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We investigate extensions for the Schrodinger equation, by considering different non-linear dependencies on the kinetic term, which admit the standard probabilistic interpretation for the wave-function. For one of the cases, we show that the Heisenberg equation is invariant in form and a modified Hamilton-Jacobi equation emerges from the semi-classical approximation for the wave-function. Furthermore, we find solutions assuming time scaling functions different from the standard case and observe nonstandard relaxation processes for the wave-function.

We analyze dynamical properties of two fractional Schrodinger equations coupled by some classes of real time independent potentials. For this set of equations, we investigate the required conditions on the equations making it possible to retain the probabilistic interpretation of their correspondent solutions when two component wave functions are considered. We observe the presence of interference between the components during the transition processes which can be either reversible or irreversible depending on the condition imposed on the potentials. The solutions for these equations are obtained in both cases of localized and non-localized coupling potentials.

Measurements of electrical properties of materials are of interest in diffusion experiments, where the mean square displacement of charge carriers plays a distinctive role. Here, an approximation of the mean square displacement is explored as a route to investigate dynamical aspects of systems from electrical conductivity, independently of the number of types of charge carriers. Within this framework, based on Kubo’s linear response theory, we focus on the variance of the center of charges displacement. As an application of this approach, firstly we considered the impedance spectroscopy of a liquid crystal sample at specific experimental conditions. In a second employment of the approach, we discussed the use of the approximate (effective) mean square displacement in relation to an equivalent circuit modeling. Our analysis highlighted the sensitivity of the method in different situations. In addition, these situations are discussed in connection with anomalous diffusion. From a theoretical point of view, the results obtained here offer a basis for making use of effective models that assume the presence of anomalous diffusion.

We investigate a family of generalized Fokker-Planck equations that contains Richardson and porous media equations as members. Considering a confining drift term that is related to an effective potential, we show that each equation of this family has a stationary solution that depends on this potential. This stationary solution encompasses several well-known probability distributions. Moreover, we verify an $H$ theorem for the generalized Fokker-Planck equations using free-energy-like functionals. We show that the energy-like part of each functional is based on the effective potential and the entropy-like part is a generalized Tsallis entropic form, which has an unusual dependence on the position and can be related to a generalization of the Kullback-Leibler divergence. We also verify that the optimization of this entropic-like form subjected to convenient constraints recovers the stationary solution. The analysis presented here includes several studies about $H$ theorems for other generalized Fokker-Planck equations as particular cases., Comment: 8 pages. Accepted for publication in Physical Review E

We propose an approach for analysing the dynamics of human postural sway using measures applied to study inhomogeneous temporal processes. Basically, we defined zero-crossings of center of pressure (COP) trajectories as events, obtained the sequence of interevent times and investigated the mean interevent time, the burstiness coefficient and the full functional form of the interevent time distribution. We applied this approach to data obtained for healthy young adults during quiet standing, under bipedal and unipedal conditions. We found that the proposed COP-based measures are able to detect differences between bipedal and unipedal postural sway temporal patterns, including the presence (or absence) of burstiness. We discussed the potential of this analysis to quantify temporal patterns of postural sway.

Concepts of statistical mechanics as well as other typical tools of physics have been largely used in the analysis of several aspects of social systems, for instance, in politics. In this work, we examine parliamentary presence utilizing data from the sessions of the 49th-54th Brazilian Chambers of Deputies (24 years, 1991-2015). For each federal deputy, we construct a random walk by considering their presence in a session as a step of unitary length and their absence as one of zero length. By using this approach, we put in evidence a quantitative description of the dynamics of the system. More specifically, we identify an anomalous diffusive process that corresponds to a robust superdiffusion, well identified with a ballistic regime. In addition, for each legislature and encompassing all its sessions, the system is modeled by a beta probability distribution, where the parliamentary presence scales with the number of sessions.

We demonstrate the influence of edge effects on the photothermal-induced phase shift measured by a homodyne quadrature laser interferometer and compare the experiments with rigorous theoretical descriptions of thermoelastic surface displacement of metals. The finite geometry of the samples is crucial in determining how the temperature is distributed across the material and how this affects the interferometer phase shift measurements. The optical path change due to the surface thermoelastic deformation and thermal lens in the surrounding air is decoded from the interferometric signal using analytical and numerical tools. The boundary/edge effects are found to be relevant to properly describe the interferometric signals. The tools developed in this study provide a framework for the study of finite size effects in heat transport in opaque materials and are applicable to describe not only the phase shift sensed by the interferometer but also to contribute to the photothermal-based technologies employing similar detection mechanisms. Prikažemo vpliv robnih učinkov na fototermalno inducirani fazni premik, ki ga merimo s homodinskim kvadraturnim laserskim interferometrom in poskuse primerjamo z eksaktnimi teoretičnimi opisi termoelastičnih površinskih premikov kovin. Končna geometrija vzorcev je ključnega pomena pri določanju, kako se temperatura porazdeli po materialu in kako to vpliva na merjenje faznega premika interferometra. Razliko optične poti zaradi površinske termoelastične deformacije in termalne leče v okoliškem zraku dekodiramo iz interferometričnega signala s pomočjo analitičnih in numeričnih orodij. Mejni/robni učinki so pomembni za pravilen opis interferometričnih signalov. Orodja, razvita v tej študiji, nudijo okvir za preučevanje učinkov zaradi končnih dimenzij pri prenosu toplote v neprozornih materialih in niso uporabna le za opisovanje faznega premika, ki ga zazna interferometer, temveč prispevajo tudi k fototermalnim tehnologijam, ki uporabljajo podobne mehanizme zaznavanja.

Researchers of several areas have reported that there are still significant gender differences in their performances within different social systems, as in science and on-line communities, for example. This paper focuses on the gender difference in politic candidature processes, investigating the effect of the electorate size upon the candidate numbers, by considering the electorate of each Brazilian city and the respective number of candidates for Mayor and City Council Member (for women and men separately). We detected a sharp gap between the number of male candidates and number of female candidates, with the disadvantage occurring to greater intent in Mayoral elections. We also found non-linear mean correspondences, allometries. The allometric exponents display values very close to the City Council Member candidatures for women and men. However, they have notable smaller values for women than for men in Mayoral candidature. This shows a political hierarchy: the most influential position is related to the greater female underrepresentation.

One of the most useful tools for distinguishing between chaotic and stochastic time series is the so-called complexity-entropy causality plane. This diagram involves two complexity measures: the Shannon entropy and the statistical complexity. Recently, this idea has been generalized by considering the Tsallis monoparametric generalization of the Shannon entropy, yielding complexity-entropy curves. These curves have proven to enhance the discrimination among different time series related to stochastic and chaotic processes of numerical and experimental nature. Here we further explore these complexity-entropy curves in the context of the R\'enyi entropy, which is another monoparametric generalization of the Shannon entropy. By combining the R\'enyi entropy with the proper generalization of the statistical complexity, we associate a parametric curve (the R\'enyi complexity-entropy curve) with a given time series. We explore this approach in a series of numerical and experimental applications, demonstrating the usefulness of this new technique for time series analysis. We show that the R\'enyi complexity-entropy curves enable the differentiation among time series of chaotic, stochastic, and periodic nature. In particular, time series of stochastic nature are associated with curves displaying positive curvature in a neighborhood of their initial points, whereas curves related to chaotic phenomena have a negative curvature; finally, periodic time series are represented by vertical straight lines., Comment: 19 pages, 6 figures; accepted for publication in Physica A

Hidden structural patterns in written texts have been subject of considerable research in the last decades. In particular, mapping a text into a time series of sentence lengths is a natural way to investigate text structure. Typically, sentence length has been quantified by using measures based on the number of words and the number of characters, but other variations are possible. To quantify the robustness of different sentence length measures, we analyzed a database containing about five hundred books in English. For each book, we extracted six distinct measures of sentence length, including number of words and number of characters (taking into account lemmatization and stop words removal). We compared these six measures for each book by using i) Pearson's coefficient to investigate linear correlations; ii) Kolmogorov--Smirnov test to compare distributions; and iii) detrended fluctuation analysis (DFA) to quantify auto-correlations. We have found that all six measures exhibit very similar behavior, suggesting that sentence length is a robust measure related to text structure., Comment: 9 pages, 5 figures, accepted for publication in Physica A

We report on the time dependent solutions of the $q-$generalized Schr\"odinger equation proposed by Nobre et al. [Phys. Rev. Lett. 106, 140601 (2011)]. Here we investigate the case of two free particles and also the case where two particles were subjected to a Moshinsky-like potential with time dependent coefficients. We work out analytical and numerical solutions for different values of the parameter $q$ and also show that the usual Schr\"odinger equation is recovered in the limit of $q\rightarrow 1$. An intriguing behavior was observed for $q=2$, where the wave function displays a ring-like shape, indicating a bind behavior of the particles. Differently from the results previously reported for the case of one particle, frozen states appear only for special combinations of the wave function parameters in case of $q=3$., Comment: Accepted for publication in Physica A

We study relaxation patterns of violent conflicts after bursts of activity. Data were obtained from available catalogs on the conflicts in Iraq, Afghanistan and Northern Ireland. We find several examples in each catalog for which the observed relaxation curves can be well described by an asymptotic power-law decay (the analog of the Omori’s law in geophysics). The power-law exponents are robust, nearly independent of the conflict. We also discuss the exogenous or endogenous nature of the shocks. Our results suggest that violent conflicts share with earthquakes and other natural and social phenomena a common feature in the dynamics of aftershocks.

Dengue infection plays a central role in our society, since it is the most prevalent vector-borne viral disease affecting humans. We statistically investigated patterns concerning the spatial spreading of dengue epidemics in Brazil, as well as their temporal evolution in all Brazilian municipalities for a period of 12 years. We showed that the distributions of cases in municipalities follow power laws persistent in time and that the infection scales linearly with the population of the municipalities. We also found that the average number of dengue cases does not have a clear dependence on the longitudinal position of municipalities. On the other hand, we found that the average distribution of cases varies with the latitudinal position of municipalities, displaying an almost constant growth from high latitudes until reaching the Tropic of Capricorn leveling to a plateau closer to the Equator. We also characterized the spatial correlation of the number of dengue cases between pairs of municipalities, where our results showed that the spatial correlation function decays with the increase of distance between municipalities, following a power-law with an exponential cut-off. This regime leads to a typical dengue traveling distance. Finally, we considered modeling this last behaviour within the framework of a Edwards-Wilkinson equation with a fractional derivative on space.

We report on a diffusive analysis of the motion of flagellate protozoa species. These parasites are the etiological agents of neglected tropical diseases: leishmaniasis caused by Leishmania amazonensis and Leishmania braziliensis, African sleeping sickness caused by Trypanosoma brucei, and Chagas disease caused by Trypanosoma cruzi. By tracking the positions of these parasites and evaluating the variance related to the radial positions, we find that their motions are characterized by a short-time transient superdiffusive behavior. Also, the probability distributions of the radial positions are self-similar and can be approximated by a stretched Gaussian distribution. We further investigate the probability distributions of the radial velocities of individual trajectories. Among several candidates, we find that the generalized gamma distribution shows a good agreement with these distributions. The velocity time series have long-range correlations, displaying a strong persistent behavior (Hurst exponents close to one). The prevalence of "universal" patterns across all analyzed species indicates that similar mechanisms may be ruling the motion of these parasites, despite their differences in morphological traits. In addition, further analysis of these patterns could become a useful tool for investigating the activity of new candidate drugs against these and others neglected tropical diseases., Accepted for publication in PLoS ONE

We study the optical transmittance of ternary mixtures of water, glycerin and ferrofluids. These mix- tures are subject to pulsed magnetic field and placed between crossed polarizers. After the magnetic field is switched off, the decay process is compared with q- exponential, stretched exponential, Mittag-Leffler, and one-sided Levy stable functions. For short time, the ex- perimental results are interpreted in terms of stretched exponentials. The parameters of this non-exponential relaxation are investigated as functions of temperature and the water, glycerin and ferrofluid concentrations.

We investigate the dynamics of many interacting bubbles in boiling water by using a laser scattering experiment. Specifically, we analyze the temporal variations of a laser intensity signal which passed through a sample of boiling water. Our empirical results indicate that the return interval distribution of the laser signal does not follow an exponential distribution; contrariwise, a heavy-tailed distribution has been found. Additionally, we compare the experimental results with those obtained from a minimalist phenomenological model, finding a good agreement., Comment: Accepted for publication in Chaos, Solitons & Fractals

A random walk-like model is considered to discuss statistical aspects of tournaments. The model is applied to soccer leagues with emphasis on the scores. This competitive system was computationally simulated and the results are compared with empirical data from the English, the German and the Spanish leagues and showed a good agreement with them. The present approach enabled us to characterize a diffusion where the scores are not normally distributed, having a short and asymmetric tail extending towards more positive values. We argue that this non-Gaussian behavior is related with the difference between the teams and with the asymmetry of the scores system. In addition, we compared two tournament systems: the all-play-all and the elimination tournaments.

The equilibrium tilt angle profile in a cell limited by two concentric cylinders filled with nematic liquid crystals is determined for strong homeotropic anchoring at the surfaces. The anchoring condition is such that the nematic director is perpendicular to the cylinder axes and a radial nonuniform electric field is applied to investigate a Freedericksz transition. The distortions induced by the field remain in the plane perpendicular to the cylinder axes, and a threshold field is analytically determined indicating a transition from a pure splay to a splay-bend conformation of the director. It is shown that this transition can be induced by the thickness of the region between the two cylinders, and can be detected even in the absence of an external field. If the maximum value of the tilt angle is assumed as an order parameter, its behavior near to the transition can be used to obtain the critical exponent, which is the same as the one obtained in the mean field approximation. These results are indications that nontrivial consequences may occur when complex fluids are subject to non-planar geometries.

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.

A family of entropy indices constructed in the framework of Tsallis entropy formalism is used to investigate ecological diversity. It represents a new perspective in ecology because a simple equation can incorporate all aspects of a diversity, from richness to dominance and can be also related to a measure of species rarity. In addition, a generalized Kullback-Leibler distance, constructed in the framework of a nonextensive formalism, is recalled and used as a measure of b diversity between two systems. These tools are applied to data relative

Several indices have been created to measure diversity, and the most frequently used are the Shannon-Wiener (H) and Simpson (D) indices along with the number of species (S) and evenness (E). Controversies about which index should be used are common in literature. However, a generalized entropy (Tsallis entropy) has the potential to solve part of these problems. Here we explore a family of diversity indices (S q ; where q is the Tsallis index) and evenness (E q ), based on Tsallis entropy that incorporates the most used indices. It approaches S when q = 0, H when q → 1 and gives D when q = 2. In general, varying the value of the Tsallis index (q), S q varies from emphasis on species richness (q 1). Similarly, E q also works as a tool to investigate diversity. In particular, for a given community, its minimum value represents the maximum deviation from homogeneity (E q = 1) for a particular q (herein named q*). It is remarkable that our analysis indicates that q* and its corresponding evenness, E q *, are negatively affected by S when using simulated data. They may represent an index related to species rarity. Furthermore, S q* (i.e. the value of S q for a specific q*) is positively affected by richness that is an important property of any diversity index. In general, our findings indicate that the indices H, D, S, S q* , E and E q* are only part of a whole set of possibilities. In addition, the ecological properties of E q* and S q* , proposed here for the first time, show promise in ecology.

We introduce an expansion approach to obtain the elastic free energy density in rubber elasticity. This new approach presents fast convergence for whole range of deformation. We apply this method to the freely jointed model and get an analytical form for the free elastic energy density in terms of the strain invariants. We use this expression to fit experimental data for PMDS network.

We consider nonlocal effects, obtained by incorporating fractional derivatives in the kinetic energy of a conventional Hamiltonian, to analyze physical properties of non-crystalline solids at very low temperature. By using thermal Green functions, we deduce some experimentally observable quantities such as the particle momentum distribution function, the particle energy distribution function, and the specific heat. The agreement between the results obtained for the specific heat and the experimental data suggests that the approach presented here may be useful as a phenomenological model to investigate thermal properties of non-crystalline solids at low temperature.

We investigate the solutions of a generalized diffusion equation which contains space and time fractional derivatives by taking an absorbent (or source) term and a linear external force into account. They are expressed in terms of the Fox H functions and, for some special cases, they are directly related to the Levy distributions. We also analyze the first passage time distribution for the case characterized by the absence of absorbent (or source) term and external force for a semi-infinite interval with absorbent boundary conditions. We also discuss the connection of the results presented here with the anomalous diffusion.

Lottery is the most famous branch among all the games of chance. By analysing data from Mega-Sena, the major lottery in Brazil, we investigated the presence of persistent behaviour in the time series of the number of winners. We found that the demand for tickets grew collectively as an exponential driven by the size of the accumulated jackpot. Finally, we identified that a stochastic model grounded on the rolling-over feature of lotteries can generate correlations qualitatively similar to those observed empirically. The model is consistent with the idea that the growth in the number of bets, motivated by the size of the expected jackpot, is a mechanism generator of correlations in an apparently random scenario.

We report on an extensive characterization of the cracking noise produced by charcoal samples when dampened with ethanol. We argue that the evaporation of ethanol causes transient and irregularly distributed internal stresses that promote the fragmentation of the samples and mimic some situations found in mining processes. The results show that, in general, the most fundamental seismic laws ruling earthquakes (Gutenberg-Richter law, unified scaling law for the recurrence times, Omori's law, productivity law and Bath's law) hold under the conditions of the experiment. Some discrepancies were also identified (a smaller exponent in Gutenberg-Richter law, a stationary behavior in the aftershock rates for long times and a double power-law relationship in productivity law) and related to the different loading condition. Our results thus corroborate to elucidate the parallel between seismic laws and fracture experiments caused by a more complex loading condition that also occurs in natural and induced seismicity (such as long-term fluid injection and gas-rock outbursts in mining processes)., Accepted for publication in PRL

Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified $q$-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.

More than a half of world population is now living in cities and this number is expected to be two-thirds by 2050. Fostered by the relevancy of a scientific characterization of cities and for the availability of an unprecedented amount of data, academics have recently immersed in this topic and one of the most striking and universal finding was the discovery of robust allometric scaling laws between several urban indicators and the population size. Despite that, most governmental reports and several academic works still ignore these nonlinearities by often analyzing the raw or the per capita value of urban indicators, a practice that actually makes the urban metrics biased towards small or large cities depending on whether we have super or sublinear allometries. By following the ideas of Bettencourt et al., we account for this bias by evaluating the difference between the actual value of an urban indicator and the value expected by the allometry with the population size. We show that this scale-adjusted metric provides a more appropriate/informative summary of the evolution of urban indicators and reveals patterns that do not appear in the evolution of per capita values of indicators obtained from Brazilian cities. We also show that these scale-adjusted metrics are strongly correlated with their past values by a linear correspondence and that they also display crosscorrelations among themselves. Simple linear models account for 31%-97% of the observed variance in data and correctly reproduce the average of the scale-adjusted metric when grouping the cities in above and below the allometric laws. We further employ these models to forecast future values of urban indicators and, by visualizing the predicted changes, we verify the emergence of spatial clusters characterized by regions of the Brazilian territory where we expect an increase or a decrease in the values of urban indicators., Comment: Accepted for publication in PLoS ONE

The spatial dynamics of criminal activities has been recently studied through statistical physics methods; however, models and results have been focused on local scales (city level) and much less is known about these patterns at larger scales such as at a country level. Here we report on a characterization of the spatial dynamics of the homicide crimes along the Brazilian territory using data from all cities (~5000) in a period of more than thirty years. Our results show that the spatial correlation function in the per capita homicides decays exponentially with the distance between cities and that the characteristic correlation length displays an acute increasing trend in the latest years. We also investigate the formation of spatial clusters of cities via a percolation-like analysis, where clustering of cities and a phase transition-like behavior describing the size of the largest cluster as a function of a homicide threshold are observed. This transition-like behavior presents evolutive features characterized by an increasing in the homicide threshold (where the transitions occur) and by a decreasing in the transition magnitudes (length of the jumps in the cluster size). We believe that our work sheds new lights on the spatial patterns of criminal activities at large scales, which may contribute for better political decisions and resources allocation as well as opens new possibilities for modeling criminal activities by setting up fundamental empirical patterns at large scales., Comment: Accepted for publication in EPL

We analyze a database comprising the impact factor (citations per recent items published) of scientific journals for a 13-year period (1992?2004). We find that i) the distribution of impact factors follows asymptotic power law behavior, ii) the distribution of annual logarithmic growth rates has an exponential form, and iii) the width of this distribution decays with the impact factor as a power law with exponent ? 0.22. The results ii) and iii) are surprising similar to those observed in the growth dynamics of organizations with complex internal structure suggesting the existence of common mechanisms underlying the dynamics of these systems. We propose a general model for such systems, an extension of the simplest model for firm growth, and compare their predictions with our empirical results.

We investigate a finite chain approximation, the non-Gaussian Tsallis distribution, to the polymeric network, which gives an improvement to the Gaussian model. This distribution presents some necessary characteristics, like a cutoff to the maximum chain length and a continuous limit to the Gaussian one for a large number of monomers. It also presents a simple quadratic structure that allows to generalize the Gaussian properties such as exact-moments calculation and Wick theorem. We obtain the free-energy density in its full tensorial structure.

We investigate the solutions of a fractional diffusion equation with radial symmetry by using the Green function approach and by taking the N -dimensional case into account. In our analysis, a spatial time-dependent diffusion coefficient is considered, i.e., D ( r , t ) = D t δ - 1 r - θ / Γ ( α ) . The presence of external forces F ( r ) = K r e with e = - 1 - θ and F ( r ) = - kr + K r e is also taken into account. In particular, we discuss the results obtained by employing boundary conditions defined on a finite interval, and afterwards the analysis is extended to a semi-infinite interval. Finally, we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work.

We analyze data sets containing the circulation of magazines and newspapers. We show that the cumulative distribution follows, in the range of large circulation, a power law behavior whose exponent is μ 1.5; and deviations from the asymptotic power law behavior can be well described by a q-exponential distribution (Zipf-Mandelbrot law) from Tsallis statistics. We also show that, in the range of large circulation, the distribution of logarithmic growth rates is consistent with an exponential; and the standard deviation of the growth rates is practically independent of the circulation (size). Moreover, we employ a model, inspired in one of the simplest model for firm growth, in order to reproduce some of our findings.

We analyze a fractional diffusion equation by taking an anisotropic case into account. In our analysis, we also consider a spatial time dependent diffusion coefficient and the presence of external forces in the system. For the cases analyzed here, we obtain exact solutions and show that the solutions have an anomalous spreading. In addition, we discuss a rich class of diffusive processes, including normal and anomalous ones described by this equation.

We analyze a multidimensional nonlinear diffusion equation taking a spatial time dependent diffusion coefficient and external forces into account. We obtain new exact classes of solutions and investigate the transverse effects induced by an external force applied in the system. We also connect the solutions found here with the distribution that emerges from the nonextensive statistics.

We analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence, the presence of external forces and time fractional derivatives. We obtain new exact classes of solutions for a linear anisotropic fractional diffusion equation and investigate the time scaling behavior and an asymptotic solution for a nonlinear anisotropic fractional diffusion equation. We connect the asymptotic solution obtained with the distribution that emerges from the nonextensive statistics to the nonlinear case. We also verify different diffusive behavior, for instance, subdiffusion and superdiffusion, in each direction.

We investigate the solutions of a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our study by considering the linear case and the nonlinear case afterward. The linear case is analyzed taking fractional time and spatial derivatives into account. In this context, we also discuss the modifications that emerge by considering a diffusion coefficient given by D (x)∝|x| −θ . For the nonlinear case accomplishing the fractional time derivative, we discuss scaling behavior of the time and the asymptotic for the solution of the nonlinear fractional diffusion equation. In this case, the connection between the asymptotic solution found here and the nonextensive Tsallis statistics is performed.

In the context of nonextensive Tsallis statistics, sensible examples are employed in order to compare the variational method proposed in Phys. Rev. Lett. 80 (1998) 218 with the one in J. Phys. A 26 (1993) L893. These examples are such that they can be solved exactly. Thus, a comparison of the exact result with the approximated ones is possible. The analysis of these examples indicates that the calculation related to the first variational method is easier to work with than the other method. In particular, when one is trying to apply the last variational method, the calculation of the approximated case can be more cumbersome than solving the exact one.

We analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our investigation by considering the linear case and the nonlinear case afterward. The linear case is discussed taking fractional time and spatial derivatives into account in a unified approach. We also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by D(x,t)=D(t)|x|−θ. For the nonlinear case, we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation.