Description of diffusive and propagative behavior on fractals

Self-similar Turing patterns: An anomalous diffusion consequence

In this work, we show that under specific anomalous diffusion conditions, chemical systems can produce well-ordered self-similar concentration patterns through diffusion-driven instability. We also find spiral patterns and patterns with mixtures of rotational symmetries. The type of anomalous diffusion discussed in this work, either subdiffusion or superdiffusion, is a consequence of the medium heterogeneity, and it is modeled through a space-dependent diffusion coefficient with a power-law functional form.

Effective degrees of freedom of a random walk on a fractal

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Reaction spreading on percolating clusters

Reaction-diffusion processes in two-dimensional percolating structures are investigated. Two different problems are addressed: reaction spreading on a percolating cluster and front propagation through a percolating channel. For reaction spreading, numerical data and analytical estimates show a power-law behavior of the reaction product as M(t)~t(d(l)), where d(l) is the connectivity dimension. In a percolating channel, a statistically stationary traveling wave develops. The speed and the width of the traveling wave are numerically computed. While the front speed is a low-fluctuating quantity and its behavior can be understood using a simple theoretical argument, the front width is a high-fluctuating quantity showing a power-law behavior as a function of the size of the channel.

Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.

Reaction spreading on graphs

We study reaction-diffusion processes on graphs through an extension of the standard reaction-diffusion equation starting from first principles. We focus on reaction spreading, i.e., on the time evolution of the reaction product M(t). At variance with pure diffusive processes, characterized by the spectral dimension d{s}, the important quantity for reaction spreading is found to be the connectivity dimension d{l}. Numerical data, in agreement with analytical estimates based on the features of n independent random walkers on the graph, show that M(t)∼t{d{l}}. In the case of Erdös-Renyi random graphs, the reaction product is characterized by an exponential growth M(t)e{αt} with α proportional to ln(k), where (k) is the average degree of the graph.

Faster diffusion across an irregular boundary

We investigate how the shape of a heat source may enhance global heat transfer at short time. An experiment is described that allows us to obtain a direct visualization of heat propagation from a prefractal radiator. We show, both experimentally and numerically, that irregularly shaped passive coolers rapidly dissipate at short times, but their efficiency decreases with time. The de Gennes scaling argument is shown to be only a large scale approximation, which is not sufficient to describe adequately the temperature distribution close to the irregular frontier. This work shows that radiators with irregular surfaces permit increased cooling of pulsed heat sources.

Different diffusive regimes, generalized Langevin and diffusion equations

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.

Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist

Single particle trajectories are investigated assuming the coexistence of two subdiffusive processes: diffusion on a fractal structure modeling spatial constraints on motion and heavy-tailed continuous time random walks representing energetic or chemical traps. The particles' mean squared displacement is found to depend on the way the mean is taken: temporal averaging over single-particle trajectories differs from averaging over an ensemble of particles. This is shown to stem from subordinating an ergodic anomalous process to a nonergodic one. The result is easily generalized to the subordination of any other ergodic process (i.e., fractional Brownian motion) to a nonergodic one. For certain parameters the ergodic diffusion on the underlying fractal structure dominates the transport yet displaying ergodicity breaking and aging.

Random walks on the Comb model and its generalizations

Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to "effective reducing" of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed.

Spatial gradients in Clovis-age radiocarbon dates across North America suggest rapid colonization from the north

A key issue in the debate over the initial colonization of North America is whether there are spatial gradients in the distribution of the Clovis-age occupations across the continent. Such gradients would help indicate the timing, speed, and direction of the colonization process. In their recent reanalysis of Clovis-age radiocarbon dates, Waters and Stafford [Waters MR, Stafford TW, Jr (2007) Science 315:1122-1126] report that they find no spatial patterning. Furthermore, they suggest that the brevity of the Clovis time period indicates that the Clovis culture represents the diffusion of a technology across a preexisting pre-Clovis population rather than a population expansion. In this article, we focus on two questions. First, we ask whether there is spatial patterning to the timing of Clovis-age occupations and, second, whether the observed speed of colonization is consistent with demic processes. With time-delayed wave-of-advance models, we use the radiocarbon record to test several alternative colonization hypotheses. We find clear spatial gradients in the distribution of these dates across North America, which indicate a rapid wave of advance originating from the north. We show that the high velocity of this wave can be accounted for by a combination of demographic processes, habitat preferences, and mobility biases across complex landscapes. Our results suggest that the Clovis-age archaeological record represents a rapid demic colonization event originating from the north.

Fractional calculus applied to the analysis of spectral electrical conductivity of clay-water system

The analysis of the low-frequency conductivity spectra of the clay-water mixtures is presented. The frequency dependence of the conductivity is shown to follow the power-law with the exponent n=0.67 before reaching the frequency-independent part. When scaled with the value of the frequency-independent part of the spectrum the conductivity spectra for samples at different water content values are shown to fit to a single master curve. It is argued that the observed conductivity dispersion is a consequence of the anomalously diffusing ions in the clay-water system. The fractional Langevin equation is then used to describe the stochastic dynamics of the single ion. The results indicate that the experimentally observed dielectric properties originate in anomalous ion transport in clay-water system characterized with time-dependent diffusion coefficient.

Generalized diffusion equation for anisotropic anomalous diffusion

Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.

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.

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.