Entropic Index Research Articles

Fractal structure of Yang-mills fields

The origin of non-extensive thermodynamics in physical systems has been under intense debate for the last decades. Recent results indicate a connection between non-extensive statistics and thermofractals. After reviewing this connection, we analyze how scaling properties of Yang-Mills theory allow the appearance of self-similar structures in gauge fields. The presence of such structures, which actually behave as fractals, allows for recurrent non-perturbative calculations of vertices. It is argued that when a statistical approach is used, the non-extensive statistics is obtained, and the Tsallis entropic index, q, is deduced in terms of the field theory parameters. The results are applied to QCD in the one-loop approximation, resulting in a good agreement with the value of q obtained experimentally.

Hollowness effect and entropy in high energy elastic scattering

This paper presents a qualitative explanation for the hollowness effect based on the inelastic overlap function, claiming this result is a consequence of fundamental thermodynamic processes. Using the Tsallis entropy, one identifies the entropic index w with the ratio of the collision energy to critical one in the total cross-section. The integrated probability density function is replaced by the inelastic overlap function, which represents the probability of occurrence of an inelastic event depending on both the collision energy and impact parameter. The Coulomb potential, as well as the confinement potential, are used as naive approaches to describe the (internal) energy of the colliding hadrons. The Coulomb potential in the impact parameter picture is not able to furnish any reliable physical result near the forward direction. However, the confinement potential in the impact parameter space results in the hollowness effect shown by the inelastic overlap function near the forward direction.

Entropy formula of N-body system

We prove a proposition that the entropy of the system composed of finite N molecules of ideal gas is the q-entropy or Havrda–Charvát–Tsallis entropy, which is also known as Tsallis entropy, with the entropic index q=frac{D(N-1)-4}{D(N-1)-2} in D-dimensional space. The indispensable infinity assumption used by Boltzmann and others in their derivation of entropy formulae is not involved in our derivation, therefore our derived formula is exact. The analogy of the N-body system brings us to obtain the entropic index of a combined system q_C formed from subsystems having different entropic indexes q_A and q_B as frac{1}{1-q_C}=frac{1}{1-q_A}+frac{1}{1-q_B}+frac{D+2}{2}. It is possible to use the number N for the physical measure of deviation from Boltzmann entropy.

Propagation and interaction of dust ion acoustic solitary waves(DIASWs) for the damped forced modified Korteweg-de-Vries-Burger equation at some critical composition of parameters

The analytical solution of dust-ion-acoustic-solitary waves (DIASWs)with the influence of external periodic force are reported in a dusty plasma model which consists of dust-ion collisions. Using reduction perturbation technique (RPT) damped modified forced Korteweg-de-Vries-Burger (DMFKdVB)equation in some critical composition is obtained. Various physical parameters i.e. dust ion collision frequency (νid0), the entropic index (q), the velocity of travelling wave (M0) ,the ratio of the densities between electrons and ions (μ), viscosity coefficient (η), frequency (w) of the external perturbation and the strength(f0) of the perturbation are observed. This study may be helpful to interpret the characteristics of DIASWs in astrophysical rings, interstellar clouds and comet tails.

Formation of the radio frequency sheath of plasma with Cairns–Tsallis electron velocity distribution

The effect of the non-Maxwellian plasma with enhanced electron tails on the properties of the radio frequency (RF) sheath is studied with a one-dimensional collisionless model, which consists of the sheath model and the equivalent circuit model. In the sheath model, electrons are assumed to obey the Cairns–Tsallis distribution. For various entropic indices q characterizing the degree of electron nonextensivity and parameter α measuring the electron nonthermality state, the electron nonextensivity and nonthermality are found to modify the potential drop across the sheath and the sheath thickness, as well as the spatiotemporal variations of the potential, the ion and electron densities inside the sheath. With the decrease in q and the increase in α, the potential drop across the sheath and the thickness increase at any time in a RF cycle as a result of the increase in superthermal electrons in the non-Maxwellian tail. The dependence of the potential drop across the sheath on q and α is deeply related to the frequency and amplitude of the disturbance current. When the electron nonextensivity and nonthermality are strengthened, the enhancement of the sheath potential drop can cause a significant increase in the ion bombardment energy on the wall, sheath power dissipation, and plasma energy flux to the wall.

Binary indices of time series complexity measures and entropy plane

Complexity of time series is an important feature of dynamical systems such as financial systems. In order to bring out the complete non-linear behavior of financial time series, non-linear tools of complexity measurement like entropy measures are indispensable. Sample entropy (SampEn) and distribution entropy (DistEn) are popular methods of assessing the complexity in various fields. However, both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets. Therefore, we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane. In order to further improve the accuracy of the research results, we take the embedding dimension m as the variable, expand the entropy point into the entropy curve for further research. Furthermore, considering the shortcomings of sample entropy and distribution entropy in practical application, we generalize them by (i) replacing sample entropy and distribution entropy with multi-scale sample entropy and multi-scale distribution entropy; (ii) generalizing Shannon entropy as Tsallis entropy. Also, scale factor τ and entropic index q are taken respectively as variables to get the entropy curves. By using the artificial data, we confirm the rationality of using entropy points and entropy curves to study the complexity of time series. Finally, we apply this method to measure the complexity of real world financial time series, the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences.

Entropic mobility index as a measure of (in)equality of opportunity

We propose a normative mobility index for the measurement of opportunity (in)equality by applying quantum entropy to transition matrices.

Acoustic emissions in compression of building materials: q-statistics enables the anticipation of the breakdown point

In this paper we present experimental results concerning Acoustic Emission (AE) recorded during cyclic compression tests on two different kinds of brittle building materials, namely concrete and basalt. The AE inter-event times were investigated through a non-extensive statistical mechanics analysis which shows that their complementary cumulative probability distributions follow q -exponential laws. The entropic index q and the relaxation parameter β q ≡ 1∕T q , obtained by fitting the experimental data, exhibit systematic changes during the various stages of the failure process, namely (q , T q ) linearly align. The T q = 0 point corresponds to the macroscopic breakdown of the material. The slope, including its sign, of the linear alignment appears to depend on the chemical and mechanical properties of the sample. These results provide an insight on the warning signs of the incipient failure of building materials and could therefore be used in monitoring the health of existing structures such as buildings and bridges.

Precursory variations of Tsallis non-extensive statistical mechanics entropic index associated with the M9 Tohoku earthquake in 2011

Applying natural time analysis, which has been introduced by the authors in 2001, to the Japanese seismic data, we find that the system enters the critical stage upon the occurrence of M = 4.2–5.0 earthquakes from 08:36 to 13:14 local time (LT) on 10 March 2011, i.e., almost one day before the M 9 Tohoku earthquake on 11 March 2011. Here, we find that just before this period the Tsallis entropic index q of non-extensive statistical mechanics started to show distinct changes. In addition an evident change of q is found upon the occurrence of the M 7.3 foreshock at 11:45 LT on 9 March 2011, which exhibits a scaling behavior with a characteristic exponent 1∕3 that conforms to Lifshitz-Slyozov-Wagner theory for phase transitions.

The relaxation processes of Pressure Stimulated Currents under the concept of Non-extensive statistical physics

This study presents the results of Pressure Stimulated Currents (PSC) analysis using non-extensive statistical physics (NESP) approach when marble and amphibolite specimens were subjected to mechanical stress up to fracture. The specimens were subjected to a constant uniaxial stress, interrupted by an abrupt step-wise stress increase (Step-Stress Technique - SST) while concurrently the PSC emission was recorded. The loading protocol involved eight sequential steps of mechanical stress at a gradually higher level until the failure of the specimens. After each step, the stress was maintained constant until the PSC signal was restored to a final low value. After the PSC restoration the next stress step was applied. The recorded PSC temporal relaxation when the stress remained constant, was analyzed under the concept of NESP based on Tsallis’ entropy and the behavior of the entropic index q was studied for various stress levels. Throughout each next loading step at higher level, the entropic index q shows a progressive increase from 1.15 to 1.40 until the stress attains about 85% of the specimens’ total strength where it reaches a maximum value close to 1.4. When the applied stress becomes higher than the 85% of the specimens’ total strength, the entropic index q gradually decreases. It should be noted that this peak along with the subsequent decline of q coexists with the excessive damage development in the specimens’ bulk due to the applied mechanical stress. Taking into consideration the fact that the entropic index q quantifies the self-organization of the system, this behavior of q, for stress levels higher than 85% of the specimens’ ultimate stress strength, clearly corresponds to dynamic processes that dominate the system and guide the upcoming fracture. The above findings advocate the use of the entropic index q as a pre-failure indicator of the upcoming specimen fracture.

Analytical solitary wave solution of the dust ion acoustic waves for the damped forced modified Korteweg-de Vries equation in q-nonextensive plasmas

Analytical solitary wave solution of the dust ion acoustic waves is studied due to the damped forced modified Korteweg-de Vries equation in an unmagnetized collisional dusty plasma consisting of negatively charged dust grain, positively charged ions, q-nonextensive electrons, and neutral particles in the presence of external periodic force. Using reductive perturbation technique, the damped forced modified Korteweg-de Vries equation is obtained for the dust ion acoustic waves. Momentum consevation law is used to obtain the dust ion acoustic solitary wave solutions in the framework of the damped forced modified Korteweg-de Vries equation. The effects of different physical parameters such as entropic index, dust ion collisional frequency, strength and frequency of the external periodic force, speed of the traveling wave and the parameter which is the ratio between the unperturbed densities of the dust ions and electrons are investigated on the analytical solution of the dust ion acoustic waves. It is observed that those parameters have significant effects on the structures of the damped forced dust ion acoustic solitary waves. The results of the present paper may have applications in laboratory and space plasma environments.

Study of Gasketed-Plate Heat Exchanger performance based on energy efficiency indexes

The aim of this work is to carry a performance analysis based on first and second law of thermodynamics for some operational configurations of feasible gasketed-plate heat exchangers. To achieve this, 40 simulations were done solving the distributed-U differential model proposed by Pinto and Gut, using an adaptive damped secant shooting method. Heat and exergy transfer effectiveness, dimensionless entropy generation, entropic potential loss and energy efficiency index were calculated when both fluids are above and below room temperature, as well as at least one fluid goes across the room temperature. Additionally, four cold fluid inlet port configuration have been studied. Our main conclusion is that countercurrent configurations have better performance than parallel flow configurations due to influence of finite temperature difference in equipment and higher NTU are reached when fluids are above room temperature for the same geometry. Since gasketed-Plate heat exchangers have applications in different industries, our work constitutes a tool and a guide to select operating and installation conditions.

Analysis of temporal and spatial distributions between earthquakes in the region of California through Non-Extensive Statistical Mechanics and its limits of validity

In this work we study the influence of considering partial sets of earthquakes data has on the temporal and spatial probability distributions of earthquakes, using data from the California region between 2003 and 2016, with different thresholds for magnitude and depth of hypocenters. For this we have considered sequences of earthquakes where we have “jump”, or pulled, a given fixed number of actually sequential earthquakes. Through the concepts of Non-Extensive Statistical Mechanics for the time interval between earthquakes, we have found that the increase of the jump between earthquakes forming the sequence, affects the non-extensive characteristic of the system in the temporal probability distribution, denoted by a change in the value of the entropic index q, for relatively small jump sizes. However, for the distance between earthquakes, we observe that the increase of jumps lets untouched the non-extensive characteristic of the system, keeping the entropic index values q approximately constant. This analysis allows not just to show the robustness of the Non-Extensive Statistical Mechanics treatments for the study of earthquakes, but also their limits of applicability with respect to jumps or data loss. At the same time, it is a very useful test for memory effects and long-range interactions.

A nonextensive insight into the stellar initial mass function

In the present paper, we propose that the stellar initial mass distributions known as IMF are best fitted by q-Weibull distributions that emerge within nonextensive statistical mechanics. As a result, we show that Salpeter's slope of ∼2.35 is replaced when a q-Weibull distribution is used. Our results point out that the nonextensive entropic index q represents a new approach for understanding the process of the star-forming and evolution of massive stars.

Associating an Entropy with Power-Law Frequency of Events.

Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy . The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy , in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature with (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.

Non-parametric application of Tsallis statistics to systems consisting of [formula omitted] hydrogen molecules

We have determined the entropy, the total energy, and the specific heat of the systems consisting of M≥3 hydrogen molecules. The calculations were conducted in the framework of the nonextensive Tsallis statistics. The relation between M and the entropic index q is given by q=1+1∕M, which results from the fact that the temperature of the nanosystems fluctuates around the temperature of the reservoir (Wilk and Włodarczyk, 2000). The electron energy states of the hydrogen molecule have been determined with the help of the Hubbard Hamiltonian, which models all two-body interactions. The Hubbard Hamiltonian integrals have been calculated by using the variational method, whereas the Wannier function has been associated with 1s Slater-type orbitals. We have included the contributions to the energy of the hydrogen molecule coming from the oscillatory (either harmonic or anharmonic), rotational and translational degrees of freedom. In addition, we have investigated the impact of the external force (F) or the magnetic field (h) on the thermodynamic parameters of the systems. In each case the noticeable deviation from the results of the classical statistical physics can be observed for the systems consisting of M<Mc∼103 molecules.

Tsallis Entropy Index q and the Complexity Measure of Seismicity in Natural Time under Time Reversal before the M9 Tohoku Earthquake in 2011.

The observed earthquake scaling laws indicate the existence of phenomena closely associated with the proximity of the system to a critical point. Taking this view that earthquakes are critical phenomena (dynamic phase transitions), here we investigate whether in this case the Lifshitz–Slyozov–Wagner (LSW) theory for phase transitions showing that the characteristic size of the minority phase droplets grows with time as is applicable. To achieve this goal, we analyzed the Japanese seismic data in a new time domain termed natural time and find that an LSW behavior is actually obeyed by a precursory change of seismicity and in particular by the fluctuations of the entropy change of seismicity under time reversal before the Tohoku earthquake of magnitude 9.0 that occurred on 11 March 2011 in Japan. Furthermore, the Tsallis entropic index q is found to exhibit a precursory increase.

Diagnostics of Rotary Equipment by Means of Entropic Indices Based on Electrical Signals

A method is proposed for determining the state of rotary mechanisms on the basis of current signals from their electrical motors, by means of entropic indices. The results indicate that the entropic indices may be used in diagnostics.

Multiplicity spectra in pp collisions at high energies in terms of Gamma and Tsallis distributions

In recent years the Tsallis statistics is gaining popularity in describing charged particle produc- tion and their properties, in particular pT spectra and the multiplicities in high energy particle collisions. Motivated by its success, an analysis of the LHC data of proton-proton collisions at ener- gies ranging from 0.9 TeV to 7 TeV in different rapidity windows for charged particle multiplicities has been done. A comparative analysis is performed in terms of the Tsallis distribution, the Gamma distribution and the shifted-Gamma distribution. An interesting observation on the inapplicability of these distributions at sqrt{s}=7 TeV in the lower rapidity windows is intriguing. The non-extensive nature of the Tsallis statistics is studied by determining the entropic index and its energy depen- dence. The analysis is extrapolated to predict the multiplicity distribution at sqrt{s}=14 TeV for one rapidity window, |y| < 1.5 with the Tsallis function.

The Complexity Measures Associated with the Fluctuations of the Entropy in Natural Time before the Deadly México M8.2 Earthquake on 7 September 2017

We analyse seismicity during the 6-year period 2012–2017 in the new time domain termed natural time in the Chiapas region where the M8.2 earthquake occurred, Mexico’s largest earthquake in more than a century, in order to study the complexity measures associated with fluctuations of entropy as well as with entropy change under time reversal. We find that almost three months before the M8.2 earthquake, i.e., on 14 June 2017, the complexity measure associated with the fluctuations of entropy change under time reversal shows an abrupt increase, which, however, does not hold for the complexity measure associated with the fluctuations of entropy in forward time. On the same date, the entropy change under time reversal has been previously found to exhibit a minimum [Physica A 506, 625–634 (2018)]; we thus find here that this minimum is also accompanied by increased fluctuations of the entropy change under time reversal. In addition, we find a simultaneous increase of the Tsallis entropic index q.