Analytical solutions for diffusion on fractal objects

Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients

We consider the Langevin equation with a multiplicative noise term that depends on time and space. The corresponding Fokker-Planck equation in the Stratonovich approach is investigated. Its exact solution is obtained for an arbitrary multiplicative noise term given by g (x,t) =D (x) T (t) , and the behaviors of probability distributions, for some specific functions of D (x) , are analyzed. We show that the asymptotic shape of the random-walk model and power-law decay obtained from other approaches can be reproduced from our solutions, by employing two simple functions for g (x,t) . In particular, for D(x) approximately /x/(-theta/2) , the physical solutions for the probability distribution in the Ito, Stratonovich, and postpoint discretization approaches can be obtained and analyzed.

Estimating the total genetic diversity of a spatial field population from a sample and implications of its dependence on habitat area

The total genetic diversity of a species is a key factor in its persistence and conservation. Because realistic sample sizes are far smaller than the total population, it is impractical to exhaustively characterize diversity of most populations. Here, we demonstrate the possibility of calculating the genetic diversity of a spatial population from a sample using genealogical models. We trace the history of a population by simulating the locations of the ancestors of a particular sample of the population backwards in time. We use this method to estimate the genetic diversity of the global population of Pseudomonas bacteria. The same results are obtained whether using a global sample or a subsample restricted to a particular geographic region (California). The results are also validated by comparing additional predictions of the model to the data. Furthermore, we use these results to show that the level of genetic diversity in a population depends strongly on the size of its habitat, much more strongly than does biodiversity as measured by the number of species. The strong dependence of diversity on habitat area has significant implications for conservation strategies.

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.

Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient

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.

Membrane lateral mobility obstructed by polymer-tethered lipids studied at the single molecule level

Obstructed long-range lateral diffusion of phospholipids (TRITC-DHPE) and membrane proteins (bacteriorhodopsin) in a planar polymer-tethered 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine bilayer is studied using wide-field single molecule fluorescence microscopy. The obstacles are well-controlled concentrations of hydrophobic lipid-mimicking dioctadecylamine moieties in the polymer-exposed monolayer of the model membrane. Diffusion of both types of tracer molecules is well described by a percolating system with different percolation thresholds for lipids and proteins. Data analysis using a free area model of obstructed lipid diffusion indicates that phospholipids and tethered lipids interact via hard-core repulsion. A comparison to Monte Carlo lattice calculations reveals that tethered lipids act as immobile obstacles, are randomly distributed, and do not self-assemble into large-scale aggregates for low to moderate tethering concentrations. A procedure is presented to identify anomalous subdiffusion from tracking data at a single time lag. From the analysis of the cumulative distribution function of the square displacements, it was found that TRITC-DHPE and W80i show normal diffusion at lower concentrations of tethered lipids and anomalous diffusion at higher ones. This study may help improve our understanding of how lipids and proteins in biomembranes may be obstructed by very small obstacles comprising only one or very few molecules.

Superdiffusion on a comb structure

We study specific properties of particles transport by exploring an exact solvable model, a so-called comb structure, where diffusive transport of particles leads to subdiffusion. A performance of the Lévy-like process enriches this transport phenomenon. It is shown that an inhomogeneous convection flow is a mechanism for the realization of the Lévy-like process. It leads to superdiffusion of particles on the comb structure. This superdiffusion is an enhanced one with an arbitrary large transport exponent, but all moments are finite. A frontier case of superdiffusion, where the transport exponent approaches infinity, is studied. The log-normal distribution with the exponentially fast superdiffusion is obtained for this case.

Description of diffusive and propagative behavior on fractals

The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification than previous works on this field. The most important result we present is the introduction of a dependence on time and space for the conductivity in fractals, which is deduced by scaling arguments and supported by computer simulations. Finally, the diffusion equation is used to introduce the possibility of reaction-diffusion processes on fractals and analyze their properties. Specifically, an analytic expression for the speed of the corresponding travelling fronts, which can be of great interest for application purposes, is derived.

Dynamical features of reaction-diffusion fronts in fractals

The speed of front propagation in fractals is studied by using (i) the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and (ii) the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration.

Speed of reaction-diffusion fronts in spatially heterogeneous media

The front speed problem for nonuniform reaction rate and diffusion coefficient is studied by using singular perturbation analysis, the geometric approach of Hamilton-Jacobi dynamics, and the local speed approach. Exact and perturbed expressions for the front speed are obtained in the limit of large times. For linear and fractal heterogeneities, the analytic results have been compared with numerical results exhibiting a good agreement. Finally we reach a general expression for the speed of the front in the case of smooth and weak heterogeneities.

Power law diffusion coefficient and anomalous diffusion: analysis of solutions and first passage time

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.

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.

Statistics of Lagrangian velocities in turbulent flows

We present a generalized Fokker-Planck equation for the joint position-velocity probability distribution of a single fluid particle in a turbulent flow. Based on a simple estimate, the diffusion term is related to the two-point two-time Eulerian acceleration-acceleration correlation. Dimensional analysis yields a velocity increment probability distribution with normal scaling v approximately t(1/2). However, the statistics need not be Gaussian.

Random walk through fractal environments

We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D(F) of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D(F)< or =2 can thus be considered as defective Levy walks. The distribution of jump increments for D(F)>2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D(F)<2, the diffusion is dominated by the finite escape rate. Diffusion for D(F)>2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.

N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients

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.

Transport with multiple-rate exchange in disordered media

We investigate transport of particles subject to exchange using the continuous-time random-walk framework. Transition is controlled by macroscale, and exchange by both macroscale and microscale disorder. A wide class of exchange mechanisms is represented using the multiple-rate exchange model. Particles are transported along random trajectories viewed as one-dimensional lattices. The solution of the transport problem is obtained in the form of the crossing-time density, h(t;L), at an exit surface L; h is dependent on two functions, g and f. g characterizes exchange controlled by microscale disorder. The joint density f is central for the solution as it relates the microscale and macroscale disorder along random trajectories. For the case of transition and exchange disorder, we show that power-law exponent eta (characterizing microscale disorder) and power-law exponents alpha(tau) and alpha(mu) (characterizing macroscale disorder), define two regions delimited by a line alpha(tau)=alpha(mu)(eta+1): One in which the asymptotic transport is dominated by transition, and one in which it is dominated by the exchange. For the case of transition disorder with uniform exchange, both transition and exchange can influence the late-time behavior of h(t). Microscale exchange processes will unconditionally influence the late-time behavior of h(t) only if eta<0. If eta>0, exchange will dominate at late time provided that transition is asymptotically Gaussian.

Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution

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.

Stochastic dynamics from the fractional Fokker-Planck-Kolmogorov equation: large-scale behavior of the turbulent transport coefficient

The formulation of the fractional Fokker-Planck-Kolmogorov (FPK) equation [Physica D 76, 110 (1994)] has led to important advances in the description of the stochastic dynamics of Hamiltonian systems. Here, the long-time behavior of the basic transport processes obeying the fractional FPK equation is analyzed. A derivation of the large-scale turbulent transport coefficient for a Hamiltonian system with 11 / 2 degrees of freedom is proposed in connection with the fractal structure of the particle chaotic trajectories. The principal transport regimes (i.e., a diffusion-type process, ballistic motion, subdiffusion in the limit of the frozen Hamiltonian, and behavior associated with self-organized criticality) are obtained as partial cases of the generalized transport law. A comparison with recent numerical and experimental studies is given.

Green's function approach to nonclassical reaction kinetics in fractal media

We present an accurate description for the mean field kinetics of the reversible recombination reaction occurring in fractal media, which is in good agreement with Monte Carlo simulation results from an intermediate to a long time region. The central dynamic quantity is the Green's function for the generalized diffusion equation that is defined in a hypothetical space whose dimensionality is given by the fracton dimension d. An exact expression for the Green's function that is valid for arbitrary values of d is presented.

Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution

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.

Percolation in sign-symmetric random fields: topological aspects and numerical modeling

The topology of percolation in random scalar fields psi(x) with sign symmetry [i.e., that the statistical properties of the functions psi(x) and -psi(x) are identical] is analyzed. Based on methods of general topology, we show that the zero set psi(x)=0 of the n-dimensional (n>/=2) sign-symmetric random field psi(x) contains a (connected) percolating subset under the condition |nablapsi(x)| not equal0 everywhere except in domains of negligible measure. The fractal geometry of percolation is analyzed in more detail in the particular case of the two-dimensional (n=2) fields psi(x). The improved Alexander-Orbach conjecture [Phys. Rev. E 56, 2437 (1997)] is applied analytically to obtain estimates of the main fractal characteristics of the percolating fractal sets generated by the horizontal "cuts," psi(x)=h, of the field psi(x). These characteristics are the Hausdorff fractal dimension of the set, D, and the index of connectivity, straight theta. We advocate an unconventional approach to studying the geometric properties of fractals, which involves methods of homotopic topology. It is shown that the index of connectivity, straight theta, of a fractal set is the topological invariant of this set, i.e., it remains unchanged under the homeomorphic deformations of the fractal. This issue is explicitly used in our study to find the Hausdorff fractal dimension of the single isolevels of the field psi(x), as well as the related geometric quantities. The results obtained are analyzed numerically in the particular case when the random field psi(x) is given by a fractional Brownian surface whose topological properties recover well the main assumptions of our consideration.

Quantitation of non-Einstein diffusion behavior of water in biological tissues by proton MR diffusion imaging: synthetic image calculations

The non-Einstein diffusion behavior of water in a model biological tissue system, intact duck embryos, has been investigated by the use of an in vivo proton pulsed-gradient spin-echo (PGSE) MR imaging technique. Multiple-frame MR images of the intact duck embryos and control solution (0.5 mM CuSO4 doped water) were acquired systematically at different diffusion times and strengths of the diffusion-sensitizing magnetic field gradients of the PGSE sequence. These raw images were then used to generate various dynamic (self-diffusion coefficient) and structural (fractal, residual attenuation, and compartment fraction) diffusion parameter maps of water in the imaging objects on the basis of different Einstein and higher order (non-Brownian, Residual, and 2-compartment) diffusion models. The self-diffusion coefficients of the body tissues of the embryos obtained from all diffusion models were significantly lower than those of the surrounding embryonic fluid. The structural diffusion parameter maps obtained from the higher order diffusion models revealed that water molecules exhibited either non-Brownian, restricted, or compartmentalized diffusion behavior in the embryonic tissues, but Einstein or Brownian diffusion behavior in the embryonic fluid and control solution. The diffusion parameter maps, both dynamic and structural, were found to provide much better contrasts than the conventional relaxation time (T1, T2, and biexponential T2) maps in separating the tissues from the surrounding embryonic fluid in the duck embryos. The mathematical models and procedures for generating the dynamic and structural diffusion parameter maps are also presented in this paper.