Non-Markovian Fokker-Planck equation: solutions and first passage time distribution

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.

Invasion percolation between two sites

We investigate the process of invasion percolation between two sites (injection and extraction sites) separated by a distance r in two-dimensional lattices of size L. Our results for the nontrapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) Pe at the extraction site. For Pe = 0, we show that the mass distribution of invaded clusters P(M) follows a power-law P(M) approximately M(-alpha) for intermediate values of the mass M, with an exponent alpha = 1.39+/-0.03. When the local pressure is set to Pe = Pc, where Pc corresponds to the site percolation threshold of the lattice topology, the distribution P(M) still displays a scaling region, but with an exponent alpha = 1.02+/-0.03. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these differences, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depend significantly on the local pressure Pe and it is consistent with the fractal dimension values reported for standard invasion percolation. Finally, we perform extensive numerical simulations to determine the effect of the lattice borders on the statistics of the invaded clusters and also to characterize the self-organized critical behavior of the invasion percolation process.

N-dimensional fractional diffusion equation and Green function approach: spatially dependent diffusion coefficient and external force

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.

Percolation on a multifractal

We investigate percolation phenomena in multifractal objects that are built in a simple way. In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. We observe that the percolation threshold for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent beta. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation.

Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption

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.

Models for correlated multifractal hypersurfaces

We discuss and implement computer approximations of fractal and multifractal hypersurfaces. These hypersurfaces consist of reconstructions of a stochastic process in the real space from randomly distributed variables in the discrete wavelet domain. The synthetic surfaces have the usual fractional Brownian motion as a particular case, and inherit the correlation structure of these fractals. We first introduce the one-dimensional version of these surfaces that obey a weak self-affine symmetry. This symmetry appears in the wavelet domain as a condition on the second moments of the probability distributions of the wavelet coefficients. Then we use these relations to define the fractals and multifractals in d dimensions. Finally, we concentrate on the generation of samples of these hypersurfaces.

Statistics of largest cluster growth through constant rate random filling of lattices

In this paper we consider a percolation model where the probability p for a site to be occupied increases linearly in time, from 0 to 1. We analyze the way the largest cluster grows in time, and in particular, we study the statistics of the "jumps" in the mass of the largest cluster, and of the time delay between those events. Different critical behaviors are observed below and above the percolation threshold. We propose a theoretical analysis, and we check our results against direct numerical simulations.

Quantifying nonstationary radioactivity concentration fluctuations near chernobyl: A complete statistical description

We analyze nonstationary 137Cs atmospheric activity concentration fluctuations measured near Chernobyl after the 1986 disaster and find three new results: (i) the histogram of fluctuations is well described by a log-normal distribution; (ii) there is a pronounced spectral component with period T=1yr, and (iii) the fluctuations are long-range correlated. These findings allow us to quantify two fundamental statistical properties of the data: the probability distribution and the correlation properties of the time series. We interpret our findings as evidence that the atmospheric radionuclide resuspension processes are tightly coupled to the surrounding ecosystems and to large time scale weather patterns.

Critical behavior of a two-species reaction-diffusion problem

We present a Monte Carlo study in dimension d=1 of the two-species reaction-diffusion process A+B-->2B and B-->A. Below a critical value rho(c) of the conserved total density rho the system falls into an absorbing state without B particles. Above rho(c) the steady state B particle density rho(st)(B) is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities rho between zero and some maximum density. From finite-size scaling we obtain the steady state exponents beta=0.435(10), nu=2.21(5), and eta=-0.606(4) for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value straight theta(')=0.30(2), in agreement with the theoretical relation straight theta(')=-eta/2 due to Van Wijland et al. [Physica A 251, 179 (1998)].