R. S. Mendes

Statistical mechanics based on Renyi entropy

In this work we show that it is possible to obtain a generalized statistical mechanics (thermostatistics) based on Renyi entropy, to be maximized with adequate constraints. The equilibrium probability distribution thus obtained has a very interesting property. Indeed, it reminds us the statistical distribution proposed by Tsallis, known to conveniently describe a variety of phenomena in nonextensive systems. Moreover, some examples are worked out in order to illustrate the main features of the herein introduced formalism.

Is re-association in folded proteins a case of nonextensivity?

Abstract Pioneering experiments by Frauenfelder and collaborators have shown that the ligands of iron (CO, O 2 ) in heme proteins such as myoglobin and hemoglobin, when photo-dissociated at temperatures up to about T ∗ (for example, T ∗ = 160 K for MbCO in glycerol-water) exhibit an anomalous (power-law, instead of exponential) rebinding behavior as a function of time. We show here that this behavior is consistent with nonextensive statistics that have been successfully applied to other physical phenomena.

q-exponential distribution in urban agglomeration.

Usually, the studies of distributions of city populations have been reduced to power laws. In such analyses, a common practice is to consider cities with more than one hundred thousand inhabitants. Here, we argue that the distribution of cities for all ranges of populations can be well described by using a q-exponential distribution. This function, which reproduces the Zipf-Mandelbrot law, is related to the generalized nonextensive statistical mechanics and satisfies an anomalous decay equation.

q-exponential, Weibull, and q-Weibull distributions: an empirical analysis

In a comparative study, the q-exponential and Weibull distributions are employed to investigate frequency distributions of basketball baskets, cyclone victims, brand-name drugs by retail sales, and highway length. In order to analyze the intermediate cases, a distribution, the q-Weibull one, which interpolates the q-exponential and Weibull ones, is introduced. It is verified that the basketball baskets distribution is well described by a q-exponential, whereas the cyclone victims and brand-name drugs by retail sales ones are better adjusted by a Weibull distribution. On the other hand, for highway length the q-exponential and Weibull distributions do not give satisfactory adjustment, being necessary to employ the q-Weibull distribution. Furthermore, the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law (q-exponential with q>1) controversy.

Perturbation and variational methods in nonextensive Tsallis statistics

A unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on Tsallis entropy is given here. In the case of the variational method, the Bogoliubov inequality is generalized in a very natural way following the Feynman proof for the usual statistical mechanics. The inequality turns out to be form-invariant with respect to the entropic index

q-distributions in complex systems: a brief review

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Regularities in football goal distributions

. The method is illustrated with a simple example in classical mechanics. The formalisms developed here are expected to be useful in the discussion of nonextensive systems.

Anomalous diffusion, nonlinear fractional Fokker–Planck equation and solutions

The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism(q-distributions) have been applied to an impressive variety of problems. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.

Distance to the Scaling Law: A Useful Approach for Unveiling Relationships between Crime and Urban Metrics

Besides complexities concerning football championships, some regularities are identified in them. These regularities refer to goal distributions by goal-players and by games. In particular, the goal distribution by goal-players is well adjusted by the Zipf–Mandelbrot law, suggesting a connection with an anomalous decay.

Time-resolved thermal mirror method: A theoretical study

We obtain new exact classes of solutions for the nonlinear fractional Fokker–Planck-like equation ∂tρ=∂x{D(x)∂μ−1xρν−F(x)ρ} by considering a diffusion coefficient D=D|x|−θ(θ∈R and D>0) and a drift force F=−k1x+kγx|x|γ−1(k1,kγ,γ∈R). Connection with nonextensive statistical mechanics based on Tsallis entropy is also discussed.