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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.

We report on a statistical analysis of the engagement in the electoral processes of all Brazilian cities by considering the number of party memberships and the number of candidates for mayor and councillor. By investigating the relationships between the number of party members and the population of voters, we have found that the functional forms of these relationships are well described by sublinear power laws (allometric scaling) surrounded by a multiplicative log-normal noise. We have observed that this pattern is quite similar to those we previously reported for the relationships between the number of candidates (mayor and councillor) and population of voters [Europhys. Lett. 96, 48001 (2011)], suggesting that similar universal laws may be ruling the engagement in the electoral processes. We also note that the power-law exponents display a clear hierarchy, where the more influential is the political position the smaller is the value of the exponent. We have also investigated the probability distributions of the number of candidates (mayor and councillor), party memberships, and voters. The results indicate that the most influential positions are characterized by distributions with very short tails, while less influential positions display an intermediate power-law decay before showing an exponential-like cutoff. We discuss the possibility that, in addition to the political power of the position, limitations in the number of available seats can also be connected with this changing of behavior. We further believe that our empirical findings point out to an under-representation effect, where the larger the city is, the larger are the obstacles for more individuals to become directly engaged in the electoral process.

We report on the dynamical behavior of defects of strength s=±1/2 in a lyotropic liquid crystal during the annihilation process. By following their positions using time-resolved polarizing microscopy technique, we present statistically significant evidence that the relative velocity between defect pairs is Gaussian distributed, antipersistent, and long-range correlated. We further show that simulations of the Lebwohl-Lasher model reproduce quite well our experimental findings.

The effects of an external force on a diffusive process subjected to a backbone structure are investigated by considering the system governed by a Fokker-Planck equation with drift terms. Our results show an anomalous spreading which may present different diffusive regimes connected to anomalous diffusion and stationary states.

We demonstrate theoretically that the presence of ions in insulating materials such as nematic liquid crystals may be responsible for the dielectric spectroscopy behavior observed experimentally. It is shown that, at low frequencies, an essentially non-Debye relaxation process takes place due to surface effects. This is accomplished by investigating the effects of the adsorption-desorption process on the electrical response of an electrolytic cell when the generation and recombination of ions is present. The adsorption-desorption is governed by a non-usual kinetic equation in order to incorporate memory effects related to a non-Debye relaxation and the roughness of the surface. The analysis is carried out by searching for solutions to the drift-diffusion equation that satisfy the Poisson equation relating the effective electric field to the net charge density. We also discuss the effect of the mobility of the ions, i.e., situations with equal and different diffusion coefficients for positive and negative ions, on the impedance and obtain an exact expression for the admittance. The model is compared with experimental results measured for the impedance of a nematic liquid crystal sample and a very good agreement is obtained.

We investigate a generalized Langevin equation (GLE) in the presence of an additive noise characterized by the mixture of the usual white noise and an arbitrary one. This scenario lead us to a wide class of diffusive processes, in particular the ones whose noise correlation functions are governed by power laws, exponentials, and Mittag-Leffler functions. The results show the presence of different diffusive regimes related to the spreading of the system. In addition, we obtain a fractional diffusionlike equation from the GLE, confirming the results for long time., (© 2012 American Physical Society)

We investigate the electrical response of Milli-Q deionized water by using a fractional diffusion equation of distributed order with the interfaces (i.e., the boundary conditions at the electrodes limiting the sample) governed by integrodifferential equations. We also consider that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies Poisson's equation. In addition, the good agreement between the experimental data and this approach evidences the presence of anomalous diffusion due to the surface effects in this system.

We report a statistical analysis of more than eight thousand songs. Specifically, we investigated the probability distribution of the normalized sound amplitudes. Our findings suggest a universal form of distribution that agrees well with a one-parameter stretched Gaussian. We also argue that this parameter can give information on music complexity, and consequently it helps classify songs as well as music genres. Additionally, we present statistical evidence that correlation aspects of the songs are directly related to the non-Gaussian nature of their sound amplitude distributions.

The non-Markovian diffusion of dispersed particles in a semi-infinite cell of an isotropic fluid limited by an adsorbing-desorbing surface is theoretically investigated. The density of dispersed particles in the bulk is a time dependent function and the time dependent density of surface particles is governed by a modified kinetic equation with a time dependent kernel. In this framework, the densities of bulk and surface particles are analytically determined, taking into account the conservation of the number of particles immersed in the sample. This system exhibits anomalous diffusion behavior as well as memory effects in the adsorption-desorption process. The results obtained here are expected to be useful to investigate the adsorption-desorption phenomena of neutral as well as charged particles in an isotropic fluid in contact with a solid substrate when the anomalous diffusion is present.

We obtain exact solutions and the survival probability for a Fokker-Planck equation subjected to the two-dimensional wedge domain. We consider a spatial dependence in the diffusion coefficient and the presence of external forces. The results show an anomalous spreading of the solution and, consequently, a nonusual behavior of the survival probability which can be connected to anomalous diffusion.

Surface effects on a diffusion process governed by a fractional diffusion equation in a confined region with spatial and time dependent boundary conditions are investigated. First, we consider the one-dimensional case with the boundary conditions rho(0,t)=Phi0(t) and rho(a,t)=Phia(t). Subsequently, the two-dimensional case in the cylindrical symmetry with rho(a,theta,t)=Phia(theta,t) and rho(b,theta,t)=Phib(theta,t) is investigated. For these cases, we also obtain exact solutions for an arbitrary initial condition by using the Green's function approach.

The adsorption-desorption phenomenon of neutral particles dissolved in an isotropic fluid is investigated by using a nonsingular kernel in the kinetic equation at the limiting surfaces. To account for the relevance of a memory effect, three types of kernels in the kinetic equation are considered. Similar kernels have been used to investigate nonexponential relaxation including several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers. A suitable choice for a temporal kernel can account for the relative importance of physisorption or chemisorption, according to the time scale governing the adsorption phenomena, and can be the key mechanism to understand the specific roles of both processes. By using a general procedure, the time evolution of the density of particles is determined in closed analytical form. The analysis is relevant in the description of the adsorption phenomena in general.

We obtain a formal solution for a large class of diffusion equations with a spatial kernel dependence in the diffusive term. The presence of this kernel represents a nonlocal dependence of the diffusive process and, by a suitable choice, it has the spatial fractional diffusion equations as a particular case. We also consider the presence of a linear external force and source terms. In addition, we show that a rich class of anomalous diffusion, e.g., the Lévy superdiffusion, can be obtained by an appropriated choice of kernel.

We investigate the solutions and first passage time distribution for an anomalous diffusion process governed by a generalized non-Markovian Fokker-Planck equation. In our analysis, we also consider the presence of external forces and absorbent (source) terms. In addition, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.

The exact dynamical evolution of the director field for splay-bend deformations, in nematic liquid crystal samples limited by inhomogeneous surfaces, is determined in the one-constant approximation. The initial conditions and boundary-value problem concerning the situation of strong anchoring at the surfaces of a sample of slab shape of thickness d is analytically solved in the presence of a time dependent external electric field, and taking into account the viscous torque. The results are used to analytically obtain the time dependence of the phase shift between the two components of a linearly polarized beam impinging perpendicularly on the sample. The analysis can be relevant to investigate the phase retardation of a nematic cell submitted to an external voltage which is lower than or in the order of the Féedericksz threshold to induce deformations in the sample.

In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.

We consider the time-fractional diffusion equation with time dependent diffusion coefficient given by (O)O(alpha)(C)(t) W (x,t) = D(alpha,gamma)(t)(gamma) [theta(2) W (x,t) /theta x(2)], where O is the Caputo operator. We investigate its solutions in the infinite and the finite domains. The mean squared displacement and the mean first passage time are also considered. In particular, for alpha = 0 , the mean squared displacement is given by approximately t(gamma) and we verify that the mean first passage time is finite for superdiffusive regimes.

We investigate an N-dimensional fractional diffusion equation with radial symmetry by using the Green function approach. We consider, in our analysis, the spatial dependence on the diffusion coefficient and the presence of an external force. In particular, we employ boundary conditions in a finite interval and after we extend it to a semi-infinite interval. We also show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the usual diffusion equation. We consider a space- and time-dependent diffusion coefficient and the presence of absorbing boundaries. We obtain analytical results for the probability distribution and the first passage time distribution for finite and semi-infinite intervals. In addition, we compare our results for the first passage time distribution with the one obtained by the usual diffusion equation with constant diffusion coefficient.

We investigate one-dimensional equations for the diffusion with a nonconstant diffusion coefficient inside the second derivative and between the derivatives. In particular, we employ the diffusion coefficient D(x) proportional to /x/(-theta)(theta in R) and a quartic potential. These diffusion equations present a rich variety of behaviors associated with different regimes. Results of two approaches are analyzed and compared. We also investigate the mean first passage time of these systems. We show that the system with the coefficient D(x) between the derivatives can produce different behaviors for the mean first passage time in comparison with those obtained by the system with the coefficient inside the derivatives.

We analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions. In particular, the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation. We also discuss some implications by considering a diffusion coefficient D(x,t)=D(t)/x/(-theta) (theta in R) and a drift force F=-k(1)(t)x+k(alpha)x/x/(alpha-1). In both situations, we relate our solutions to those obtained within the maximum entropy principle by using the Tsallis entropy.

Ubiquitous phenomena exist in nature where, as time goes on, a crossover is observed between different diffusion regimes (e.g., anomalous diffusion at early times which becomes normal diffusion at long times, or the other way around). In order to focus on such situations we have analyzed particular relevant cases of the generalized Fokker-Planck equation integral dgamma(')tau(gamma('))[ partial differential (gamma('))rho(x,t)]/ partial differential t(gamma('))= integral dmu(')dnu'D(mu('),nu('))[ partial differential (mu('))[rho(x,t)](nu('))]/ partial differential x(mu(')), where tau(gamma(')) and D(mu('),nu(')) are kernels to be chosen; the choice tau(gamma('))=delta(gamma(')-1) and D(mu('),nu('))=delta(mu(')-2)delta(nu(')-1) recovers the normal diffusion equation. We discuss in detail the following cases: (i) a mixture of the porous medium equation, which is connected with nonextensive statistical mechanics, with the normal diffusion equation; (ii) a mixture of the fractional time derivative and normal diffusion equations; (iii) a mixture of the fractional space derivative, which is related with Lévy flights, and normal diffusion equations. In all three cases a crossover is obtained between anomalous and normal diffusions. In cases (i) and (iii), the less diffusive regime occurs for short times, while at long times the more diffusive regime emerges. The opposite occurs in case (ii). The present results could be easily extended to more complex situations (e.g., crossover between two, or even more, different anomalous regimes), and are expected to be useful in the analysis of phenomena where nonlinear and fractional diffusion equations play an important role. Such appears to be the case for isolated long-ranged interaction Hamiltonians, which along time can exhibit a crossover from a longstanding metastable anomalous state to the usual Boltzmann-Gibbs equilibrium one. Another illustration of such crossover occurs in active intracellular transport.

An N-dimensional nonlinear Fokker-Planck equation is investigated here by considering the time dependence of the coefficients, where drift-controlled and source terms are present. We exhibit the exact solution based on the generalized Gaussian function related to the Tsallis statistics. Furthermore, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained by changing the time dependence of the coefficients.

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.

In the non-extensive Tsallis scenario, Page's conjecture for the average entropy of a subsystem [Phys. Rev. Lett. 71, 1291 (1993)] as well as its demonstration are generalized, i.e., when a pure quantum system, whose Hilbert space dimension is mn, is considered, the average Tsallis entropy of an m-dimensional subsystem is obtained. This demonstration is expected to be useful to study systems where the usual entropy does not give satisfactory results.

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.

We investigate the escape behavior of systems governed by the one-dimensional nonlinear diffusion equation theta(t)rho=theta(x)[theta(x)Urho]+Dtheta(x)2rho(nu), where the potential of the drift, U(x), presents a double well and D,nu are real parameters. For systems close to the steady state, we obtain an analytical expression of the mean first-passage time, yielding a generalization of Arrhenius law. Analytical predictions are in very good agreement with numerical experiments performed through integration of the associated Ito-Langevin equation. For nu not equal to 1, important anomalies are detected in comparison to the standard Brownian case. These results are compared to those obtained numerically for initial conditions far from the steady state.

The nonlinear diffusion equation partial delta rho/delta t=D Delta rho(nu) is analyzed here, where Delta[triple bond](1/r(d-1))(delta/delta r)r(d-1-theta) delta/delta r, and d, theta, and nu are real parameters. This equation unifies the anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [ theta>(1-nu)d], "normal" diffusion [theta=(1-nu)d] and superdiffusion [theta<(1-nu)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.