Extensive Statistical Mechanics Research Articles

Observational evidence of non-extensive behavior of seismic coda waves

The last decades a lot of progress has been made in studying Earth and planetary Physics based on the science of complexity and the application of generalized formalisms of statistical physics and entropy. Towards this direction, investigation of the dynamical features of seismic waves is of fundamental importance since they could provide information on the statistical properties of the lithosphere.The non extensive statistical mechanics (NESM) seems to be a promising framework for studying complex earthquake systems that exhibit long-range interactions and memory effects. In this work, coda waves are studied from the point of view of non extensive statistical mechanics. Our aim is to demonstrate the applicability of Non extensive statistical mechanics to the fluctuations of coda amplitudes and to define the entropic q parameters obtained in the coda waves recorded by seismological stations in the South Aegean, and generated by three strong earthquake events in the front of the Hellenic Arc.Coda waves are interpreted as scattered seismic waves by heterogeneities in the earth (cracks, faults, velocity and/or density anomalies, etc.). Due to the nature of those scatterers distribution we follow the NESM approach and study the probability distribution functions of the increments of seismic coda waves generated by seismic events recorded in South Aegean and generated by three strong events.We defined the increment function X(t)=V(t+1)−Vt, where V(t) is the measured horizontal ground velocity. We proceed by analyzing the normalized increments X(t) constructing the probability density function PDFp(x), where x=X−XσX, σX being the standard deviation of X(t) and <X> the observed mean value on the same data set. The results of the present work suggests that the seismic coda waves increments deviate from the Gaussian shape and their respective probability density function px could be adequately described by a q-Gaussian with entropic q-values in the range of1.35<q<2.02 with an error of δq=±0.05.This observation implies that seismic coda wave complexity originating by multiscattering effects, can be described by Tsallis entropy terms. The entropic q-values estimated vary along each seismic path for each of the three strong earthquake events analyzed. This observation implies that the value of q is determined by the particular nature of path between the source and the recorded station.

A Boltzmann-Tsallis approach towards cell survival curve in Radiobiology

This paper outlines a phenomenological approach towards cell survival curve at low dose using tools of extensive Statistical Mechanics and nonextensive Statistical Mechanics. An Ising chain model is developed for the cell survival curve and the canonical ensemble formalism based on Boltzmann Gibbs statistic and Tsallis statistic is presented. The resulting cell survival curve shows excellent agreement with the experimental data and the physical parameters from our Tsallis model (N’, q) can be shown to provide clear classification between healthy and cancerous cells. In this paper, we also provides possible biophysical interpretation to the (N’, q) parameters where N’ is representative of the amount of repairable DNA content in the nucleus and q represents the degree of correlation in DNA damage. Overall, this is the first time a Statistical Mechanics approach is used in Radiobiology, and could present a new perspective.

Thermodynamic properties of graphene using the static fluctuation approximation (SFA)

The thermodynamic properties of two-dimensional graphene nanosystems are investigated using the static fluctuation approximation (SFA). These properties are analyzed using both extensive and nonextensive statistical mechanics. It is found that these properties are less sensitive to temperature when using nonextensive — in contrast to extensive — statistical mechanics. It is also noted that the mean internal energy and the specific heat behave as a power law, Tα, at T &lt; 8 eV; whereas they go to the classical limit for the two-dimensional ideal gas at T &gt; 8 eV. The results are presented in a set of figures and one table. The roles played by the number of particles and the entropy parameter q are underlined. Whenever possible, comparisons are made to previous studies. It is concluded that Boltzmann–Gibbs statistics are not valid for some cases, and that SFA results are in good agreement with those obtained within other formalisms.

On application of non—extensive statistical mechanics to studying ecological diversity

The concept of Tsallis entropy provides an extension of thermodynamics and statistical physics. In the ecology, Tsallis entropy is proposed to be a new class of diversity indices Sq which covers many common diversity indices found in ecological literature. As a new statistical model for the Whittaker plots describing species abundance distribution, the truncated exponential distribution is used to calculate the diversity and evenness indices. The obtained results in new model are graphically compared with those in previous publication in the same field of interests, and shows a good agreement. A further development of a thermodynamic theory of ecological systems that is consistent with entropic approach of statistical physics is motivated.

Complexity of Economical Systems

In this study new theoretical concepts are described concerning the interpretation of economical complex dynamics. In addition a summary of an extended algorithm of nonlinear time series analysis is provided which is applied not only in economical time series but also in other physical complex systems (e.g. [22, 24]). In general, Economy is a vast and complicated set of arrangements and actions wherein agents—consumers, firms, banks, investors, government agencies—buy and sell, speculate, trade, oversee, bring products into being, offer services, invest in companies, strategize, explore, forecast, compete, learn, innovate, and adapt. As a result the economic and financial variables such as foreign exchange rates, gross domestic product, interest rates, production, stock market prices and unemployment exhibit large-amplitude and aperiodic fluctuations evident in complex systems. Thus, the Economics can be considered as spatially distributed non-equilibrium complex system, for which new theoretical concepts, such as Tsallis non extensive statistical mechanics and strange dynamics, percolation, nonGaussian, multifractal and multiscale dynamics related to fractional Langevin equations can be used for modeling and understanding of the economical complexity locally or globally. Kewwords: Economical Dynamics, Time series analysis, Tsallis non-extensive statistics, Fractal Dynamics, Non-equilibrium dynamics, Turbulence.

Study on Traditional Extensive Statistical Mechanics in Non-extension System

Study on Traditional Extensive Statistical Mechanics in Non-extension System

Study on Extensive Statistical Mechanics by Nonextensivity

This paper proposes a phase rule for non-extensive systems and asserts that the thermodynamic limit commonly used in extensive statistical mechanics is no longer applicable when considering non-extensive quantities. In addition, this paper revises some of the concepts and conclusions arising in extensive statistical mechanics.

Entropic lattice Boltzmann representations required to recover Navier-Stokes flows

There are two disparate formulations of the entropic lattice Boltzmann scheme: one of these theories revolves around the analog of the discrete Boltzmann H function of standard extensive statistical mechanics, while the other revolves around the nonextensive Tsallis entropy. It is shown here that it is the nonenforcement of the pressure tensor moment constraints that lead to extremizations of entropy resulting in Tsallis-like forms. However, with the imposition of the pressure tensor moment constraint, as is fundamentally necessary for the recovery of the Navier-Stokes equations, it is proved that the entropy function must be of the discrete Boltzmann form. Three-dimensional simulations are performed which illustrate some of the differences between standard lattice Boltzmann and entropic lattice Boltzmann schemes, as well as the role played by the number of phase-space velocities used in the discretization.

Insurance pricing in small size markets

In analogy with standard derivation of Tsallis factor in non extensive statistical mechanics, we find the wealth distribution for an economic agent in a conservative exchange market. Tsallis entropic index distinguish between two different regimes, the large and small size market. The Pareto like wealth distribution is obtained in the case of small size market. We consider the insurance market as an example of conservative exchange market and suggest a new method for insurance pricing based on the wealth distribution in the market. We generalize the Esscher transform for the insurance pricing and simulate a real case of the car insurance. The results show the initial wealth of an insurer company in a small size market should be greater than a threshold value.

Extensive statistical mechanics based on nonadditive entropy: Canonical ensemble

The original canonical ensemble formalism for the nonextensive entropy thermostatistics is reconsidered. It is shown that the unambiguous connection of the statistical mechanics with the equilibrium thermodynamics is provided if the entropic parameter $1/(q-1)$ is an extensive variable of the state. Based on a particular example of the perfect gas, it is proved that the Tsallis thermostatistics meets all the requirements of equilibrium thermodynamics in the thermodynamic limit. In particular, the entropy of the system is extensive and the temperature is intensive. However, for finite systems both the Tsallis and Boltzmann-Gibbs entropies are nonextensive. The equivalence of the canonical and microcanonical ensembles of Tsallis thermostatistics in the thermodynamic limit is established. The issue associated with physical interpretation of the entropic variable is discussed in detail.

Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics

We explain the ubiquity and extremely slow evolution of non-Gaussian out-of-equilibrium distributions for the Hamiltonian mean-field model, by means of traditional kinetic theory. Deriving the Fokker-Planck equation for a test particle, one also unambiguously explains and predicts striking slow algebraic relaxation of the momenta autocorrelation, previously found in numerical simulations. Finally, angular anomalous diffusion are predicted for a large class of initial distributions. Non-extensive statistical mechanics is shown to be unnecessary for the interpretation of these phenomena.

The spacetime associated with galactic dark matter halos

We show how an adequate post--Newtonian generalization can be obtained for Newtonian dark matter halos associated with an empiric density profile. Applying this approach to halos that follow from the well known numerical simulations of Navarro, Frenk and White (NFW), we derive all dynamical variables and show that NFW halos approximatelly follow an ideal gas type of equation of state which fits very well to a polytropic relation in the region outside the core. This fact suggests that ``outer'' regions of NFW halos might be related to equilibrium states in the non--extensive Statistical Mechanics formalism proposed by Tsallis.

Jeans’ criterion in nonextensive statistical mechanics

The Jeans’ gravitational instability in nonextensive statistical mechanics is studied and a general form of the generalized Jeans’ criterion is obtained that is related to the q-function c q=∑ i p i q . In this approach, the nonextensive model of classical ideal gas is applied to the Jeans’ problem instead of the ordinary one in extensive statistical mechanics and the generalized critical wavelength to describe the gravitational instability is deduced. This nonextensive modification of the Jeans’ criterion leads to a new critical length that depends not only on the nonextensive parameter q but also on the dimension D and the total particle numbers N of the system. When q→1−, the Jeans’ length is perfectly recovered. We also give the nonextensive parameter q a physical interpretation that q=1 represents an isothermal process of the gas, corresponding to the state of complete mixing, but 0< q<1 is nonisothermal, corresponding to the state of incomplete mixing, it measures the degree of mixing.

Renormalization-Group Calculation of Local Magnetizations and Correlations: Random-Bond, Random-Field, and Spin-Glass Systems

We show that extensive statistical mechanics information can be obtained by the calculation of local densities such as local magnetizations and neighbor correlations, made possible by renormalization-group theory. This is illustrated with quenched random-bond, random-field, and spin-glass systems, in two and three dimensions. The different ordering behaviors thus become vividly and quantitatively apparent.