Ernesto P. Borges

A possible deformed algebra and calculus inspired in nonextensive thermostatistics

We present a deformed algebra related to the q-exponential and the q-logarithm functions that emerge from nonextensive statistical mechanics. We also develop a q-derivative (and consistently a q-integral) for which the q-exponential is an eigenfunction. The q-derivative and the q-integral have a dual nature, that is also presented.

On a q-generalization of circular and hyperbolic functions

A generalization of the circular and hyperbolic functions is proposed, based on the Tsallis statistics and on a consistent generalization of the Euler formula. Some properties of the presently proposed q-trigonometry are then investigated. The generalized functions are exact solutions of a nonlinear oscillator. Original circular and hyperbolic relations are recovered as the limiting case.

A family of nonextensive entropies

A generalized nonextensive two-parameter entropy is developed, along lines which unify current nonextensive frameworks. It recovers, as particular cases, the Tsallis and symmetric entropies, as well as the Boltzmann-Gibbs entropy. The properties of the new (q, q′)-entropy are analysed.

Fluxes of cosmic rays: a delicately balanced stationary state

The analysis of cosmic rays fluxes reveals a knee slightly below 1016 eV and an ankle close to 1019 eV . Their physical origins remain up to now quite enigmatic, essentially because no elementary process is known which occurs at energies close to 1016 eV . We propose a phenomenological approach along the lines of nonextensive statistical mechanics, a formalism which contains Boltzmann-Gibbs statistical mechanics as a particular case. The knee then appears as a crossover between two fractal-like thermal regimes, the crossover being caused by process occurring at energies ten million times lower than that of the knee, in the region of the quark hadron transition (≃ 109 eV ). This opens the door to an unexpected standpoint for further clarifying the phenomenon. PACS number: 95.30.-k, 95.35.+d, 95.85.Ry, 05.90.+m Cosmic rays fascinate since long. They provide galactic, extragalactic and cosmological

Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos

We consider nonequilibrium probabilistic dynamics in logisticlike maps x(t+1)=1-a|x(t)|(z), (z>1) at their chaos threshold: We first introduce many initial conditions within one among W>>1 intervals partitioning the phase space and focus on the unique value q(sen)<1 for which the entropic form S(q) identical with (1- summation operator Wp(q)(i))/(q-1) linearly increases with time. We then verify that S(q(sen))(t)-S(q(sen))( infinity ) vanishes like t(-1/[q(rel)(W)-1]) [q(rel)(W)>1]. We finally exhibit a new finite-size scaling, q(rel)( infinity )-q(rel)(W) proportional, variant W(-|q(sen)|). This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.

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) stan-dard 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 statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.

Mixing and equilibration: protagonists in the scene of nonextensive statistical mechanics

After a brief review of the present status of nonextensive statistical mechanics, we present a conjectural scenario where mixing (characterized by the entropic index qmix⩽1) and equilibration (characterized by the entropic index qeq⩾1) play central and inter-related roles, and appear to determine a priori the values of the relevant indices of the formalism. Boltzmann–Gibbs statistical mechanics is recovered as the qmix=qeq=1 particular case.

Empirical nonextensive laws for the county distribution of total personal income and gross domestic product

We analyze the cumulative distribution of total personal income of USA counties, and gross domestic product of Brazilian, German and United Kingdom counties, and also of world countries. We verify that generalized exponential distributions, related to nonextensive statistical mechanics, describe almost the whole spectrum of the distributions (within acceptable errors), ranging from the low region to the middle region, and, in some cases, up to the power-law tail. The analysis over about 30 years (for USA and Brazil) shows a regular pattern of the parameters appearing in the present phenomenological approach, suggesting a possible connection between the underlying dynamics of (at least some aspects of) the economy of a country (or of the whole world) and nonextensive statistical mechanics. We also introduce two additional examples related to geographical distributions: land areas of counties and land prices, and the same kind of equations adjust the data in the whole range of the spectrum.

Negative specific heat in a Lennard-Jones-like gas with long-range interactions

We study, through molecular dynamics, a conservative two-dimensional Lennard-Jones-like gas (with attractive potential ∝r−α). We consider the effect of the range index α of interactions, number of particles, total energy and particle density. We detect negative specific heat when the interactions become long-ranged (0⩽α/d<1).

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-Pascals triangles, based on q-sum, whose elements are q-numbers.