Diffusion crossover between q statistics and Boltzmann-Gibbs statistics in the classical inertial α-XY ferromagnet.

We study the angular diffusion in a classical d-dimensional inertial XY model with interactions decaying with the distance between spins as r^{-α}, with α⩾0. After a very short-time ballistic regime, with σ_{θ}^{2}∼t^{2}, a superdiffusive regime, for which σ_{θ}^{2}∼t^{α_{D}}, with α_{D}≃1.45 is observed, whose duration covers an initial quasistationary state and its transition to a second plateau characterized by the Boltzmann-Gibbs temperature T_{BG}.

Generalization of the Gauss map: A jump into chaos with universal features.

The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behavior, and it generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map, which is given by x_{t+1}=1/x_{t}^{α}-[1/x_{t}^{α}] where α≥0 is a parameter and x_{t}∈[0,1] (t=0,1,2,3,..

Criticality in the duration of the quasistationary state of the d-dimensional α-Heisenberg ferromagnet.

The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial α-Heisenberg model, i.e., N three-dimensional rotators whose interactions decay with distance r_{ij} as 1/r_{ij}^{α} (α≥0), is studied through first-principle molecular dynamics.

Reminiscences of Half a Century of Life in the World of Theoretical Physics.

Selma Lagerlöf said that culture is what remains when one has forgotten everything we had learned. Without any warranty, through ongoing research tasks, that I will ever attain this high level of wisdom, I simply share here reminiscences that have played, during my life, an important role in my incursions in science, mainly in theoretical physics. I end by presenting some perspectives for future developments.

de Broglie-Bohm analysis of a nonlinear membrane: From quantum to classical chaos.

Within the de Broglie-Bohm theory, we numerically study a generic two-dimensional anharmonic oscillator including cubic and quartic interactions in addition to a bilinear coupling term. Our analysis of the quantum velocity fields and trajectories reveals the emergence of dynamical vortices. In their vicinity, fingerprints of chaotic behavior such as unpredictability and sensitivity to initial conditions are detected.

First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model.

The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, Th and Tl (Th>Tl), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (i=2,⋯,L-1) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit L→∞, scaling with the temperature, as κ(T)∼T-2.

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 (currently referred to as ), and its validity has been experimentally verified in countless situations.

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).

Senses along Which the Entropy Is Unique.

The Boltzmann-Gibbs-von Neumann-Shannon 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.

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.

Medical Applications of Nonadditive Entropies.

The Boltzmann-Gibbs additive entropy SBG=-k∑ipilnpi and associated statistical mechanics were generalized in 1988 into nonadditive entropy Sq=k1-∑ipiqq-1 and nonextensive statistical mechanics, respectively. Since then, a plethora of medical applications have emerged. In the present review, we illustrate them by briefly presenting image and signal processings, tissue radiation responses, and modeling of disease kinetics, such as for the COVID-19 pandemic.

Entropy Optimization, Generalized Logarithms, and Duality Relations.

Several generalizations or extensions of the Boltzmann-Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies Sq≡k1-∑ipiqq-1(q∈R;S1=SBG≡-k∑ipilnpi) have harvested the largest number of successful applications. The specific structural features of the Sq thermostatistics, therefore, are worthy of close scrutiny.

Relativistic gas: Lorentz-invariant distribution for the velocities.

In 1911, Jüttner proposed the generalization, for a relativistic gas, of the Maxwell-Boltzmann distribution of velocities. Here, we want to discuss, among others, the Jüttner probability density function (PDF). Both the velocity space and, consequently, the momentum space are not flat in special relativity.

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 ∝ (with q = 4 / 3 and = 10 / 3) optimizing, under simple constraints, the nonadditive entropy with a specific geographic growth random model based on preferential attachment through exponentially distributed weighted links, being the characteristic weight.

Finite-size scaling of quasi-stationary-state temperature.

We numerically study, from first principles, the temperature T_{QSS} and duration t_{QSS} of the longstanding initial quasi-stationary state of the isolated d-dimensional classical inertial α-XY ferromagnet with two-body interactions decaying as 1/r_{ij}^{α} (α≥0). It is shown that this temperature T_{QSS} (defined proportional to the kinetic energy per particle) depends, for the long-range regime 0≤α/d≤1, on (α,d,U,N) with numerically negligible changes for dimensions d=1,2,3, with U the energy per particle and N the number of particles. We verify the finite-size scaling T_{QSS}-T_{∞}∝1/N^{β} where T_{∞}≡2U-1 for U≲U_{c}, and β appears to depend sensibly only on α/d.

Along the Lines of Nonadditive Entropies: -Prime Numbers and -Zeta Functions.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n-s=∏pprime11-p-s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡-k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1-q-11-q(ln1z=lnz). It is already known that this function paves the way for the emergence of a -generalized algebra, using -numbers defined as ⟨x⟩q≡elnqx, which recover the number for q=1.

Criticality in the duration of quasistationary state.

The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial α-XY model, i.e., N planar rotators whose interactions decay with the distance r_{ij} as 1/r_{ij}^{α} (α≥0), is studied through first-principles molecular dynamics.

Reply to Pessoa, P.; Arderucio Costa, B. Comment on "Tsallis, C. Black Hole Entropy: A Closer Look. 2020, , 17".

In the present Reply we restrict our focus only onto the main erroneous claims by Pessoa and Costa in their recent Comment ( , , 1110).

Quasi-stationary-state duration in the classical d-dimensional long-range inertial XY ferromagnet.

A classical α-XY inertial model, consisting of N two-component rotators and characterized by interactions decaying with the distance r_{ij} as 1/r_{ij}^{α} (α≥0) is studied through first-principle molecular-dynamics simulations on d-dimensional lattices of linear size L (N≡L^{d} and d=1,2,3). The limits α=0 and α→∞ correspond to infinite-range and nearest-neighbor interactions, respectively, whereas the ratio α/d>1 (0≤α/d≤1) is associated with short-range (long-range) interactions. By analyzing the time evolution of the kinetic temperature T(t) in the long-range-interaction regime, one finds a quasi-stationary state (QSS) characterized by a temperature T_{QSS}; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature T_{BG} (as predicted within the BG theory), with T_{BG}>T_{QSS}.

Connecting complex networks to nonadditive entropies.

Boltzmann-Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space-time entanglement. Its generalization based on nonadditive q-entropies adequately handles a wide class of such systems.

Exploring the Neighborhood of -Exponentials.

The -exponential form eqx≡[1+(1-q)x]1/(1-q)(e1x=ex) is obtained by optimizing the nonadditive entropy Sq≡k1-∑ipiqq-1 (with S1=SBG≡-k∑ipilnpi, where BG stands for Boltzmann-Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms.

Black Hole Entropy: A Closer Look.

In many papers in the literature, author(s) express their perplexity concerning the fact that the ( 3 + 1 ) black-hole 'thermodynamical' entropy appears to be proportional to its area and not to its volume, and would therefore seemingly be nonextensive, or, to be more precise, subextensive. To discuss this question on more clear terms, a non-Boltzmannian entropic functional noted S δ was applied [Tsallis and Cirto, Eur. Phys.

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 S B G started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems.

Nonadditive Entropies and Complex Systems.

An entropic functional is said if it satisfies, for any two probabilistically independent systems and , that S ( A + B ) = S ( A ) + S ( B ) [...

Scaling properties of d-dimensional complex networks.

The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located d-dimensional networks. In this paper, we study the scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r_{ij}^{-α_{A}} (α_{A}≥0).