new network formation algorithm, in which the mean degree scales sub-linearly with a number of nodes in the network (the scaling depends on a single parameter). By changing this parameter we are able to continuously transition from short-range to long-range interactions in the agent-based model.

“Lagrangian Temperature”: Derivation and Physical Meaning for Systems Described by Kappa Distributions

The paper studies the “Lagrangian temperature” defined through the entropy maximization in the canonical ensemble, which is the negative inverse Lagrangian multiplier corresponding to the constraint of internal energy. The Lagrangian temperature is derived for systems out of thermal equilibrium described by kappa distributions such as space plasmas. The physical meaning of temperature is manifested by the equivalency of two different definitions, that is, through Maxwell’s kinetic theory and Clausius’ thermodynamics. The equivalency of the two definitions is true either for systems at thermal equilibrium described by Maxwell distributions or for systems out of thermal equilibrium described by kappa distributions, and gives the meaning of the actual temperature, that is, the real or measured temperature. However, the third definition, that of the Lagrangian temperature, coincides with the primary two definitions only at thermal equilibrium, and thus, in the general case of systems out of thermal equilibrium, it does not represent the actual temperature, but it is rather a function of this. The paper derives and examines the exact expression and physical meaning of the Lagrangian temperature, showing that it has essentially different content to what is commonly thought. This is achieved by: (i) maximizing the entropy in the continuous description of energy within the general framework of non-extensive statistical mechanics, (ii) using the concept of the “N-particle” kappa distribution, which is governed by a special kappa index that is invariant of the degrees of freedom and the number of particles, and (iii) determining the appropriate scales of length and speed involved in the phase-space microstates. Finally, the paper demonstrates the behavior of the Lagrangian against the actual temperature in various datasets of space plasmas.

Carnot cycle for interacting particles in the absence of thermal noise.

A thermodynamic formalism is developed for a system of interacting particles under overdamped motion, which has been recently analyzed within the framework of nonextensive statistical mechanics. It amounts to expressing the interaction energy of the system in terms of a temperature θ, conjugated to a generalized entropy s(q), with q = 2. Since θ assumes much higher values than those of typical room temperatures T ≪ θ, the thermal noise can be neglected for this system (T/θ ≃ 0). This framework is now extended by the introduction of a work term δW which, together with the formerly defined heat contribution (δ Q = θ ds(q)), allows for the statement of a proper energy conservation law that is analogous to the first law of thermodynamics. These definitions lead to the derivation of an equation of state and to the characterization of s(q) adiabatic and θ isothermic transformations. On this basis, a Carnot cycle is constructed, whose efficiency is shown to be η = 1-(θ(2)/θ(1)), where θ(1) and θ(2) are the effective temperatures of the two isothermic transformations, with θ(1)>θ(2). The results for a generalized thermodynamic description of this system open the possibility for further physical consequences, like the realization of a thermal engine based on energy exchanges gauged by the temperature θ.

Nonextensive nuclear liquid–gas phase transition

We study an effective relativistic mean-field model of nuclear matter with arbitrary proton fraction at finite temperature in the framework of nonextensive statistical mechanics, characterized by power-law quantum distributions. We investigate the presence of thermodynamic instability in a warm and asymmetric nuclear medium and study the consequent nuclear liquid–gas phase transition by requiring the Gibbs conditions on the global conservation of baryon number and electric charge fraction. We show that nonextensive statistical effects play a crucial role in the equation of state and in the formation of mixed phase also for small deviations from the standard Boltzmann–Gibbs statistics.

The thermodynamic stability criterion in a self-gravitational system with phase transition

In this paper, we analyze the heat capacity in a self-gravitational system, where there are two phases, in the framework of nonextensive statistical mechanics. We define one value which is the ratio of gas mass to total mass of the system. When the mass ratio is less than a critical value, the heat capacity of the system is positive, so the system is stable. If it is higher than this critical value, the system will be unstable. Then the thermodynamic instability, coupled with the gravitational instability, stimulates the evolution of the system. The liquefaction of the gas in the system, due to the high pressure and temperature inside it, leads to the decrease of gas mass ratio. This will continue until the system becomes stable. Therefore, one can find that phase transition plays an important role in maintaining the stability of such system.

Dynamical analogy between economical crisis and earthquake dynamics within the nonextensive statistical mechanics framework

The field of study of complex systems considers that the dynamics of complex systems are founded on universal principles that may be used to describe a great variety of scientific and technological approaches of different types of natural, artificial, and social systems. Several authors have suggested that earthquake dynamics and the dynamics of economic (financial) systems can be analyzed within similar mathematical frameworks. We apply concepts of the nonextensive statistical physics, on time-series data of observable manifestations of the underlying complex processes ending up with these different extreme events, in order to support the suggestion that a dynamical analogy exists between a financial crisis (in the form of share or index price collapse) and a single earthquake. We also investigate the existence of such an analogy by means of scale-free statistics (the Gutenberg–Richter distribution of event sizes). We show that the populations of: (i) fracto-electromagnetic events rooted in the activation of a single fault, emerging prior to a significant earthquake, (ii) the trade volume events of different shares/economic indices, prior to a collapse, and (iii) the price fluctuation (considered as the difference of maximum minus minimum price within a day) events of different shares/economic indices, prior to a collapse, follow both the traditional Gutenberg–Richter law as well as a nonextensive model for earthquake dynamics, with similar parameter values. The obtained results imply the existence of a dynamic analogy between earthquakes and economic crises, which moreover follow the dynamics of seizures, magnetic storms and solar flares.