arXiv subjects

Ernesto P. Borges

Publications and source records attributed to Ernesto P. Borges.

At least 19 recordsLinked to original sources

arXiv2023

Nonlinear dynamical systems: Time reversibility {\it versus} sensitivity to the initial conditions

Time reversal of vast classes of phenomena has direct implications with predictability, causality and the second principle of thermodynamics. We analyze in detail time reversibility of a paradigmatic dissipative nonlinear dynamical system, namely the logistic map $x_{t+1}=1-ax_t^2$. A close relation is revealed between time reversibility and the sensitivity to the initial conditions. Indeed, depending on the initial condition and the size of the time series, time reversal can enable the recovery, within a small error bar, of past information when the Lyapunov exponent is non-positive, notably at the Feigenbaum point (edge of chaos), where weak chaos is known to exist. Past information is gradually lost for increasingly large Lyapunov exponent (strong chaos), notably at $a=2$ where it attains a large value. These facts open the door to diverse novel applications in physicochemical, astronomical, medical, financial, and other time series.

nlin.CD↗

arXiv2023

Time evolution of nonadditive entropies: The logistic map

Due to the second principle of thermodynamics, the time dependence of entropy for all kinds of systems under all kinds of physical circumstances always thrives interest. The logistic map $x_{t+1}=1-a x_t^2 \in [-1,1]\;(a\in [0,2])$ is neither large, since it has only one degree of freedom, nor closed, since it is dissipative. It exhibits, nevertheless, a peculiar time evolution of its natural entropy, which is the additive Boltzmann-Gibbs-Shannon one, $S_{BG}=-\sum_{i=1}^W p_i \ln p_i$, for all values of $a$ for which the Lyapunov exponent is positive, and the nonadditive one $S_q= \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ with $q=0.2445\dots$ at the edge of chaos, where the Lyapunov exponent vanishes, $W$ being the number of windows of the phase space partition. We numerically show that, for increasing time, the phase-space-averaged entropy overshoots above its stationary-state value in all cases. However, when $W\to\infty$, the overshooting gradually disappears for the most chaotic case ($a=2$), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point ($a=1.4011\dots$). Consequently, the stationary-state entropy value is achieved from {\it above}, instead of from {\it below}, as it could have been a priori expected. These results raise the question whether the usual requirements -- large, closed, and for generic initial conditions -- for the second principle validity might be necessary but not sufficient.

cond-mat.stat-mech↗

arXiv2021

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 $ζ(s)\equiv\sum_{n=1}^\infty n^{-s}=\prod_{p\,prime} \frac{1}{1- p^{-s}}$, 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 $\mathbb{R}(z)=1/2$ axis. The nonadditive entropy $S_q=k\sum_ip_i\ln_q(1/p_i)\;(q\in\mathbb{R})$ involves the function $\ln_q z\equiv\frac{z^{1-q}-1}{1-q}\;(\ln_1 z=\ln z)$ that is interconnected to a $q$-generalized algebra, using $q$-numbers defined as $\langle x\rangle_q\equiv e^{\ln_q x}$ ($\langle x\rangle_1\equiv x$). The $q$-prime numbers are then defined as the $q$-natural numbers $\langle n\rangle_q\equiv e^{\ln_q n}\;(n=1,2,3,\dots)$, where $n$ is a prime number $p=2,3,5,7,\dots$ We show that, for any value of $q$, infinitely many $q$-prime numbers exist; for $q\le1$ they diverge for increasing prime number, whereas they converge for $q>1$; the standard prime numbers are recovered for $q=1$. For $q\le 1$, we generalize the $ζ(s)$ function as follows: $ζ_q(s)\equiv\langleζ(s)\rangle_q$ ($s\in\mathbb{R}$). We show that this function appears to diverge at $s=1+0$, $\forall q$. Also, we alternatively define, for $q\le 1$, $ζ_q^Σ(s)\equiv\sum_{n=1}^\infty\frac{1}{\langle n\rangle_q^s}=1+\frac{1}{\langle 2\rangle_q^s}+\dots $ and $ζ_q^Π(s)\equiv\prod_{p\,prime}\frac{1}{1-\langle p\rangle_q^{-s}}=\frac{1}{1-\langle 2\rangle_q^{-s}}\frac{1}{1-\langle 3\rangle_q^{-s}}\frac{1}{1-\langle 5\rangle_q^{-s}}\cdots$, which, for $q<1$, generically satisfy $ζ_q^Σ(s)<ζ_q^Π(s)$, in variance with the $q=1$ case, where of course $ζ_1^Σ(s)=ζ_1^Π(s)$.

math.NT↗

arXiv2021

Deformed mathematical objects stemming from the $q$-logarithm function

Generalized numbers, arithmetic operators and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of $q$-logarithm/$q$-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity and distributivity, and also a pair of linear/nonlinear derivatives are observed within each class. Two entropic functionals emerge from the formalism, one of them is the nonadditive Tsallis entropy.

math.GM↗

arXiv2020

Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the H-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.

cond-mat.stat-mech↗

arXiv2019

Algebraic structures and deformed Schrödinger equations from groups entropies

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schrödinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the $q$-deformed (standard) Schrödinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.

math-ph↗

arXiv2018

A unified time scale for quantum chaotic regimes

We present a generalised time scale for quantum chaos dynamics, motivated by nonextensive statistical mechanics. It recovers, as particular cases, the relaxation (Heisenberg) and the random (Ehrenfest) time scales. Moreover, we show that the generalised time scale can also be obtained from a nonextensive version of the Kolmogorov-Sinai entropy by considering the graininess of quantum phase space and a generalised uncorrelation between subsets of the phase space. Lyapunov and regular regimes for the fidelity decay are obtained as a consequence of a nonextensive generalisation of the $m$th point correlation function for a uniformly distributed perturbation in the classical limit.

quant-ph↗

arXiv2018

Fisher metric from relative entropy group

In this work we consider the Fisher metric which results from the Hessian of the relative entropy group, that we called Fisher metric group, and we obtain the corresponding ones to the Boltzmann-Gibbs, Tsallis, Kaniadakis and Abe-Borges-Roditi classes. We prove that the scalar curvature of the Fisher metric group results a multiple of the standard Fisher one, with the factor of proportionality given by the local properties of the entropy group. For the Tsallis class, the softening and strengthening of the scalar curvature is illustrated with the $2D$ correlated model, from which their associated indexes for the canonical ensemble of a pair of interacting harmonic oscillators, are obtained.

math-ph↗

arXiv2018

A position-dependent mass harmonic oscillator and deformed space

We consider canonically conjugated generalized space and linear momentum operators $\hat{x}_q$ and $ \hat{p}_q$ in quantum mechanics, associated to a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter $q$. A canonical transformation $(\hat{x}, \hat{p}) \rightarrow (\hat{x}_q, \hat{p}_q)$ leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space $(x, p)$ may be expressed in terms of the deformed (dual) $q$-derivative. We revisit the problem of a $q$-deformed oscillator in both classical and quantum formalisms. Particularly, this canonical transformation leads a particle with position-dependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces $(x,p)$ and $(x_q, p_q)$ are analyzed for different values of the deformation parameter. Lastly, we compare the results of the problem in classical and quantum formalisms through the principle of correspondence and the WKB approximation.

quant-ph↗

arXiv2017

Statistical characterization of discrete conservative systems: The web map

We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional $S_{BG}[p(x)] = -k\int dx\,p(x) \ln p(x)$. In contrast, possible ergodicity breakdown and transitory sticky dynamical behavior drag the map into the realm of generalized $q$-statistics, based on the nonadditive entropic functional $S_q[p(x)]=k\frac{1-\int dx\,[p(x)]^q}{q-1}$ ($q \in {\cal R}; S_1=S_{BG}$). We statistically describe the system (probability distribution of the sum of successive iterates, sensitivity to the initial condition, and entropy production per unit time) for typical values of the parameter that controls the ergodicity of the map. For small (large) values of the external parameter $K$, we observe $q$-Gaussian distributions with $q=1.935\dots$ (Gaussian distributions), like for the standard map. In contrast, for intermediate values of $K$, we observe a different scenario, due to the fractal structure of the trajectories embedded in the chaotic sea. Long-standing non-Gaussian distributions are characterized in terms of the kurtosis and the box-counting dimension of chaotic sea.

cond-mat.stat-mech↗

arXiv2016

Statistical characterization of the standard map

The standard map, paradigmatic conservative system in the $(x,p)$ phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter $K$. A detailed numerical description is achieved in the present paper. More precisely, for large values of $K$, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=q_{\text{stat}}=q_{\text{rel}}=1$, where $q_{\text{ent}}$ is the $q$-index for which the nonadditive entropy $S_q \equiv k \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $S_1=S_{BG} \equiv -k\sum_{i=1}^W p_i \ln p_i$) grows linearly with time before achieving its $W$-dependent saturation value; $q_{\text{sen}}$ characterizes the time increase of the sensitivity $ξ$ to the initial conditions, i.e., $ξ\sim e_{q_{\text{sen}}}^{λ_{q_{\text{sen}}} \,t}\;(λ_{q_{\text{sen}}}>0)$, where $e_q^z \equiv[1+(1-q)z]^{1/(1-q)}$; $q_{\text{stat}}$ is the index associated with the $q_{\text{stat}}$-Gaussian distribution of the time average of successive iterations of the $x$-coordinate; finally, $q_{\text{rel}}$ characterizes the $q_{\text{rel}}$-exponential relaxation with time of the entropy $S_{q_{\text{ent}}}$ towards its saturation value. In remarkable contrast, for small values of $K$, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with $q$-statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=0$, $q_{\text{stat}} \simeq 1.935$, and $q_{\text{rel}} \simeq1.4$. The situation corresponding to intermediate values of $K$, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG or $q$-statistical behavior are observed.

cond-mat.stat-mech↗

arXiv2015

The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics

As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) standard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistics.

cond-mat.stat-mech↗

arXiv2014

Nonextensive Statistical Analysis of Meteor Showers and Lunar Flashes

The distribution of meteor magnitudes is usually supposed to be described by power-laws. However, this relationship is not able to model the whole data set, and the parameters are considered to be dependent on the magnitude intervals. We adopt a statistical distribution derived from Tsallis nonextensive statistical mechanics which is able to model the whole magnitude range. We combined meteor data from various sources, ranging from telescopic meteors to lunar impactors. Our analysis shows that the probability distribution of magnitudes of IMO and MORP data are similar. The distribution of IMO visual magnitudes indicates that $2.4\pm0.5%$ of the meteors of a shower may be telescopic ($m> 6$). We note that the distribution of duration of lunar flashes follows a power-law, and a comparison with the distribution of meteor showers suggests the occurrence of observational bias. The IMO sporadic meteor distribution also seems to be influenced by observational factors.

astro-ph.EP↗

arXiv2013

Generalized space and linear momentum operators in quantum mechanics

We propose a modification of a recently introduced generalized translation operator, by including a $q$-exponential factor, which implies in the definition of a Hermitian deformed linear momentum operator $\hat{p}_q$, and its canonically conjugate deformed position operator $\hat{x}_q$. A canonical transformation leads the Hamiltonian of a position-dependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase space. The equation of motion for the classical phase space may be expressed in terms of the generalized dual $q$-derivative. A position-dependent mass confined in an infinite square potential well is shown as an instance. Uncertainty and correspondence principles are analyzed.

math-ph↗

arXiv2012

Nonextensive local composition models in theories of solutions

Thermodynamic models present binary interaction parameters, based on the Boltzmann weight. Discrepancies from experimental data lead to empirically consider temperature dependence of the parameters, but these modifications keep unchanged the exponential nature of the equations. We replace the Boltzmann weight by the nonextensive Tsallis weight, and generalize three models for nonelectrolyte solutions that use the local composition hypothesis, namely Wilson's, NRTL, and UNIQUAC models. The proposed generalizations present a nonexponential dependence on the temperature, and relies on a theoretical basis of nonextensive statistical mechanics. The $q$-models present one extra binary parameter $q_{ij}$, that recover the original cases in the limit $q_{ij} \to 1$. Comparison with experimental data is illustrated with two examples of the activity coefficient of ethanol, infinitely diluted in toluene, and in decane.

physics.chem-ph↗

arXiv2011

Nonextensive distributions of rotation periods and diameters of asteroids

Context. To investigate the distribution of rotation periods of asteroids from different regions of the Solar System and distribution of diameters of near-Earth asteroids (NEAs). Aims. Verify if nonextensive statistics satisfactorily describes the data. Methods. Light curve data was taken from Planetary Database System (PDS) with Rel $\ge 2$. Taxonomic class and region of the Solar System was also considered. Data of NEA were taken from Minor Planet Center. Results. The rotation periods of asteroids follow a $q$-Gaussian with $q=2.6$ regardless of taxonomy, diameter or region of the Solar System of the object. The distribution of rotation periods is influenced by observational bias. The diameters of NEAs are described by a $q$-exponential with $q=1.3$. According to this distribution, there are expected to be $994 \pm 30$ NEAs with diameters greater than 1 km.

astro-ph.EP↗

arXiv2011

Generalising the logistic map through the $q$-product

We investigate a generalisation of the logistic map as $ x_{n+1}=1-ax_{n}\otimes_{q_{map}} x_{n}$ ($-1 \le x_{n} \le 1$, $0 1$ at the edge of chaos, particularly at the first critical point $a_c$, that depends on the value of $q_{map}$. Bifurcation diagrams, sensitivity to initial conditions, fractal dimension and rate of entropy growth are evaluated at $a_c(q_{map})$, and connections with nonextensive statistical mechanics are explored.

nlin.CD↗

arXiv2009

Some properties of deformed $q$-numbers

Nonextensive statistical mechanics has been a source of investigation in mathematical structures such as deformed algebraic structures. In this work, we present some consequences of $q$-operations on the construction of $q$-numbers for all numerical sets. Based on such a construction, we present a new product that distributes over the $q$-sum. Finally, we present different patterns of $q$-Pascal's triangles, based on $q$-sum, whose elements are $q$-numbers.

math-ph↗