E. K. Lenzi

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.

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.

Anomalous diffusion: nonlinear fractional Fokker–Planck equation

Abstract We discuss the anomalous diffusion associated with a nonlinear fractional Fokker–Planck equation with a diffusion coefficient D∝|x|−θ ( θ∈ R ). Two classes of exact solutions are found. The first one is a modified porous medium equation and corresponds to integer derivatives and a drift force F∝x|x|α−1 ( α∈ R ). The second one corresponds to fractional space derivative in the absence of external drift. The connection with nonextensive statistical mechanics is also discussed in both cases.

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

Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell

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Fractional Diffusion Equation and Impedance Spectroscopy of Electrolytic Cells

. 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 electrical response of an electrolytic cell in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order is analyzed. The boundary conditions at the electrodes limiting the sample are described by an integro-differential equation governing the kinetic at the interface. The analysis is carried out by supposing that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies the Poissons equation. The results cover a rich variety of scenarios, including the ones connected to anomalous diffusion.

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

The influence of the ions on the electrochemical impedance of a cell is calculated in the framework of a complete model in which the fractional drift-diffusion problem is analytically solved. The resulting distribution of the electric field inside the sample is determined by solving Poissons equation. The theoretical model to determine the electrical impedance we are proposing here is based on the fractional derivative of distributed order on the diffusion equation. We argue that this is the more convenient and physically significant approach to account for the enormous variety of the diffusive regimes in a real cell. The frequency dependence of the real and imaginary parts of the impedance are shown to be very similar to the ones experimentally obtained in a large variety of electrolytic samples.

BLACKBODY RADIATION IN NONEXTENSIVE TSALLIS STATISTICS : EXACT SOLUTION

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.

N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients

We report on a quantitative analysis of relationships between the number of homicides, population size and ten other urban metrics. By using data from Brazilian cities, we show that well-defined average scaling laws with the population size emerge when investigating the relations between population and number of homicides as well as population and urban metrics. We also show that the fluctuations around the scaling laws are log-normally distributed, which enabled us to model these scaling laws by a stochastic-like equation driven by a multiplicative and log-normally distributed noise. Because of the scaling laws, we argue that it is better to employ logarithms in order to describe the number of homicides in function of the urban metrics via regression analysis. In addition to the regression analysis, we propose an approach to correlate crime and urban metrics via the evaluation of the distance between the actual value of the number of homicides (as well as the value of the urban metrics) and the value that is expected by the scaling law with the population size. This approach has proved to be robust and useful for unveiling relationships/behaviors that were not properly carried out by the regression analysis, such as the non-explanatory potential of the elderly population when the number of homicides is much above or much below the scaling law, the fact that unemployment has explanatory potential only when the number of homicides is considerably larger than the expected by the power law, and a gender difference in number of homicides, where cities with female population below the scaling law are characterized by a number of homicides above the power law.