A. A. Tateishi

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Characterization of river flow fluctuations via horizontal visibility graphs

We report on a large-scale characterization of river discharges by employing the network framework of the horizontal visibility graph. By mapping daily time series from 141 different stations of 53 Brazilian rivers into complex networks, we present a useful approach for investigating the dynamics of river flows. We verified that the degree distributions of these networks were well described by exponential functions, where the characteristic exponents are almost always larger than the value obtained for random time series. The faster-than-random decay of the degree distributions is an another evidence that the fluctuation dynamics underlying the river discharges has a long-range correlated nature. We further investigated the evolution of the river discharges by tracking the values of the characteristic exponents (of the degree distribution) and the global clustering coefficients of the networks over the years. We show that the river discharges in several stations have evolved to become more or less correlated (and displaying more or less complex internal network structures) over the years, a behavior that could be related to changes in the climate system and other man-made phenomena.

Universal patterns in sound amplitudes of songs and music genres.

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.

Investigating the interplay between mechanisms of anomalous diffusion via fractional Brownian walks on a comb-like structure

The comb model is a simplified description for anomalous diffusion under geometric constraints. It represents particles spreading out in a two-dimensional space where the motions in the x-direction are allowed only when the y coordinate of the particle is zero. Here, we propose an extension for the comb model via Langevin-like equations driven by fractional Gaussian noises (long-range correlated). By carrying out computer simulations, we show that the correlations in the y-direction affect the diffusive behavior in the x-direction in a non-trivial fashion, resulting in a quite rich diffusive scenario characterized by usual, superdiffusive or subdiffusive scaling of second moment in the x-direction. We further show that the long-range correlations affect the probability distribution of the particle positions in the x-direction, making their tails longer when noise in the y-direction is persistent and shorter for anti-persistent noise. Our model thus combines and allows the study/analysis of the interplay between different mechanisms of anomalous diffusion (geometric constraints and long-range correlations) and may find direct applications for describing diffusion in complex systems such as living cells.

Anomalous diffusion and transport in heterogeneous systems separated by a membrane

Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.

Green function for a non-Markovian Fokker-Planck equation: comb-model and anomalous diffusion

We investigate solutions, by using the Green function approach, for a system governed by a non-Markovian Fokker-Planck equation and subjected to a Comb structure. This structure consists of the axis of structure as the backbone and fingers which are attached perpendicular to the axis. For this system, we consider an arbitrary initial condition, in the presence of time dependent diffusion coefficients and spatial fractional derivative, and analyze the connection to the anomalous diffusion.

Exact propagator for a Fokker-Planck equation, first passage time distribution, and anomalous diffusion

We obtain an exact form for the propagator of the Fokker-Planck equation ∂tρ = ∂xD(x)∂xρ −∂x(F(x, t)ρ), with D(x)=D|x|−η in presence of the external force F(x,t)=−k(t)x+K/x|x|−η. Using the results found here, we also investigate the mean square displacement, survival probability, and first passage time distribution. In addition, we discuss the connection of these results with anomalous diffusion phenomena.