143 research outputs found
Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions
We present the main features of the mathematical theory generated by the Îș-deformed exponential function exp_Îș (x) with 0 †Îș < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The Îș-mathematics has its roots in special relativity and furnishes the theoretical foundations of the Îș-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the Îș-algebra, we present the associated Îș-differential and Îș-integral calculus. Then, we obtain the corresponding Îș-exponential and Îș-logarithm functions and give the Îș-version of the main functions of the ordinary mathematics
Finite-Size Effects on Return Interval Distributions for Weakest-Link-Scaling Systems
The Weibull distribution is a commonly used model for the strength of brittle
materials and earthquake return intervals. Deviations from Weibull scaling,
however, have been observed in earthquake return intervals and in the fracture
strength of quasi-brittle materials. We investigate weakest-link scaling in
finite-size systems and deviations of empirical return interval distributions
from the Weibull distribution function. We use the ansatz that the survival
probability function of a system with complex interactions among its units can
be expressed as the product of the survival probability functions for an
ensemble of representative volume elements (RVEs). We show that if the system
comprises a finite number of RVEs, it obeys the -Weibull distribution.
We conduct statistical analysis of experimental data and simulations that show
good agreement with the -Weibull distribution. We show the following:
(1) The weakest-link theory for finite-size systems involves the
-Weibull distribution. (2) The power-law decline of the
-Weibull upper tail can explain deviations from the Weibull scaling
observed in return interval data. (3) The hazard rate function of the
-Weibull distribution decreases linearly after a waiting time , where is the Weibull modulus and is the system size
in terms of representative volume elements. (4) The -Weibull provides
competitive fits to the return interval distributions of seismic data and of
avalanches in a fiber bundle model. In conclusion, using theoretical and
statistical analysis of real and simulated data, we show that the
-Weibull distribution is a useful model for extreme-event return
intervals in finite-size systems.Comment: 33 pages, 11 figure
Nonlinear Kinetics on Lattices based on the Kinetic Interaction Principle
Master equations define the dynamics that govern the time evolution of
various physical processes on lattices. In the continuum limit, master
equations lead to Fokker-Planck partial differential equations that represent
the dynamics of physical systems in continuous spaces. Over the last few
decades, nonlinear Fokker-Planck equations have become very popular in
condensed matter physics and in statistical physics. Numerical solutions of
these equations require the use of discretization schemes. However, the
discrete evolution equation obtained by the discretization of a Fokker-Planck
partial differential equation depends on the specific discretization scheme. In
general, the discretized form is different from the master equation that has
generated the respective Fokker-Planck equation in the continuum limit.
Therefore, the knowledge of the master equation associated with a given
Fokker-Planck equation is extremely important for the correct numerical
integration of the latter, since it provides a unique, physically motivated
discretization scheme. This paper shows that the Kinetic Interaction Principle
(KIP) that governs the particle kinetics of many body systems, introduced in
[G. Kaniadakis, Physica A, 296, 405 (2001)], univocally defines a very simple
master equation that in the continuum limit yields the nonlinear Fokker-Planck
equation in its most general form.Comment: 26 page
Wave Propagation And Landau-Type Damping In Liquids
Intermolecular forces are modeled by means of a modified Lennard-Jones
potential, introducing a distance of minimum approach, and the effect of
intermolecular interactions is accounted for with a self consistent field of
the Vlasov type. A Vlasov equation is then written and used to investigate the
propagation of perturbations in a liquid. A dispersion relation is obtained and
an effect of damping, analogous to what is known in plasmas as "Landau
damping", is found to take place.Comment: 13 pages, 3 figures, SigmaPhi 2011 conferenc
Energy from Negentropy of Non-Cahotic Systems
: Negative contribution of entropy (negentropy) of a non-cahotic system, representing the
potential of work, is a source of energy that can be transferred to an internal or inserted subsystem.
In this case, the system loses order and its entropy increases. The subsystem increases its energy
and can perform processes that otherwise would not happen, like, for instance, the nuclear fusion
of inserted deuterons in liquid metal matrix, among many others. The role of positive and negative
contributions of free energy and entropy are explored with their constraints. The energy available to
an inserted subsystem during a transition from a non-equilibrium to the equilibrium chaotic state,
when particle interaction (element of the system) is switched off, is evaluated. A few examples are
given concerning some non-ideal systems and a possible application to the nuclear reaction screening
problem is mentioned
Weakest-link scaling and extreme events in finite-sized systems
Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture strength of brittle materials, maximal annual rainfall, wind speed and earthquake return times. We investigate two new distributions that exhibit weakest-link scaling, i.e., a Weibull generalization known as the Îș-Weibull and a modified gamma probability function that we propose herein. We show that in contrast with the Weibull and the modified gamma, the hazard function of the Îș -Weibull is non-extensive, which is a signature of inter-dependence between the links. We also investigate the impact of heterogeneous links, modeled by means of a stochastic Weibull scale parameter, on the observed probability distribution
Advances in statistical physics
The organization of this Conference settled the foundation of a series hold up to following a three-year cycle. The founding declaration, undersigned by more than 100 physicists, is now deposited at the Orthodox Academy of Crete and will move to the sites of the future editions of the Conference. The text of the founding declaration is inspired by the book E, rows 269-275, of the Homerus Odyssey as a metaphor of the knowledge willing pursued by the humanity. The objective of the Conference was to provide the opportunity for interaction and cross-fertilization betwee
The k-statistics approach to epidemiology
A great variety of complex physical, natural and artificial systems are
governed by statistical distributions, which often follow a standard
exponential function in the bulk, while their tail obeys the Pareto power law.
The recently introduced -statistics framework predicts distribution
functions with this feature. A growing number of applications in different
fields of investigation are beginning to prove the relevance and effectiveness
of -statistics in fitting empirical data. In this paper, we use
-statistics to formulate a statistical approach for epidemiological
analysis. We validate the theoretical results by fitting the derived
-Weibull distributions with data from the plague pandemic of 1417 in
Florence as well as data from the COVID-19 pandemic in China over the entire
cycle that concludes in April 16, 2020. As further validation of the proposed
approach we present a more systematic analysis of COVID-19 data from countries
such as Germany, Italy, Spain and United Kingdom, obtaining very good agreement
between theoretical predictions and empirical observations. For these countries
we also study the entire first cycle of the pandemic which extends until the
end of July 2020. The fact that both the data of the Florence plague and those
of the Covid-19 pandemic are successfully described by the same theoretical
model, even though the two events are caused by different diseases and they are
separated by more than 600 years, is evidence that the -Weibull model
has universal features.Comment: 15 pages, 1 table, 5 figure
Quantum Computation and Information: Multi-Particle Aspects
This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers
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