Influence of thermal fluctuations on bosonic correlations and the ac Stark effect in two-level atoms: A superstatistical perspective

We study the influence of thermal fluctuations on the two-time correlation functions of bosonic baths within a superstatistics framework by assuming that fluctuations follow the gamma distribution. We further establish a connection between superstatistics and Tsallis nonadditive thermodynamics by introducing a temperature-renormalizing parameter. Our results show that, for an Ohmic model, the system's correlation functions exhibit diverse time-dependent behaviors, with the real and imaginary parts displaying enhancement or suppression depending on temperature and fluctuation strength. Additionally, we analyze the impact of these fluctuations on the quantum master equation of a damped two-level atom coupled to an out-of-equilibrium radiation bath. We demonstrate that while the equation's algebraic structure remains intact, the coupling constants are modified by the fluctuation parameters and cavity volume. Specifically, we observe that the ac Stark effect undergoes significant modifications, with fluctuations influencing the transition between repulsive and attractive energy levels.

Nonadditive Entropies and Nonextensive Statistical Mechanics

The centennial Boltzmann-Gibbs statistical mechanics [...].

Data-driven discovery of Tsallis-like distribution using symbolic regression in high-energy physics

The application of atificial intelligence (AI) in fundamental physics has faced limitations due to its inherently uninterpretable nature, which is less conducive to solving physical problems where natural phenomena are expressed in human-understandable language, i.e. mathematical equations. Fortunately, there exists a form of interpretable AI that aligns seamlessly with this requirement, namely, symbolic regression (SR), which learns mathematical equations directly from data. We introduce a groundbreaking application of SR on actual experimental data with an unknown underlying model, representing a significant departure from previous applications, which are primarily limited to simulated data. This application aims to evaluate the reliability of SR as a bona fide scientific discovery tool. SR is applied on transverse-momentum-dependent distributions of charged hadrons measured in high-energy-physics experiments. The outcome underscores the capability of SR to derive an analytical expression closely resembling the Tsallis distribution. The latter is a well-established and widely employed functional form for fitting measured distributions across a broad spectrum of hadron transverse momentum. This achievement is among the first instances where SR demonstrates its potential as a scientific discovery tool. It holds promise for advancing and refining SR methods, paving the way for future applications on experimental data.

Rapidity and Energy Dependencies of Temperatures and Volume Extracted from Identified Charged Hadron Spectra in Proton-Proton Collisions at a Super Proton Synchrotron (SPS)

The standard (Bose-Einstein/Fermi-Dirac, or Maxwell-Boltzmann) distribution from the relativistic ideal gas model is used to study the transverse momentum (pT) spectra of identified charged hadrons (π-, π+, K-, K+, p¯, and p) with different rapidities produced in inelastic proton-proton (pp) collisions at a Super Proton Synchrotron (SPS). The experimental data measured using the NA61/SHINE Collaboration at the center-of-mass (c.m.) energies s=6.3, 7.7, 8.8, 12.3, and 17.3 GeV are fitted well with the distribution. It is shown that the effective temperature (Teff or T), kinetic freeze-out temperature (T0), and initial temperature (Ti) decrease with the increase in rapidity and increase with the increase in c.m. energy. The kinetic freeze-out volume (V) extracted from the π-, π+, K-, K+, and p¯ spectra decreases with the rapidity and increase with the c.m. energy. The opposite tendency of V, extracted from the p spectra, is observed to be increasing with the rapidity and decreasing with the c.m. energy due to the effect of leading protons.

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional [Formula: see text]. The index [Formula: see text] and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Senses along Which the Entropy Sq Is Unique

The Boltzmann-Gibbs-von Neumann-Shannon additive entropy SBG=-k∑ipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics-as currently referred to-grounded on the nonadditive entropy Sq=k1-∑ipiqq-1 as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. Sq plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity-plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy Sq unique? The present effort is dedicated to a-surely non exhaustive-mathematical answer to this basic question.

Acoustic Emissions in Rock Deformation and Failure: New Insights from Q-Statistical Analysis

We propose a new statistical analysis of the Acoustic Emissions (AE) produced in a series of triaxial deformation experiments leading to fractures and failure of two different rocks, namely, Darley Dale Sandstone (DDS) and AG Granite (AG). By means of q-statistical formalism, we are able to characterize the pre-failure processes in both types of rocks. In particular, we study AE inter-event time and AE inter-event distance distributions. Both of them can be reproduced with q-exponential curves, showing universal features that are observed here for the first time and could be important in order to understand more in detail the dynamics of rock fractures.

Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann–Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature.

Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space

Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order N≥2, which maximizes Tsallis q=3 entropy rather than the ordinary q=1 Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant αel=1/137 is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for N=3.

Enthusiasm and Skepticism: Two Pillars of Science—A Nonextensive Statistics Case

Science and its evolution are based on complex epistemological structures. Two of the pillars of such a construction definitively are enthusiasm and skepticism, both being ingredients without which solid knowledge is hardly achieved and certainly not guaranteed. Our friend and colleague Jean Willy André Cleymans (1944–2021), with his open personality, high and longstanding interest for innovation, and recognized leadership in high-energy physics, constitutes a beautiful example of the former. Recently, Joseph I. Kapusta has generously and laboriously offered an interesting illustration of the latter pillar, in the very same field of physics, concerning the very same theoretical frame, namely, nonextensive statistical mechanics and the nonadditive q-entropies on which it is based. I present here a detailed analysis, point by point, of Kapusta’s 19 May 2021 talk and, placing the discussion in a sensibly wider and updated perspective, I refute his bold conclusion that indices q have no physical foundation.

Superstatistics and quantum entanglement in the isotropic spin-1/2 XX dimer from a nonadditive thermodynamics perspective

In this paper, the impact of temperature fluctuations in the entanglement of two qubits described by a spin-1/2 XX model is studied. To describe the out-of-equilibrium situation, superstatistics is used with fluctuations given by a χ^{2}-distribution function, and its free parameters are chosen in such a way that resembles the nonadditive Tsallis thermodynamics. In order to preserve the Legendre structure of the thermal functions, particular energy constraints are imposed on the density operator and the internal energy. Analytical results are obtained using an additional set of constraints after a parametrization of the physical temperature. We show that the well-known parametrization may lead to undesirable values of the physical temperature so that by analyzing the entropy as a function of energy, the correct values are found. The quantum entanglement is obtained from the concurrence and is compared with the case when the Tsallis restrictions are not imposed on the density operator.

Fractal Structures of Yang–Mills Fields and Non-Extensive Statistics: Applications to High Energy Physics

In this work, we provide an overview of the recent investigations on the non-extensive Tsallis statistics and its applications to high energy physics and astrophysics, including physics at the Large Hadron Collider (LHC), hadron physics, and neutron stars. We review some recent investigations on the power-law distributions arising in high energy physics experiments focusing on a thermodynamic description of the system formed, which could explain the power-law behavior. The possible connections with a fractal structure of hadrons is also discussed. The main objective of the present work is to delineate the state-of-the-art of those studies and show some open issues that deserve more careful investigation. We propose several possibilities to test the theory through analyses of experimental data.

Similarities in Multiparticle Production Processes in pp Collisions as Imprints of Nonextensive Statistics

The transverse momentum pT spectra of both hadrons and jets produced in pp collisions at beam energies from hundreds GeV to a few TeV exhibit power-law behavior of 1/pTn at high pT, with similar power indices n. The respective nonextensivity parameters for transverse momenta distributions and the global nonextensivity parameter obtained from multiplicities are compared. In particular, data on kaon to pion (charged particles) multiplicity ratio are analyzed, assuming that the reaction occurs in some nonextensive environment. The values of the corresponding nonextensivity parameters were found to be similar, strongly indicating the existence of a common mechanism behind all these observables.

Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity (q=1 corresponds to the BG theory). Such transforms have the form q↦(aq+1−a)/[(1+a)q−a], where a is a real number; the particular cases a=−1 and a=0 yield, respectively, q↦(2−q) and q↦1/q, currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.

Beyond Boltzmann-Gibbs-Shannon in Physics and Elsewhere

The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy SBG started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.

Testing nonextensive statistics in relativistic heavy-ion collisions

Numerical solutions of the nonlinear Fokker-Planck equation (FPE) which has been associated with nonextensive q-statistics show that the available data on rapidity distributions for stopping in relativistic heavy-ion collisions cannot be reproduced with any permitted value of the nonextensivity parameter (1 < q < 1.5). This casts doubt on the nonextensivity concept that is widely used in relativistic heavy-ion physics.

Fractal Structure and Non-Extensive Statistics

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.

Generalized statistical mechanics of cosmic rays: Application to positron-electron spectral indices

Cosmic ray energy spectra exhibit power law distributions over many orders of magnitude that are very well described by the predictions of q-generalized statistical mechanics, based on a q-generalized Hagedorn theory for transverse momentum spectra and hard QCD scattering processes. QCD at largest center of mass energies predicts the entropic index to be [Formula: see text]. Here we show that the escort duality of the nonextensive thermodynamic formalism predicts an energy split of effective temperature given by Δ [Formula: see text] MeV, where T H is the Hagedorn temperature. We carefully analyse the measured data of the AMS-02 collaboration and provide evidence that the predicted temperature split is indeed observed, leading to a different energy dependence of the e+ and e- spectral indices. We also observe a distinguished energy scale E* ≈ 50 GeV where the e+ and e- spectral indices differ the most. Linear combinations of the escort and non-escort q-generalized canonical distributions yield excellent agreement with the measured AMS-02 data in the entire energy range.

Route from discreteness to the continuum for the Tsallis q-entropy

The existence and exact form of the continuum expression of the discrete nonlogarithmic q-entropy is an important open problem in generalized thermostatistics, since its possible lack implies that nonlogarithmic q-entropy is irrelevant for the continuous classical systems. In this work, we show how the discrete nonlogarithmic q-entropy in fact converges in the continuous limit and the negative of the q-entropy with continuous variables is demonstrated to lead to the (Csiszár type) q-relative entropy just as the relation between the continuous Boltzmann-Gibbs expression and the Kullback-Leibler relative entropy. As a result, we conclude that there is no obstacle for the applicability of the q-entropy to the continuous classical physical systems.