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. The simultaneous presence of the off-diagonal -kxy and nonlinear terms leads to robust quantum chaos very analogous to its classical version.

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.

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.

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.

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 L 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-2.25, where T=(Th+Tl)/2. Moreover, the thermal conductance, σ(L,T)≡κ(T)/L, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to σ(L,T)=Aexpq(-Bxη), where A and B are constants, x=L0.475T, q=2.28±0.04, and η=2.88±0.04.

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.

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 ß 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.

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. The velocity space corresponds to the Lobachevsky one, which has a negative curvature. This curvature induces a specific power for the Lorentz factor in the PDF, affecting the Jüttner normalization constant in one, two, and three dimensions. Furthermore, Jüttner distribution, written in terms of a more convenient variable, the rapidity, presents a curvature change at the origin at sufficiently high energy, which does not agree with our computational dynamics simulations of a relativistic gas. However, in one dimension, the rapidity satisfies a simple additivity law. This allows us to obtain, through the central limit theorem, a new, Lorentz-invariant, PDF whose curvature at the origin does not change for any energy value and which agrees with our computational dynamics simulations data. Also, we perform extensive first-principle simulations of a one-dimensional relativistic gas constituted by light and heavy particles.

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. In the present work, we analyze one of these features, according to which the q-logarithm function lnqx≡x1-q-11-q(ln1x=lnx) associated with the Sq entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations.

Reply to Pessoa, P.; Arderucio Costa, B. Comment on "Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17".

In the present Reply we restrict our focus only onto the main erroneous claims by Pessoa and Costa in their recent Comment (Entropy 2020, 22, 1110).

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-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 z 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 q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζq∑(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζq∏(s)≡∏pprime11-⟨p⟩q-s=11-⟨2⟩q-s11-⟨3⟩q-s11-⟨5⟩q-s⋯, which, for q<1, generically satisfy ζq∑(s)<ζq∏(s), in variance with the q=1 case, where of course ζ1∑(s)=ζ1∏(s).

Exploring the Neighborhood of q-Exponentials.

The q-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. We first discuss departures from q-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding q-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing Sq as particular case.

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. 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.

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. J. C 73, 2487 (2013)] to this complex system which exhibits the so-called area-law. However, some nontrivial physical points still remain open, which we revisit now. This discussion is also based on the fact that the well known Bekenstein-Hawking entropy can be expressed as being proportional to the event horizon area divided by the square of the Planck length.

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies.

We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α ≥ 0 ), where the limit α = 0 ( α → ∞ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 ≤ α / d ≤ 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ ∼ N - κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.

Nonadditive Entropies and Complex Systems.

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

Noisy coupled logistic maps in the vicinity of chaos threshold.

We focus on a linear chain of N first-neighbor-coupled logistic maps in the vicinity of their edge of chaos in the presence of a common noise. This model, characterised by the coupling strength ϵ and the noise width σmax, was recently introduced by Pluchino et al. [Phys. Rev. E 87, 022910 (2013)]. They detected, for the time averaged returns with characteristic return time τ, possible connections with q-Gaussians, the distributions which optimise, under appropriate constraints, the nonadditive entropy, Sq, basis of nonextensive statistics mechanics. Here, we take a closer look on this model, and numerically obtain probability distributions which exhibit a slight asymmetry for some parameter values, in variance with simple q-Gaussians. Nevertheless, along many decades, the fitting with q-Gaussians turns out to be numerically very satisfactory for wide regions of the parameter values, and we illustrate how the index q evolves with (N,τ,ϵ,σmax). It is nevertheless instructive on how careful one must be in such numerical analysis. The overall work shows that physical and/or biological systems that are correctly mimicked by this model are thermostatistically related to nonextensive statistical mechanics when time-averaged relevant quantities are studied.

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.

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^{ß} where T_{∞}≡2U-1 for U≲U_{c}, and ß appears to depend sensibly only on α/d. Our numerical results indicate that neither the scaling with N of T_{QSS} nor that of t_{QSS} depend on U.