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This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo's derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry. [ABSTRACT FROM AUTHOR]

Scale-free networks constitute a fast-developing field that has already provided us with important tools to understand natural and social phenomena. From biological systems to environmental modifications, from quantum fields to high energy collisions, or from the number of contacts one person has, on average, to the flux of vehicles in the streets of urban centres, all these complex, non-linear problems are better understood under the light of the scale-free network's properties. A few mechanisms have been found to explain the emergence of scale invariance in complex networks, and here we discuss a mechanism based on the way information is locally spread among agents in a scale-free network. We show that the correct description of the information dynamics is given in terms of the q-exponential function, with the power-law behaviour arising in the asymptotic limit. This result shows that the best statistical approach to the information dynamics is given by Tsallis Statistics. We discuss the main properties of the information spreading process in the network and analyse the role and behaviour of some of the parameters as the number of agents increases. The different mechanisms for optimization of the information spread are discussed. [ABSTRACT FROM AUTHOR]

Entropy is a great tool in thermodynamics and statistical physics. This Special Issue is devoted to the mathematical background and physical motivation behind using non-additive entropy formulas, which are, in most cases, still group entropies, defined as expectation values, and incorporate, in one way or another, nontrivial correlations in the investigated systems and processes. The concept of entropy has been changing swiftly since C. Tsallis and, before him, a series of mathematicians and informatics experts proposed various generalizations of Boltzmann-Gibbs entropy. [Extracted from the article]

The Fokker-Planck Equation (FPE) is a fundamental tool for the investigation of kinematic aspects of a wide range of systems. The Plastino-Plastino Equation (PPE) is the correct generalization describing the kinematic evolution of complex systems consistent with q -statistics. In the present work, we use this particular problem to compare the results obtained with the FPE and the PPE, and discuss the different aspects of the dynamical evolution of the system according to the solutions for each equation. We observe clear differences in the solutions for all the cases studied here and discuss possible experimental investigations that can indicate which of those equations better describes the heavy-quark kinematics in the medium. The results obtained here have implications in the study of anomalous diffusion in porous and granular media, in Cosmology and Astrophysics. The obtained results reinforce the validity of the relation (q − 1) − 1 = (11 / 3) N c − (4 / 3) (N f / 2) , where N c and N f are, respectively, the number of colours and the effective number of flavours. This equation was recently established in the context of a fractal approach to QCD in the non-perturbative regime. [ABSTRACT FROM AUTHOR]

The nonextensive statistics proposed by Tsallis has found wide applicability, being present even in the description of experimental data from high energy collisions. A system with a fractal structure in its energy-momentum space, named thermofractal, was shown to be described thermodynamically by the nonextensive statistics. Due to the many common features between thermofractals and Hagedorn’s fireballs, this system offers the possibility of investigating the origins of nonextensivity in hadronic physics and in QCD. In this regard, the investigation of the scaling properties of thermofractals through the renormalization group equation, known as Callan–Symanzik equation, can be an interesting approach. [ABSTRACT FROM AUTHOR]

In this work, we study the formal connections between the non-linear Fokker-Planck Equation associated with the non-additive entropy and the Boltzmann Equation with the non-additive correlation functional. The collisional term following the q -algebra is adopted. In the derivation of the non-additive Fokker-Planck Equation, two constraints are imposed on the final result: i) that the entropic index q is a characteristic parameter of the non-additive systems with a value that does not change with time, and ii) that for q → 1 a smooth transition for the standard Fokker-Planck Equation is obtained. [ABSTRACT FROM AUTHOR]

We study the applications of nonextensive Tsallis statistics to high energy and hadron physics. These applications include studies of p p collisions, equation of state of QCD, as well as Bose–Einstein condensation. We also analyze the connections of Tsallis statistics with thermofractals, and address some of the conceptual aspects of the fractal approach, which are expressed in terms of the renormalization group equation and the self-energy corrections to the parton mass. We associate these well-known concepts with the origins of the fractal structure in the quantum field theory. [ABSTRACT FROM AUTHOR]

The nonadditive entropy introduced by Tsallis in 1988 has been used in different fields and generalizes the Boltzmann entropy, extending the possibilities of the application of the statistical methods developed in the context of Mechanics. Here, we investigate one of the last points of the theory that is still under discussion: the source term of the nonextensive transport equation. Based on a simple system, we show that the nonadditivity is a direct consequence of the phase space topology and derive the source term that leads to the nonextensive transport equation. [ABSTRACT FROM AUTHOR]

This study focuses on the presence of (multi)fractal structures in hadronic matter through the momentum distributions of mesons produced in proton-proton collisions between 23 GeV and 63 GeV. The analysis demonstrates that the q-exponential behaviour of the particle momentum distributions is consistent with fractal characteristics, exhibiting fractal structures in confined hadronic matter with features similar to those observed in the deconfined quark-gluon plasma (QGP) regime. Furthermore, the systematic analysis of meson production in hadronic collisions at energies below 1 TeV suggests that specific fractal parameters are universal, independently of confinement or deconfinement, while others may be influenced by the quark content of the produced meson. These results pave the way for further research exploring the implications of fractal structures on various physical distributions. [ABSTRACT FROM AUTHOR]

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that provided evidence of the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan–Symanzik equation. [ABSTRACT FROM AUTHOR]

This study investigates the interconnections between the traditional Fokker-Planck Equation (FPE) and its fractal counterpart (FFPE), utilizing fractal derivatives. By examining the continuous approximation of fractal derivatives in the FPE, it derives the Plastino-Plastino Equation (PPE), which is commonly associated with Tsallis Statistics. This work deduces the connections between the entropic index and the geometric quantities related to the fractal dimension. Furthermore, it analyzes the implications of these relationships on the dynamics of systems in fractal spaces. To assess the effectiveness of both equations, numerical solutions are compared within the context of complex systems dynamics, specifically examining the behaviours of quark-gluon plasma (QGP). The FFPE provides an appropriate description of the dynamics of fractal systems by accounting for the fractal nature of the momentum space, exhibiting distinct behaviours compared to the traditional FPE due to the system's fractal nature. The findings indicate that the fractal equation and its continuous approximation yield similar results in studying dynamics, thereby allowing for interchangeability based on the specific problem at hand. [ABSTRACT FROM AUTHOR]

In this work we revisit the MIT bag model to describe quark matter within both the usual Fermi-Dirac and the Tsallis statistics. We verify the effects of the non-additivity of the latter by analysing two different pictures: the first order phase transition of the QCD phase diagram and stellar matter properties. While the QCD phase diagram is visually affected by the Tsallis statistics, the resulting effects on quark star macroscopic properties are barely noticed. [ABSTRACT FROM AUTHOR]

The midrapidity transverse momentum distributions of the charged pions, kaons, protons, and antiprotons, measured by ALICE Collaboration at ten centrality classes of Pb + Pb collisions at s n n   = 5.02 TeV in the Large Hadron Collider (LHC, CERN, Switzerland), are successfully analyzed using combined minimum χ2 fits with a thermodynamically non-consistent, as well as thermodynamically consistent, Tsallis function with transverse flow. The extracted non-extensivity parameter q decreases systematically for all considered particle species with increasing Pb + Pb collision centrality, suggesting an increase in the degree of system thermalization with an increase in collision centrality. The results for q suggest quite a large degree of thermalization of quark–gluon plasma (QGP) created in central Pb + Pb collisions at s n n   = 5.02 TeV with the average number of participant nucleons 〈 N p a r t 〉 > 160. The obtained significantly different growth rates of transverse flow velocity, 〈 β T 〉 , in regions 〈 N p a r t 〉 < 71 ± 7 and 〈 N p a r t 〉 > 71 ± 7 with the temperature parameter T0 remaining constant within uncertainties in region 〈 N p a r t 〉 > 71 ± 7 probably indicates that 〈 N p a r t 〉 ≈ 71 ± 7 (corresponding to 〈 d N c h / d η 〉 ≈ 251 ± 20) is a threshold border value for a crossover transition from a dense hadronic state to the QGP phase (or mixed phase of QGP and hadrons) in Pb + Pb collisions at s n n = 5.02 TeV. The threshold border value for transverse flow velocity 〈 β T 〉 ≈ 0.46 ± 0.03 (corresponding to 〈 N p a r t 〉 ≈ 71 ± 7), estimated by us in Pb + Pb collisions at s n n = 5.02 TeV, agrees well with the corresponding border value 〈 β T 〉 ≈ 0.44 ± 0.02, recently obtained in Xe + Xe collisions at s n n   = 5.44 TeV, and with almost constant 〈 β T 〉 values extracted earlier in the Beam Energy Scan (BES) program of the Relativistic Heavy-Ion Collider (RHIC, Brookhaven, GA, USA) in central Au + Au collisions in the s n n   = 7.7 − 39 GeV energy range, where the threshold for QGP production is achieved. The correlations between extracted T0 and 〈 β T 〉 parameters are found to be greatly different in regions 〈 β T 〉 < 0.46 and 〈 β T 〉 > 0.46, which further supports our result obtained for the threshold border value in Pb + Pb collisions at s n n   = 5.02 TeV. [ABSTRACT FROM AUTHOR]

Particle production yields measured in central Au-Au collision at RHIC are obtained with free Fermi and Bose gases and also with a replacement of these statistics by non-extensive statistics. For the latter calculation, a set of different parameters was used with values of the Tsallis parameter q chosen between 1.01 and 1.25, with 1.16 generating the best agreement with experimental data, an indication that non-extensive statistics may be one of the underlying features in heavy ion-collisions. [ABSTRACT FROM AUTHOR]

The present paper reports on the methods of the systematic analysis of the high-energy collision distributions—in particular, those adopted by Jean Cleymans. The analysis of data on high-energy collisions, using non-extensive statistics, represents an important part of Jean Cleymans scientific activity in the last decade. The methods of analysis, developed and employed by Cleymans, are discussed and compared with other similar methods. As an example, analyses of a set of the data of proton-proton collisions at the center-of-mass energies, s = 0.9 and 7 TeV, are provided applying different methods and the results obtained are discussed. This line of research has the potential to enlarge our understanding of strongly interacting systems and to be continued in the future. [ABSTRACT FROM AUTHOR]

The non extensive thermodynamics of an ideal gas composed by bosons and/or fermions is derived from its partition function for systems with finite chemical potentials. It is shown that the thermodynamical quantities derived in the present work are in agreement with those obtained in previous works when μ ≤ m . However some inconsistencies of previous references are corrected when μ > m . A discontinuity in the first derivatives of the partition function and its effects are discussed in detail. We show that at similar conditions, the non extensive statistics provide a harder equation of state than that provided by the Boltzmann–Gibbs statistics. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we report results of a quantitative analysis of the effects of neutrons on DNA, and, specifically, the production of simple and double breaks of plasmid DNA in aqueous solutions with different concentrations of free-radical scavengers. The radiation damage to DNA was evaluated by electrophoresis through agarose gels. The neutron and gamma doses were measured separately with thermoluminescent detectors. In this work, we have also demonstrated usefulness of a new system for positioning and removing samples in channel BH#3 of the IEA-R1 reactor at the Instituto de Pesquisas Energéticas e Nucleares (Brazil) without necessity of interrupting the reactor operation. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a new mechanical samples positioning system that allows the safe placement and removal of biological samples for prolonged irradiation, in a nuclear reactor during full-power continuous operation. Also presented herein the materials of construction and operating principles. Additionally, this sample positioning system is compared with an existing pneumatic and automated transfer system, already available at the research reactors. The system consists of a mechanical arm with a claw, which can deliver the samples for irradiations without reactor shutdown. It was installed in the IEA-R1 research reactor at Instituto de Pesquisas Energéticas e Nucleares (IPEN), Sao Paulo, Brazil, and for the past 5years, the system has successfully operated and allowed the conducting of important experiments. As a result of its introduction, the facility has been in a position to positively respond to the increased demand in studies of biology, medicine, physics, engineering, detector/dosimeter calibrations, etc. It is one example of the appropriated technologies that save energy and resources. [ABSTRACT FROM AUTHOR]