Showing papers on "Anomalous diffusion published in 2003"
TL;DR: It is shown by FCS that Golgi resident membrane proteins move subdiffusively in the endoplasmic reticulum and the Golgi apparatus in vivo and can be ruled out that the observed anomalous diffusion is a result of the complex topology of the membrane.
271 citations
TL;DR: Front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights is studied, replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay.
Abstract: The use of reaction-diffusion models rests on the key assumption that the diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand reactive systems in the presence of this type of non-Gaussian diffusion. Here we study front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay, x(-(1+alpha)). Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, x(-alpha), of the front's tail.
208 citations
TL;DR: It is shown that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.
Abstract: We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.
195 citations
TL;DR: In this article, the self-similarity and the independent-increments principles are used to extend the notion of diffusion process to the class of Levy-stable processes, which results in fractional-order partial space-time differential equations of diffusion.
Abstract: Stochastic principles for constructing the process of anomalous diffusion are considered, and corresponding models of random processes are reviewed. The self-similarity and the independent-increments principles are used to extend the notion of diffusion process to the class of Levy-stable processes. Replacing the independent-increments principle with the renewal principle allows us to take the next step in generalizing the notion of diffusion, which results in fractional-order partial space–time differential equations of diffusion. Fundamental solutions to these equations are represented in terms of stable laws, and their relationship to the fractality and memory of the medium is discussed. A new class of distributions, called fractional stable distributions, is introduced.
144 citations
TL;DR: In this paper, the authors point out a connection between anomalous transport in an optical lattice and Tsallis' generalized statistics and show that the momentum equation for the semiclassical Wigner function which describes atomic motion in the optical potential, belongs to a class of transport equations recently studied by Borland.
Abstract: We point out a connection between anomalous transport in an optical lattice and Tsallis' generalized statistics. Specifically, we show that the momentum equation for the semiclassical Wigner function which describes atomic motion in the optical potential, belongs to a class of transport equations recently studied by Borland [Phys. Lett. A 245, 67 (1998)]. The important property of these ordinary linear Fokker-Planck equations is that their stationary solutions are exactly given by Tsallis distributions. An analytical expression of the Tsallis index q in terms of the microscopic parameters of the quantum-optical problem is given and the spatial coherence of the atomic wave packets is discussed.
129 citations
TL;DR: It is observed, for the weak-binding case of 1-dodecylamine on mica, that anomalous diffusion occurs, leading to nearly fractal deposition patterns, in nanometer scale direct deposition processes utilizing dip-pen nanolithography (DPN).
Abstract: We report the first observation of anomalous diffusion in nanometer scale direct deposition processes utilizing dip-pen nanolithography (DPN). DPN permits quite general nanostructure patterns to be drawn on flat surfaces. Here we demonstrate experimentally, and discuss theoretically, the situation in which the molecular ink in DPN binds weakly to the surface. We observe, for the weak-binding case of 1-dodecylamine on mica, that anomalous diffusion occurs, leading to nearly fractal deposition patterns.
118 citations
01 Jan 2003
TL;DR: In this paper, a generalization of the Cauchy problem to a generalized diffusion process called fractional diffusion process is presented. But the authors do not consider the problem of anomalous diffusion in a continuous time random walk.
Abstract: A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α e (0, 2] and skewness θ (|θ| ≤ min {α, 2 - α}), and the first-order time derivative with a Caputo derivative of order β e (0, 1]. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.
115 citations
Book•
01 Jan 2003
TL;DR: In this article, the authors present a tutorial for the prediction of long-term memory time series using fractional diffusion processes and wavelet estimation for the Hurst Parameter in stable processes.
Abstract: Theory.- Prediction of Long-Memory Time Series: A Tutorial Review.- Fractional Brownian Motion and Fractional Gaussian Noise.- Scaling and Wavelets: An Introductory Walk.- Wavelet Estimation for the Hurst Parameter in Stable Processes.- From Stationarity to Self-similarity, and Back: Variations on the Lamperti Transformation.- Fractal Sums of Pulses and a Practical Challenge to the Distinction Between Local and Global Dependence.- Applications.- Supra-diffusion.- Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk.- First Passage Distributions for Long Memory Processes.- Non-Gaussian Statistics and Anomalous diffusion in Porous Media.- Directed Transport in AC-Driven Hamiltonian Systems.- Patterns and Correlations in Economic Phenomena Uncovered Using Concepts of Statistical Physics.- Semiparametric Modeling of Stochastic and Deterministic Trends and Fractional Stationarity.- Interaction Models for Common Long-Range Dependence in Asset Prices Volatility.- Long Memory and Economic Growth in the World Economy Since the 19th Century.- Correlations and Memory in Neurodynamical Systems.- Long Range Dependence in Human Sensorimotor Coordination.- Scaling and Criticality in Large-Scale Neuronal Activity.- Long-Range Dependence in Heartbeat Dynamics.- Multifractals: From Modeling to Control of Broadband Network Traffic.
114 citations
TL;DR: In this paper, the main problems associated with the determination and interpretation of molecular diffusion in zeolites are discussed, and it is shown that the diffusivities may most decisively depend on the relevant space and time scales of observation, as well as on the physical state under which the measurements are carried out.
Abstract: A review is given on the main problems associated with the determination and interpretation of molecular diffusion in zeolites. It is shown that the diffusivities may most decisively depend on the relevant space and time scales of observation, as well as on the physical state under which the measurements are carried out. Special emphasis is given to the microscopic techniques and their most recent evidence on the existence of transport resistances distributed over the intracrystalline space.
111 citations
TL;DR: It is shown that the essential thermodynamic properties of the heat channel can be obtained from the diffusion properties ofThe underlying particles, which move between two heat baths according to some dynamical process.
Abstract: We consider heat conduction in a 1D dynamical channel. The channel consists of an ensemble of noninteracting particles, which move between two heat baths according to some dynamical process. We show that the essential thermodynamic properties of the heat channel can be obtained from the diffusion properties of the underlying particles. Emphasis is put on the conduction under anomalous diffusion conditions.
83 citations
TL;DR: In this article, the evolution of vibrational wave packets built from the normal modes of cytochrome c, myoglobin and green fluorescent protein is investigated, and anomalous subdiffusion is characterized by an exponent, ν, that is related to the spectral dimension, d, and fractal dimension, D, of the protein.
Abstract: The evolution of vibrational wave packets built from the normal modes of cytochrome c, myoglobin and green fluorescent protein is investigated. Vibrational energy flow in these proteins is found to exhibit anomalous subdiffusion, a consequence of trapping of energy by spatially localized normal modes contained in the wave packet. Anomalous subdiffusion is characterized by an exponent, ν, that is related to the spectral dimension, d, and fractal dimension, D, of the protein. The dispersion relation describing variation of the protein’s normal mode frequencies with wave number is also characterized by an exponent, a, that is related to d and D. Values of the exponent, a, computed for the three proteins are consistent with the computed values for ν. The values of D obtained from ν, a, and d for each protein are the same within computational error, and close to the mass fractal dimension computed for each protein, all values falling in the range D=2.3±0.2. We find also that relaxation of the center of ener...
Book•
01 Jan 2003
TL;DR: In this article, anomalous diffusion on fractal networks is discussed. But the authors do not discuss the effect of spin waves in the Anderson transition on the dynamics of the fractal network.
Abstract: 1. Introduction.- 2. Fractals.- 3. Percolating Networks as Random Fractals.- 4. Multifractals.- 5. Anomalous Diffusion on Fractal Networks.- 6. Atomic Vibrations of Percolating Networks.- 7. Scaling Arguments for Dynamic Structure Factors.- 8. Spin Waves in Diluted Heisenberg Antiferromagnets.- 9. Anderson Transition.- 10. Multifractals in the Anderson Transition.- Appendices.- A. Multifractality of the HRN Model.- B. Spectral Dimensions for Deterministic Fractals.- B.1 Sierpinski Gasket.- B.2 Mandelbrot-Given Fractal.- C. Diffusion and Dynamics on Networks.- C.1 Atomic Vibrations.- C.2 Spin Waves in Diluted Ferro- and Antiferromagnets.- C.3 Superconducting Networks.- D. Wigner Distributions.- References.
TL;DR: In this article, a continuous time random walk model with long-tailed waiting time density that approaches a Gaussian distribution in the continuum limit is presented with long time tails and diffusion equations with a fractional time derivative are in general not asymptotically equivalent.
Abstract: A continuous time random walk model is presented with long-tailed waiting time density that approaches a Gaussian distribution in the continuum limit. This example shows that continuous time random walks with long time tails and diffusion equations with a fractional time derivative are in general not asymptotically equivalent.
TL;DR: In this paper, the authors studied the front propagation of reactive fields in systems whose diffusive behavior is anomalous (both superdiffusive and subdiffusive), and they showed that the probability distribution of the transport process follows the scaling relation given by the Flory argument even in the presence of super (or sub) diffusion.
TL;DR: In this article, a new set of distributions, called fractional stable distributions, is described, which arise as solutions to fractional-order partial differential equations that represent a generalization of the ordinary diffusion equation to the case of anomalous diffusion.
Abstract: A new set of distributions, called fractional stable distributions, is described. A subset of this set is represented by stable laws, while its particular case is the Gaussian distribution. These distributions arise as solutions to fractional-order partial differential equations that represent a generalization of the ordinary diffusion equation to the case of anomalous diffusion. The properties of multidimensional fractional stable densities are described, their expressions in terms of special functions are presented, and physical problems that lead to these densities are discussed.
TL;DR: In this article, the authors obtained new exact classes of solutions for the nonlinear fractional Fokker-Planck-like equation by considering a diffusion coefficient D = D |x| −θ (θ∈ R and D > 0) and a drift force F =−k 1 x+ k γ x|x| γ−1 (k 1, k ε,γ∈R ).
Abstract: We obtain new exact classes of solutions for the nonlinear fractional Fokker–Planck-like equation ∂tρ=∂x{D(x)∂μ−1xρν−F(x)ρ} by considering a diffusion coefficient D= D |x| −θ (θ∈ R and D >0) and a drift force F=−k 1 x+ k γ x|x| γ−1 (k 1 , k γ ,γ∈ R ) . Connection with nonextensive statistical mechanics based on Tsallis entropy is also discussed.
TL;DR: 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.
Abstract: 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 Levy 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.
TL;DR: Molecular-dynamics results on water confined in a silica pore are reviewed and discussed in connection with experiments performed on water in Vycor and with studies of water in contact with proteins.
Abstract: Molecular-dynamics results on water confined in a silica pore are reviewed and discussed in connection with experiments performed on water in Vycor and with studies of water in contact with proteins. The properties of confined water are studied as a function of both temperature and hydration level. The interaction of water in the film close to the substrate with the silica atoms induces a strong distortion of the hydrogen bond network. At high hydration levels a double dynamical regime is observed. At low hydration an anomalous diffusion is found upon supercooling with a transition from a Brownian to a non-Brownian regime on approaching the substrate in agreement with results found in studies of water in contact with globular proteins.
TL;DR: In this paper, a simple model for diffusion of a particle with initial velocity coupled to a non-Ohmic environment over a barrier is proposed, and an exact expression for the passing probability of the particle over a parabolic potential barrier is obtained.
Abstract: A simple model for diffusion of a particle with initial velocity coupled to a non-Ohmic environment over a barrier is proposed. An exact expression for the passing probability of the particle over a parabolic potential barrier is obtained. It is shown that in the case of subdiffusion, more than half of the particles cannot overcome the barrier, no matter what the initial velocity is. However, in the case of superdiffusion, a minimum critical velocity needed for passing over the barrier is found, which can lead to a strong friction and make the passing probability increase with kinetic energy, but still slower than in the case of normal diffusion. Effects of quantum diffusion and zero point fluctuation are also discussed. The model is applied to solve a long-standing problem in nuclear fusion reactions, though the mechanism found in this study can have a rather broad application.
TL;DR: Atomic force microscopy is used to detect and measure the amount of the solid phase in supported bilayers that contain coexisting fluid- and solid-phase lipids andSolid-phase domains in bilayers have been shown to a...
Abstract: Fluorescence photobleaching recovery (FPR) is commonly used to measure lipid and protein diffusion in cellular membranes. Typically, a model wherein diffusion is constant with time and the mean-squared displacement is directly proportional to time is used to analyze the results; however, in nonhomogeneous systems such as cellular membranes, anomalous subdiffusion may occur. In anomalous subdiffusion, the diffusion coefficient, D, decreases with time and thus the mean-squared displacement is proportional to some power of time less than 1. Although theory predicts that diffusion can be anomalous through protein interactions or obstruction, the complex composition of cellular membranes has made the actual origin and consequences of anomalous diffusion in phospholipid bilayers unclear. In this study, we use atomic force microscopy to detect and measure the amount of the solid phase in supported bilayers that contain coexisting fluid- and solid-phase lipids. Solid-phase domains in bilayers have been shown to a...
TL;DR: In this article, the theory of the standard field gradient pulse sequences for diffusion studies with the aid of NMR is developed, and the advantages and disadvantages are weighed up with respect to the experimental limits.
Abstract: Starting from basic differential equations for the description of ordinary and anomalous diffusion the theory of the standard field gradient pulse sequences for diffusion studies with the aid of NMR is developed. Anomalous diffusion can be due to an “obstruction effect” or to a “trapping effect” depending on the geometry and nature of the system matrix confining the diffusing particles. The NMR methods under consideration refer both to pulsed and steady field gradients, and both to laboratory and rotating-frame variants. The advantages and disadvantages are weighed up. More recent results refer to the attenuation of so-called multiple (or nonlinear) echoes by diffusion, and the use of internal field gradients for diffusion studies in fluid filled porous media. The diffusion of ordinary or laser-polarized gases is discussed for applications to porous media again. On a much longer time scale, isotope interdiffusion can be used as a technique for the elucidation of anomalous diffusion on percolation networks. A further and very distinct example of anomalous displacement characteristics was predicted for entangled polymers. Corresponding studies are reviewed and discussed with respect to the experimental limits. Finally, spin diffusion by flip-flop processes of dipolar coupled spins is shown to be a competitive mechanism if Brownian diffusion is strongly hindered by obstacles.
15 Mar 2003
TL;DR: In this article, the authors investigate the influence of geometrical configurations on the recovered diffusion coefficient and find that this parameter can vary by factors of at least 2-6 depending on alignment of sample and focus, and on membrane topography.
Abstract: Fluorescence correlation spectroscopy (FCS) is a widely used technique for the measurement of diffusion coefficients and concentrations. The standard deviation (SD) of FCS in solution is normally smaller than 1–5%, but it is often 50–100% or larger on cell membranes. This effect is usually attributed to the non-homogeneous structure of biological membranes that leads to variations of the diffusion coefficient at different membrane sites. Here we perform experiments and simulations to investigate the influence of geometrical configurations on the recovered diffusion coefficient. We find that this parameter can vary by factors of at least 2–6 depending on alignment of sample and focus, and on membrane topography. Misalignment and membrane topography increase the SD of membrane measurements and make it impossible to decide whether free diffusion or anomalous diffusion is measured. We propose a routine to optimally perform FCS measurements on membranes and to identify the source of the deviations.
TL;DR: The generalized trap model with parameter alpha is considered and it is obtained that the large scale effective model at low temperature does not depend on alpha in any dimension, so that the only observables sensitive to alpha are those that measure the "local persistence," such as the probability to remain exactly in the same trap during a time interval.
Abstract: We study in detail the dynamics of the one-dimensional symmetric trap model via a real-space renormalization procedure which becomes exact in the limit of zero temperature. In this limit, the diffusion front in each sample consists of two delta peaks, which are completely out of equilibrium with each other. The statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit T-->0. We first compute disorder averages of one-time observables, such as the diffusion front, the thermal width, the localization parameters, the two-particle correlation function, and the generating function of thermal cumulants of the position. We then study aging and subaging effects: our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two-time correlations. We also extend the real-space renormalization group method to include systematic corrections to the previous zero temperature procedure via a series expansion in T. We then consider the generalized trap model with parameter alpha in [0,1] and obtain that the large scale effective model at low temperature does not depend on alpha in any dimension, so that the only observables sensitive to alpha are those that measure the "local persistence," such as the probability to remain exactly in the same trap during a time interval. Finally, we extend our approach at a scaling level for the trap model in d=2 and obtain the two relevant time scales for aging properties.
TL;DR: In this paper, the mobility of water molecules confined in a silica pore is studied by computer simulation in the low hydration regime, where most of the molecules reside close to the hydrophilic substrate.
Abstract: The mobility of water molecules confined in a silica pore is studied by computer simulation in the low hydration regime, where most of the molecules reside close to the hydrophilic substrate. A layer analysis of the single particle dynamics of these molecules shows an anomalous diffusion with a sublinear behaviour over a long period. This behaviour is strictly connected to the long time decay of the residence time distribution analogous to water in contact with proteins.
TL;DR: In this paper, anomalous diffusion for one-dimensional systems described by a generalized Langevin equation is studied and it is shown that superdiffusive systems can be divided into two classes: normal and fast.
Abstract: We study anomalous diffusion for one-dimensional systems described by a generalized Langevin equation. We show that superdiffusive systems can be divided into two classes: normal and fast. For fast superdiffusion we prove that the Fluctuation-Dissipation Theorem does not hold. As a result, the system acquires an effective temperature. This effective temperature is a signature of metastability found in many complex systems such as spin-glass and granular material.
TL;DR: The approach provides a description of the structure of the thermal packet sample by sample, and a full statistical characterization of the important traps at a given order in micro, and explains how these results apply to the Sinai diffusion with bias by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models.
Abstract: We study the localization properties of the anomalous diffusion phase x approximately t(micro) with 0 0 and corresponds to the Golosov localization. For finite micro we thus generalize the usual real space renormalization method to allow for the spreading of the thermal packet over many renormalized valleys. Our construction allows one to compute exact series expansions in micro for all observables: to compute observables at order micro (n), it is sufficient to consider in each sample a spreading of the thermal packet onto at most (1+n) traps. So our approach provides a description of the structure of the thermal packet sample by sample, and a full statistical characterization of the important traps at a given order in micro. For the directed trap model, we show explicitly up to order micro(2) how to recover the exact expressions for the diffusion front, the thermal width, and the localization parameter Y2. We then use our method to derive exact results for the localization parameters Y(k) for arbitrary k, the correlation function of two particles, and the generating function of thermal cumulants. We then explain how these results apply to the Sinai diffusion with bias by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models. Finally we study the internal structure of the effective "traps" for the Sinai model via path-integral methods.
01 Jan 2003-Nuclear Instruments & Methods in Physics Research Section B-beam Interactions With Materials and Atoms
TL;DR: In this article, an anomalous diffusion (AD) equation in terms of fractional derivatives is used to describe cosmic ray propagation, and the energy dependence of spectral exponent of observed particles in different regimes is discussed.
Abstract: We consider the propagation of galactic cosmic rays under assumption that the interstellar medium is a fractal one. An anomalous diffusion (AD) equation in terms of fractional derivatives is used to describe cosmic ray propagation. The anomaly in the model results from large free paths (“Levy flights”) of particles between galactic inhomogeneities, and from the fact that a particle stays in a trap for a long time. Asymptotical solutions of the AD equation for point instantaneous and for impulse sources with inverse power spectrum relating to supernova bursts are found. They cover both subdiffusive and superdiffusive regimes, and are expressed in terms of the stable distributions. The energy dependence of spectral exponent of observed particles in the different regimes is discussed.
TL;DR: In this article, the reorientational and translational dynamics of the polymer, anion and cation species in polymer electrolyte systems consisting of a cross-linked poly(ethylene oxide−propylene oxide) random copolymer (poly(EO−PO)) doped with LiN(SO2CF3)2 at two different cross-linking densities were studied using multinuclear NMR relaxation and pulsed field gradient spin−echo (PGSE) diffusion measurements.
Abstract: The reorientational and translational dynamics of the polymer, anion and cation species in polymer electrolyte systems consisting of a cross-linked poly(ethylene oxide−propylene oxide) random copolymer (poly(EO−PO)) doped with LiN(SO2CF3)2 at two different cross-linking densities were studied using multinuclear NMR relaxation and pulsed field gradient spin−echo (PGSE) diffusion measurements. Analysis of the data provided exquisite molecular level insight into the individual behaviors of the species. In particular, it was found that in the presence of salt the polymer chains form hyperstructures and that the lithium ions undergo curvilinear diffusion along the chains within these hyperstructures. The anions, on the other hand, diffuse independently of both the polymer and lithium ions, but nevertheless undergo anomalous diffusion due to the effects of diffusing through the microheterogeneous polymer hyperstructure network. The motional behaviors of the three species were reasonably insensitive to the degre...
TL;DR: It is shown 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 systemwith the coefficient inside the derivatives.
Abstract: 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.
TL;DR: For space-time fractional diffusion equations a theory of discrete-space discrete-time random walks, analogous to the theory of continuous-timerandom walks, is presented.