Boltzmann-Gibbs Statistical Mechanics Research Articles

Energy distributions of Frenkel–Kontorova-type atomic chains: Transition from conservative to dissipative dynamics

We investigate energy distributions of Frenkel–Kontorova-type atomic chains generated from large number of independent identically distributed (iid) random initial atomic positionings under two cases. In the first case, atoms at the free-end chains without dissipation (conservative case) are only coupled to one other atom, whereas each atom inside the bulk is coupled to its 2 nearest neighbors. Here, atoms located at the chain are all at the same type. Such kind of systems can be modelled by conservative standard map. We show that, when the coupling is non-linear (which leads chaotic arrangement of the atoms) for energy distribution, the Boltzmann–Gibbs statistical mechanics is constructed, namely, exponential form emerges as Boltzmann factor P(E)∝e−βE. However, when the coupling is linear (which leads linear arrangement of the atoms) the Boltzmann–Gibbs statistical mechanics fails and the exponential distribution is replaced by a q-exponential form, which generalizes the Boltzmann factor as P(E)∝eq−βqE=[1−(1−q)βqE]1/(1−q). We also show for each type of atom localization with N number of atoms, β (or βq) values can be given as a function of 1/N. In the second case, although the couplings among the atoms are exactly the same as the previous case, atoms located at the chain are now considered as being at different types. We show that, for energy distribution of such linear chains, each of the distributions corresponding to different dissipation parameters (γ) are in the q-exponential form. Moreover, we numerically verify that βq values can be given as a linear function of 1/∑n=1N(1−γ)(n−2). On the other hand, although energy distributions of the chaotic chains for different dissipation parameters are in exponential form, a linear scaling between β and γ values cannot be obtained. This scaling is possible if the energies of the chains are scaled with 1/(1−γ)−N. For both cases, clear data collapses among distributions are evident.

Assessing structural integrity of non-homogeneous systems by means of Acoustic Emissions and Non-Extensive Statistical Mechanics

Taking advantage of the interevent time intervals, namely the time intervals between two successive acoustic events recorded during mechanical loading of structural elements, it is attempted to detect indices warning about upcoming failure. The innovative aspect of the study is that the analysis is implemented in the frame of Non-Extensive Statistical Mechanics, a discipline founded on a class of entropies violating the additivity principle, which is the cornerstone of the Boltzmann-Gibbs Statistical Mechanics. The specimens used for the experiments were marble blocks simulating either fragmented and restored epistyles or mutually interconnected intact epistyles of the Parthenon Temple on the Acropolis of Athens. The specimens consisted of three mate­rials (marble, metallic connectors and cementitious pastes) exhibiting, thus, a strongly non-homogeneous nature. The entropic index, i.e., the parameter quantifying the degree of non-additivity, was used for the analysis of the ex­pe­rimental data. The results were considered in juxtaposition to the re­spective ones from experimental protocols with specimens of macroscopically homo­geneous nature. It was concluded that the temporal evolution of the entropic index provides very good insight into the level of damage ac­cumu­lated in the loaded structure, independently of whether the structure is homogeneous or not, providing an interesting pre-failure indicator.

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)'.

Loewner time conversion for q-generalized stochastic dynamics

Generalized statistical mechanics based on q-Gaussian has been demonstrated to be an effective theoretical framework for the analysis of non-equilibrium systems. Since q-generalized (non-extensive) statistical mechanics reduces the nonlinearity in the system into deformed entropy and probability distributions, we introduce an alternative method based on the time conversion method using the Loewner equation by investigating the statistical physical properties of one-dimensional stochastic dynamics described by the Langevin equation with multiplicative noise. We demonstrate that a randomized time transformation using Loewner time enables the conversion of the multiplicative Langevin dynamics into an equilibrium system obeying a conventional microcanonical ensemble. For the equilibrium Langevin system after the Loewner time conversion, the fluctuation-dissipation relation and path integral fluctuation theorem were discussed to derive the response function under a nonlinear perturbation and an extended Jarzynski equality. The present results suggest the efficacy of the introducing randomized time for analyzing non-equilibrium systems, and indicate a novel connection between q-generalized (non-extensive) and Boltzmann–Gibbs statistical mechanics.

Neural complexity through a nonextensive statistical–mechanical approach of human electroencephalograms

The brain is a complex system whose understanding enables potentially deeper approaches to mental phenomena. Dynamics of wide classes of complex systems have been satisfactorily described within q-statistics, a current generalization of Boltzmann-Gibbs (BG) statistics. Here, we study human electroencephalograms of typical human adults (EEG), very specifically their inter-occurrence times across an arbitrarily chosen threshold of the signal (observed, for instance, at the midparietal location in scalp). The distributions of these inter-occurrence times differ from those usually emerging within BG statistical mechanics. They are instead well approached within the q-statistical theory, based on non-additive entropies characterized by the index q. The present method points towards a suitable tool for quantitatively accessing brain complexity, thus potentially opening useful studies of the properties of both typical and altered brain physiology.

A Non-Extensive Statistical Physics View of the Temporal Properties of the Recent Aftershock Sequences of Strong Earthquakes in Greece

Greece is one of Europe’s most seismically active areas. Seismic activity in Greece has been characterized by a series of strong earthquakes with magnitudes up to Mw = 7.0 over the last five years. In this article we focus on these strong events, namely the Mw6.0 Arkalochori (27 September 2021), the Mw6.3 Elassona (3 March 2021), the Mw7.0 Samos (30 October 2020), the Mw5.1 Parnitha (19 July 2019), the Mw6.6 Zakynthos (25 October 2018), the Mw6.5 Kos (20 July 2017) and the Mw6.1 Mytilene (12 June 2017) earthquakes. Based on the probability distributions of interevent times between the successive aftershock events, we investigate the temporal evolution of their aftershock sequences. We use a statistical mechanics model developed in the framework of Non-Extensive Statistical Physics (NESP) to approach the observed distributions. NESP provides a strictly necessary generalization of Boltzmann–Gibbs statistical mechanics for complex systems with memory effects, (multi)fractal geometries, and long-range interactions. We show how the NESP applicable to the temporal evolution of recent aftershock sequences in Greece, as well as the existence of a crossover behavior from power-law (q ≠ 1) to exponential (q = 1) scaling for longer interevent times. The observed behavior is further discussed in terms of superstatistics. In this way a stochastic mechanism with memory effects that can produce the observed scaling behavior is demonstrated. To conclude, seismic activity in Greece presents a series of significant earthquakes over the last five years. We focus on strong earthquakes, and we study the temporal evolution of aftershock sequences of them using a statistical mechanics model. The non-extensive parameter q related with the interevent times distribution varies between 1.62 and 1.71, which suggests a system with about one degree of freedom.

Response of fiber reinforced concrete while entering into the critical stage: An attempt to detect pre-failure indicators in terms of non-extensive statistical mechanics

The main goal of the present study is to explore the potentialities of Non-Extensive Statistical Mechanics (NESM) to provide signals that could be considered as pre-failure indicators. In this direction concrete beams, either unreinforced or reinforced with short fibers, were submitted to three-point bending. The acoustic signals detected during loading were analyzed in terms of their interevent time intervals and the respective cumulative probability. The experimental data were then properly fitted by means of the q-exponential function providing the temporal evolution of the entropic index q throughout the duration of each experiment. The evolution of q was studied in juxtaposition to that of the load applied as well as to that of some additional parameters used to describe the acoustic activity, namely the b-values and the F-, P- and d-functions. The analysis indicated that at load levels approaching the load-carrying capacity of the specimens, the response of both the unreinforced and the fiber reinforced beams corresponds to q-values approaching 2. It is thus highlighted that, according to NESM principles, the damage processes at the specific load levels are characterized by the existence of a number of non-independent subsystems with long-range mutual interactions and memory effects, which cannot be properly described by the concepts of the traditional Boltzmann-Gibbs Statistical Mechanics. On the contrary, description in terms of NESM provided useful information about the damage evolution during the very last loading stages before macroscopic fracture. In addition, the analysis provided indications that the onset of decrease of the q-values, towards the critical limit of 1, could be considered as a potential pre-failure index, warning about upcoming entrance of the system into its critical stage, namely that of impending fracture.

Probabilistic models with nonlocal correlations: Numerical evidence of [formula omitted]-Large Deviation Theory

The correlated probabilistic model introduced and analytically discussed in Hanel et al. (2009) is based on a self-dual transformation of the index q which characterizes a current generalization of Boltzmann–Gibbs statistical mechanics, namely nonextensive statistical mechanics, and yields, in the N→∞ limit, a Q-Gaussian distribution for any chosen value of Q∈[1,3). We show here that, by properly generalizing that self-dual transformation, it is possible to obtain an entire family of such probabilistic models, all of them yielding Qc-Gaussians (Qc≥1) in the N→∞ limit. This family turns out to be isomorphic to the Hanel et al model through a specific monotonic transformation Qc(Q). Then, by following along the lines of Tirnakli et al (2022), we numerically show that this family of correlated probabilistic models provides further evidence towards a q-generalized Large Deviation Theory (LDT), consistently with the Legendre structure of thermodynamics. The present analysis deepens our understanding of complex systems (with global correlations among their elements), supporting the conjecture that generic models whose attractors under summation of N strongly-correlated random variables are Q-Gaussians, might always be concomitantly associated with q-exponentials in the LDT sense.

The spectrum of Hawking radiation in Tsallis statistical mechanics

Hawking radiation is one of the cores in modern gravitational theory. Several articles have calculated the spectrum of Hawking radiation in Boltzmann–Gibbs statistical mechanics. However, based on recent researches, gravitational systems cannot be studied by the standard statistical mechanics. In this article, we calculate the modification to the spectrum of Hawking radiation in Tsallis statistical mechanics. We obtain the modified Stefan–Boltzmann’s law and modified power of Hawking radiation. We confirm the conclusion proposed by Giddings, namely, the radiation should originate from the effective radius, which extends well outside the horizon of black-hole. The lifetime of black hole and the effect of large q are discussed as well.

Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry.

In the realm of Boltzmann-Gibbs statistical mechanics, there are three well known isomorphic connections with random geometry, namely, (i) the Kasteleyn-Fortuin theorem, which connects the λ → 1 limit of the λ-state Potts ferromagnet with bond percolation, (ii) the isomorphism, which connects the λ → 0 limit of the λ-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism, which connects the n → 0 limit of the n-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism appears to emerge connecting the energy q-exponential distribution ∝ e (with q = 4 / 3 and β ω = 10 / 3) optimizing, under simple constraints, the nonadditive entropy S with a specific geographic growth random model based on preferential attachment through exponentially distributed weighted links, ω being the characteristic weight.

Foundations of statistical mechanics for unstable interactions.

Traditional Boltzmann-Gibbs statistical mechanics does not apply to systems with unstable interactions, because for such systems the conventional thermodynamic limit does not exist. In unstable systems the ground state energy does not have an additive lower bound, i.e., no lower bound linearly proportional to the number N of particles or degrees of freedom. In this article unstable systems are studied whose ground state energy is bounded below by a regularly varying function of N with index σ≥1. The index σ≥1 of regular variation introduces a classification with respect to stability. Stable interactions correspond to σ=1. A simple example for an unstable system with σ=2 is an ideal gas with a nonvanishing constant two-body potential. The foundations of statistical physics are revisited, and generalized ensembles are introduced for unstable interactions in such a way that the thermodynamic limit exists. The extended ensembles are derived by identifying and postulating three basic properties as extended foundations for statistical mechanics: first, extensivity of thermodynamic systems, second, divisibility of equilibrium states, and third, statistical independence of isolated systems. The traditional Boltzmann-Gibbs postulate, resp. the hypothesis of equal a priori probabilities, is identified as a special case of the extended ensembles. Systems with unstable interactions are found to be thermodynamically normal and extensive. The formalism is applied to ideal gases with constant many-body potentials. The results show that, contrary to claims in the literature, stability of the interaction is not a necessary condition for the existence of a thermodynamic limit. As a second example the formalism is applied to the Curie-Weiss-Ising model with strong coupling. This model has index of stability σ=2. Its thermodynamic potentials [originally obtained in R. Hilfer, Physica A 320, 429 (2003)10.1016/S0378-4371(02)01585-6] are confirmed up to a trivial energy shift. The strong coupling model shows a thermodynamic phase transition of order 1 representing a novel mean-field universality class. The disordered high temperature phase collapses into the ground state of the system. The metastable extension of the high temperature free energy to low temperatures ends at absolute zero in a phase transition of order 1/2. Between absolute zero and the critical temperature of the first order transition all fluctuations are absent.

On the role of Barrow's fractal entropy in the loop quantum gravity Barbero-Immirzi parameter

It is claimed that, using the Boltzmann-Gibbs statistical mechanics together with the Bekenstein-Hawking area entropy law, it is possible to determine a value for the Barbero-Immirzi parameter, . In this letter, we have shown that if we use Barrow's entropy rather than Bekenstein-Hawking's area entropy law then the Barbero-Immirzi parameter will depend on the Δ parameter. As a consequence, the loop quantum gravity area element will also depend on the Δ parameter.

Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation

<abstract> Lévy-type behaviors are widely involved in anomalous thermal transport, yet generic investigations based on the mathematical descriptions of the confined Lévy flights are still lacking. In the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, the Lévy-Fokker-Planck equation is connected to near-equilibrium thermal transport. In this work, we show that thermal transport dominated by the confined Lévy flights will be paired with an anomaly, namely that the local effective thermal conductivity is nonlocal. It is demonstrated that the near-equilibrium assumption is not unconditionally valid, which relies on several thermodynamic restrictions expressed by the probability density function (PDF). It is illustrated that the Lévy-Fokker-Planck equation based on the Caputo operator will give rise to two signatures of anomalous thermal transport, the power-law size-dependence of the global effective thermal conductivity and nonlinear boundary asymptotics of the stationary temperature profile. These anomalies are interrelated with each other, and their quantitative relations can be considered as criteria for Lévy-based thermal transport. </abstract>

Reparametrizations and metric structures in thermodynamic phase space

We investigate the consequences of reparametrizations in the geometric description of thermodynamics analyzing the effects on the thermodynamic phase space. It is known that the contact and Riemannian structures of the thermodynamic phase space are related to thermodynamic equilibrium and statistical fluctuations in the Boltzmann–Gibbs statistical mechanics. The physical motivation for this analysis rests upon the possibility of having, instead of a direct control of the intensive parameters determining the state of the corresponding physical reservoirs, the control of a set of differentiable functions of the original variables. Likewise, we consider a set of differentiable functions of the extensive variables accounting for the possibility of not having direct access to the original variables. We find that the effect of reparametrizations on the thermodynamic phase space can be codified, in geometrical terms, in its contact and Riemannian structures. In particular, we single out a rank-two tensor that enters in the definition of the metric which geometrically comprises the information about such reparametrizations. We notice that even if these geometric structures are modified by the reparametrizations, the metric structure on the thermodynamic space of equilibrium states is preserved.

Barrow fractal entropy and the black hole quasinormal modes

Using the Boltzmann-Gibbs statistical mechanics together with the quasinormal modes of the black holes and the Bekenstein-Hawking area entropy law, we can determine univocally the lowest possible value for the spin j, which is jmin=1, in the framework of the Loop Quantum Gravity theory. Subsequently, the value of Immirzi parameter is given by γ=ln⁡3/(2π2). In this paper, we have demonstrated that if we use the Barrow formulation for the black hole entropy then the minimum value of the label j depends on the value, which characterizes the fractal structure of the black hole surface called Δ parameter, and may have values other than jmin=1.

Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities.

The concept of duality of probability distributions constitutes a fundamental “brick” in the solid framework of nonextensive statistical mechanics—the generalization of Boltzmann–Gibbs statistical mechanics under the consideration of the q-entropy. The probability duality is solving old-standing issues of the theory, e.g., it ascertains the additivity for the internal energy given the additivity in the energy of microstates. However, it is a rather complex part of the theory, and certainly, it cannot be trivially explained along the Gibb’s path of entropy maximization. Recently, it was shown that an alternative picture exists, considering a dual entropy, instead of a dual probability. In particular, the framework of nonextensive statistical mechanics can be equivalently developed using q- and 1/q- entropies. The canonical probability distribution coincides again with the known q-exponential distribution, but without the necessity of the duality of ordinary-escort probabilities. Furthermore, it is shown that the dual entropies, q-entropy and 1/q-entropy, as well as, the 1-entropy, are involved in an identity, useful in theoretical development and applications.

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 ( corresponds to the BG theory). Such transforms have the form , where a is a real number; the particular cases and yield, respectively, and , 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.

Black holes quasinormal modes, Loop Quantum Gravity Immirzi parameter and nonextensive statistics

It is argued that, using the black hole area entropy law together with the Boltzmann-Gibbs statistical mechanics and the quasinormal modes of the black holes, it is possible to determine univocally the lowest possible value for the spin j in the context of the Loop Quantum Gravity theory which is jmin=1. Consequently, the value of Immirzi parameter is given by γ=ln⁡3/(2π2). In this paper, we have shown that if we use Tsallis microcanonical entropy rather than Boltzmann-Gibbs framework then the minimum value of the label j depends on the nonextensive q-parameter and may have values other than jmin=1.

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 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.

Validity and failure of the Boltzmann weight

The dynamics and thermostatistics of a classical inertial XY model, characterized by long-range interactions, are investigated on d-dimensional lattices (d = 1, 2, and 3), through molecular dynamics. The interactions between rotators decay with the distance rij like , where α → ∞ and respectively correspond to the nearest-neighbor and infinite-range interactions. We verify that the momenta probability distributions are Maxwellians in the short-range regime, whereas q-Gaussians emerge in the long-range regime. Moreover, in this latter regime, the individual energy probability distributions are characterized by long tails, corresponding to q-exponential functions. The present investigation strongly indicates that, in the long-range regime, central properties fall out of the scope of Boltzmann-Gibbs statistical mechanics, depending on d and α through the ratio .