Anomalous Diffusion

The subject of this paper is the evolution of Browman particles m disordered environments The "Ariadne's clew" we follow is understanding of the general statistical mechanisms which may generate "anomalous" (non-Brownlan) diffusion laws, this allows us to develop simple arguments to obtain a qualitative (but often qmte accurate) picture of most situations Several analytical techniques -such as the Green function formalism and renormahzatlon group methods-are also exposed Care is devoted to the problem of sample to sample fluctuations, particularly acute here We consider the specific effects of a bias on anomalous diffusion, and discuss the generahzat~ons of Einstein's relation m the presence of disorder An effort is made to illustrate the theoreucal models by describing many physical situations where anomalous d~ffuslon laws have been-or could be -observed (1 16) ~>2 ~GAU$SIAN J -P Bouchaud and A Georges, Anomalous d(fuston m dzsordered media 1 2 2 2 Polymer adsorption and self-avoiding Levy flights The structure of an adsorbed polymer is shown m fig 1 4 it Is made of pomts m &rect contact with an attractive wall, separated by large loops in the bulk The main point is that the *~ This is of course only a rough approximation, since complicated correlations between successive lumps exist in this deterministic motion They are, however, very likely to be short ranged and thus will not change the time dependence of X, (see section 1 3) A Comgho, M Daoud and H M DonsKer ana a varadhan, Lommun IJure Appl Math 32 (1979) 721 P Le Doussal, Thesis, Umv Pans 6 (1987), unpubhshed P Le Doussal, Phys Rev B 39 (1988) 881 P Le Doussal and J Machta, Phys Rev B 40 (1989) 9427 P Le Doussal, Phys Rev Lett 62 (1989) 3097 P LeDoussal, private communication, and m preparation P G Doyle and

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.

... LETTERS VOLUME 82 3 MAY 1999 NUMBER 18 Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach Ralf Metzler,1, * Eli Barkai,2, and Joseph Klafter1, 1 ... For the forcefree case, a subdiffusive behavior is recovered ...

The lateral mobility of lipids in phospholipid membranes has attracted numerous experimental and theoretical studies, inspired by the model of Singer and Nicholson (1972. Science, 175:720-731) and the theoretical description by Saffman and Delbrück (1975. Proc. Natl. Acad. Sci. USA. 72:3111-3113). Fluorescence recovery after photobleaching (FRAP) is used as the standard experimental technique for the study of lateral mobility, yielding an ensemble-averaged diffusion constant. Single-particle tracking (SPT) and the recently developed single-molecule imaging techniques now give access to data on individual displacements of molecules, which can be used for characterization of the mobility in a membrane. Here we present a new type of analysis for tracking data by making use of the probability distribution of square displacements. The potential of this new type of analysis is shown for single-molecule imaging, which was employed to follow the motion of individual fluorescence-labeled lipids in two systems: a fluid-supported phospholipid membrane and a solid polymerstabilized phospholipid monolayer. In the fluid membrane, a high-mobility component characterized by a diffusion constant of 4.4 microns2/s and a low-mobility component characterized by a diffusion constant of 0.07 micron2/s were identified. It is proposed that the latter characterizes the so-called immobile fraction often found in FRAP experiments. In the polymer-stabilized system, diffusion restricted to corrals of 140 nm was directly visualized. Both examples show the potentials of such detailed analysis in combination with single-molecule techniques: with minimal interference with the native structure, inhomogeneities of membrane mobility can be resolved with a spatial resolution of 100 nm, well below the diffraction limit.

We have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation. This model equation governs the evolution for the probability density function that describes anomalously diffusing particles. Anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur. The implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement. The accuracy of the scheme is O(Dx 2 ) in the spatial grid size and O(Dt 1 + c ) in the fractional time step, where 0 6 1 À c < 1 is the order of the fractional derivative and c = 1 is standard diffusion. We have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 < c 6 1.

The fractional di usion equation is solved for di erent boundary value problems, these being absorbing and re ecting boundaries in half-space and in a box. Thereby, the method of images and the Fourier-Laplace transformation technique are employed. The separation of variables is studied for a fractional di usion equation with a potential term, describing a generalisation of an escape problem through a uctuating bottleneck. The results lead to a further understanding of the fractional framework in the description of complex systems which exhibit anomalous di usion.

We measured individual trajectories of fluorescently labeled telomeres in the nucleus of eukaryotic cells in the time range of 10(-2)-10(4)sec by combining a few acquisition methods. At short times the motion is subdiffusive with r2 approximately talpha and it changes to normal diffusion at longer times. The short times diffusion may be explained by the reptation model and the transient diffusion is consistent with a model of telomeres that are subject to a local binding mechanism with a wide but finite distribution of waiting times. These findings have important biological implications with respect to the genome organization in the nucleus.

The viscoelastic properties of the cytoplasm of living yeast cells were investigated by studying the motion of lipid granules naturally occurring in the cytoplasm. A large frequency range of observation was obtained by a combination of video-based and laser-based tracking methods. At time scales from 10 ÿ4 to 10 2 s, the granules typically perform subdiffusive motion with characteristics different from previous measurements in living cells. This subdiffusive behavior is thought to be due to the presence of polymer networks and membranous structures in the cytoplasm. Consistent with this hypothesis, we observe that the motion becomes less subdiffusive upon actin disruption.

Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[À(bD)]-where D is the apparent diffusion coefficient (mm 2 /s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential modelexp[À(bD) a ], where a is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.

We review recent evidence illustrating the fundamental difference between cytoplasmic and test tube biochemical kinetics and thermodynamics, and showing the breakdown of the law of mass action and power-law approximation in in vivo conditions. Simulations of biochemical reactions in non-homogeneous media show that as a result of anomalous diffusion and mixing of the biochemical species, reactions follow a fractal-like kinetics. Consequently, the conventional equations for biochemical pathways fail to describe the reactions in in vivo conditions. We present a modification to fractal-like kinetics following the Zipf-Mandelbrot distribution which will enable the modelling and analysis of biochemical reactions occurring in crowded intracellular environments. r

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).

Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.

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 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 ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . Such evolution equation implies for the flux a fractional Fick's law which accounts for spatial and temporal non-locality. 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 that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

Cajal bodies (CBs) are nuclear organelles that contain factors required for splicing, ribosome biogenesis and transcription. Our previous analysis in living cells showed that CBs are dynamic structures. Here, we show that CB mobility is described by anomalous diffusion and that bodies alternate between association with chromatin and diffusion within the interchromatin space. CB mobility increases after ATP depletion and inhibition of transcription, suggesting that the association of CB and chromatin requires ATP and active transcription. This behaviour is fundamentally different from the ATP-dependent mobility observed for chromatin and suggests that a novel mechanism governs CB, and possibly other, nuclear body dynamics.

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.

Details about molecular membrane dynamics in living cells, such as lipid-protein interactions, are often hidden from the observer because of the limited spatial resolution of conventional far-field optical microscopy. The superior spatial resolution of stimulated emission depletion (STED) nanoscopy can provide new insights into this process. The application of fluorescence correlation spectroscopy (FCS) in focal spots continuously tuned down to 30 nm in diameter distinguishes between free and anomalous molecular diffusion due to, for example, transient binding of lipids to other membrane constituents, such as lipids and proteins. We compared STED-FCS data recorded on various fluorescent lipid analogs in the plasma membrane of living mammalian cells. Our results demonstrate details about the observed transient formation of molecular complexes. The diffusion characteristics of phosphoglycerolipids without hydroxyl-containing headgroups revealed weak interactions. The strongest interactions were observed with sphingolipid analogs, which showed cholesterol-assisted and cytoskeletondependent binding. The hydroxyl-containing headgroup of gangliosides, galactosylceramide, and phosphoinositol assisted binding, but in a much less cholesterol-and cytoskeleton-dependent manner. The observed anomalous diffusion indicates lipid-specific transient hydrogen bonding to other membrane molecules, such as proteins, and points to a distinct connectivity of the various lipids to other membrane constituents. This strong interaction is different from that responsible for forming cholesterol-dependent, liquid-ordered domains in model membranes.

To explore the character of transport in a plasma turbulence model with avalanche transport, the motion of tracer particles has been followed. Both the time evolution of the moments of the distribution function of the tracer particle radial positions, ͉͗r(t)Ϫr(0)͉ n ͘, and their finite scale Lyapunov number are used to determine the anomalous diffusion exponent,. The numerical results show that the transport mechanism is superdiffusive with an exponent close to 0.88Ϯ0.07. The distribution of the exit times of particles trapped into stochastic jets is also determined. These particles have the lowest separation rate at the low resonant surfaces.

Subdiffusion and its causes in both in vivo and in vitro lipid membranes have become the focus of recent research. We report apparent subdiffusion, observed via single particle tracking (SPT), in a homogeneous system that only allows normal diffusion (a DMPC monolayer in the fluid state). The apparent subdiffusion arises from slight errors in finding the actual particle position due to noise inherent in all experimental SPT systems. A model is presented that corrects this artifact, and predicts the time scales after which the effect becomes negligible. The techniques and results presented in this paper should be of use in all SPT experiments studying normal and anomalous diffusion.

A computer algorithm for the visualization of sample paths of anomalous diffusion processes is developed. It is based on the stochastic representation of the fractional Fokker-Planck equation describing anomalous diffusion in a nonconstant potential. Monte Carlo methods employing the introduced algorithm will surely provide tools for studying many relevant statistical characteristics of the fractional Fokker-Planck dynamics.

A mathematical method called subordination broadens the applicability of the classical advection-dispersion equation for contaminant transport. In this method the time variable is randomized to represent the operational time experienced by different particles. In a highly heterogeneous aquifer the operational time captures the fractal properties of the medium. This leads to a simple, parsimonious model of contaminant transport that exhibits many of the features (heavy tails, skewness, and non-Fickian growth rate) typically seen in real aquifers. We employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives. Applied to a onedimensional approximation of the MADE-2 data set, the model shows excellent agreement.

We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

Anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas. To model this transport mechanism, fractional order space derivative operators can be used. Here, the numerical properties of partial differential equations of fractional order a; 1 6 a 6 2, are studied. Two numerical schemes, an explicit and a semi-implicit one, are used in solving these equations. Two different discretization methods of the fractional derivative operator have also been used. The accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order.

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by distributed-order equations. In the present paper we consider different forms of distributed-order fractional kinetic equations and investigate the effects described by different classes of such equations. In

We present a general-purpose model for biomolecular simulations at the molecular level that incorporates stochasticity, spatial dependence, and volume exclusion, using diffusing and reacting particles with physical dimensions. To validate the model, we first established the formal relationship between the microscopic model parameters (time-step, move length, and reaction probabilities) and the macroscopic coefficients for diffusion and reaction rate. We then compared simulation results with Smoluchowski theory for diffusion-limited irreversible reactions and the best available approximation for diffusion-influenced reversible reactions. To simulate the volumetric effects of a crowded intracellular environment, we created a virtual cytoplasm composed of a heterogeneous population of particles diffusing at rates appropriate to their size. The particle size distribution was estimated from the relative abundance, mass and stoichiometries of protein complexes using an experimentally derived proteome catalogue from E. coli K12. Simulated diffusion constants exhibited anomalous behaviour as a function of time and crowding. Although significant, the volumetric impact of crowding on diffusion cannot fully account for retarded protein mobility in vivo, suggesting that other biophysical factors are at play. The simulated effect of crowding on barnase:barstar dimerization, an experimentally characterized example of a bimolecular association reaction, reveals a biphasic time course, indicating that crowding exerts different effects over different time scales. These observations illustrate that quantitative realism in biosimulation will depend to some extent on meso-scale phenomena that are not currently well understood.

The use of reaction-diffusion models rests on the key assumption that the underlying 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 the dynamics of reactive systems in the presence of this type of non-Gaussian diffusion. Here we present a study of 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, whose fundamental solutions are Levy α-stable distributions. Numerical simulation of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of fronts and a universal power law decay, x −α , of the tail, where α, the index of the Levy distribution, is the order of the fractional derivative.

1] We introduce a new conceptual model for longitudinal and transverse diffusion of moving bed particles under weak bed load transport. For both rolling/sliding and saltating modes the model suggests that the particle motion is diffusive and comprises at least three ranges of temporal and spatial scales with different diffusion regimes: (1) the local range (ballistic diffusion), (2) the intermediate range (normal or anomalous diffusion), and the global range (subdiffusion). The local range corresponds to ballistic particle trajectories between two successive collisions with the static bed particles. The intermediate range corresponds to particle trajectories between two successive periods of rest. These trajectories consist of many local trajectories and may include tens or hundreds of collisions with the bed. The global range of scales corresponds to particle trajectories consisting of many intermediate trajectories, just as intermediate trajectories consist of many local trajectories. Our data from the Balmoral Canal (the intermediate range) and Drake et al. 's [1988] data from the Duck Creek (the global range) provide strong support for this conceptual model and identify anomalous diffusion regimes for the intermediate range (superdiffusion) and the global range (subdiffusion).

... Originally described as the “battling” of (dust) particles seen against the sunlight in dark hallways of houses by Roman poet-philosopher Titus Lucretius Carus [5], re-discovered by Dutch physicist-physician Jan Ingenhousz [6] as the jittery motion of fine charcoal dust on an ...

The statistical properties of fast protein -water motions are analyzed by dynamic neutron scattering experiments. Using isotopic exchange, one probes either protein or water hydrogen displacements. A moment analysis of the scattering function in the time domain yields modelindependent information such as time-resolved mean square displacements and the Gauss-deviation. From the moments, one can reconstruct the displacement distribution. Hydration water displays two dynamical components, related to librational motions and anomalous diffusion along the protein surface. Rotational transitions of side chains, in particular of methyl groups, persist in the dehydrated and in the solventvitrified protein structure. The interaction with water induces further continuous protein motions on a small scale. Water acts as a plasticizer of displacements, which couple to functional processes such as open-closed transitions and ligand exchange. D

The superdiffusion behavior, i.e. < x 2 (t) >∼ t 2ν , with ν > 1/2, in general is not completely characherized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e. < |x(t)| q >∼ t qν(q) where ν(2) > 1/2 and qν(q) is not a linear function of q. This feature is different from the weak superdiffusion regime, i.e. ν(q) = const > 1/2, as in random shear flows. The strong anomalous diffusion can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in 2d time-dependent incompressible velocity fields, 2d symplectic maps and 1d intermittent maps. Typically the function qν(q) is piecewise linear. This corresponds to two mechanisms: a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions. In order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space. In the presence of the strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution, i.e. P (x, t) = t −ν F (x/t ν). This implies that the effective equation at large scale and long time for P (x, t), cannot obey neither the usual Fick equation nor other linear equations involving temporal and/or spatial fractional derivatives.

An analytical model for drillstring torque and drag is generated using a soft model. The soft model does not integrate all parameters affecting the drillstring behavior although some other researchers have taken the stiffness into account in the soft model. The torque and drag calculated by finite element method (FEM) is more accurate than the analytical model. The FEM can generate results that the analytical method cannot. This is because that the FEM takes both stiffness and some complicated boundary conditions into account. The program developed and presented in this paper can be used for torque and drag analysis in vertical, directional, horizontal, and other complicated wells under different drilling operational modes. Three examples on the calculation of torque and hookload are presented. The comparison between the analytical and numerical models was done, and the results were also compared with field data. The comparison shows that the result from FEM in one example matches the field data well but the analytical model doesn't get good results no matter how the friction coefficient is adjusted. The calculation of the contact force between the drillstring and the wellbore wall was conducted using the FEM. The calculated contact force estimates where the contact happens, as well as brings information about the location of possible drillstring sticking. The analytical model cannot do this, because the entire drillstring is in contact with the low side of the hole. The FEM program presented in this paper will enforce real-time analysis correctness of torque and drag together with the analytical model. The value of these torque and drag models could play an important role in real-time and planning of future drilling operations.

The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker-Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevintype equation, which is driven by a Lévy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker-Planck equation.

We present a numerical study of classical particles diffusing on a solid surface. The particles' motion is modeled by an underdamped Langevin equation with ordinary thermal noise. The particle-surface interaction is described by a periodic or a random two-dimensional potential. The model leads to a rich variety of different transport regimes, some of which correspond to anomalous diffusion such as has recently been observed in experiments and Monte Carlo simulations. We show that this anomalous behavior is controlled by the friction coefficient and stress that it emerges naturally in a system described by ordinary canonical Maxwell-Boltzmann statistics.

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0, 1) . The fundamental solution for the Cauchy problem is interpreted as a probability density of a selfsimilar non-Markovian stochastic process related to a phenomenon of subdiffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.

1] One way to study the mechanism of gravel bed load transport is to seed the bed with marked gravel tracer particles within a chosen patch and to follow the pattern of migration and dispersal of particles from this patch. In this study, we invoke the probabilistic Exner equation for sediment conservation of bed gravel, formulated in terms of the difference between the rate of entrainment of gravel into motion and the rate of deposition from motion. Assuming an active layer formulation, stochasticity in particle motion is introduced by considering the step length (distance traveled by a particle once entrained until it is deposited) as a random variable. For step lengths with a relatively thin (e.g., exponential) tail, the above formulation leads to the standard advection-diffusion equation for tracer dispersal. However, the complexity of rivers, characterized by a broad distribution of particle sizes and extreme flood events, can give rise to a heavy-tailed distribution of step lengths. This consideration leads to an anomalous advection-diffusion equation involving fractional derivatives. By identifying the probabilistic Exner equation as a forward Kolmogorov equation for the location of a randomly selected tracer particle, a stochastic model describing the temporal evolution of the relative concentrations is developed. The normal and anomalous advection-diffusion equations are revealed as its long-time asymptotic solution. Sample numerical results illustrate the large differences that can arise in predicted tracer concentrations under the normal and anomalous diffusion models. They highlight the need for intensive data collection efforts to aid the selection of the appropriate model in real rivers.

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Lévy process, we generate and display sample paths for some special cases.

In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the M-Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast types.

The best high-resolution records of climate over the past few hundred millennia are derived from ice cores retrieved from Greenland and Antarctica. The interpretation of these records relies on the assumption that the trace constituents used as proxies for past climate have undergone only modest post-depositional migration. Many of the constituents are soluble impurities found principally in unfrozen liquid that separates the grain boundaries in ice sheets. This phase behaviour, termed premelting, is characteristic of polycrystalline material. Here we show that premelting influences compositional diffusion in a manner that causes the advection of impurity anomalies towards warmer regions while maintaining their spatial integrity. Notwithstanding chemical reactions that might fix certain species against this prevailing transport, we find that-under conditions that resemble those encountered in the Eemian interglacial ice of central Greenland (from about 125,000 to 115,000 years ago)-...

The long-term limit motions of individual heavy-tailed (power-law) particle jumps that characterize anomalous diffusion may have different scaling rates in different directions. Operator stable motions [Y(t):t&gt; or =0] are limits of d-dimensional random jumps that are scale-invariant according to c(H)Y(t)=Y(ct), where H is a dxd matrix. The eigenvalues of the matrix have real parts 1/alpha(j), with each positive alpha(j)&lt; or =2. In each of the j principle directions, the random motion has a different Fickian or super-Fickian diffusion (dispersion) rate proportional to t(1/alpha(j)). These motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions. Subsets of the generalized fractional operator include (i) a fractional Laplacian with a single order alpha and a general directional mixing measure m(straight theta); and (ii) a fractional Laplacian with uniform mixing measure (the Riesz ...

To offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.

We demonstrate that the Fokker-Planck equation can be generalized into a 'Fractional Fokker-Planck' equation, i.e. an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions due to a Lévy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Lévy stable source to the classical gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non trivial fractional operator which corresponds to the possible asymmetry of the Lévy stable source. Both of them cannot be obtained by scaling arguments. The (mono-) scaling behaviors of the Fractional Fokker-Planck equation and of its solutions are analysed and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward physical interpretation of the parameters of Lévy stable distributions. Furthermore, with the help of important examples, we show the applicability of the Fractional Fokker-Planck equation in physics.

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 spacetime 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 ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (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.

In this paper, we briefly review the progress made in the mathematical modeling of biofilms over the last 30 years. Biofilms constitute a spectrum of dynamical microorganisms, whose interaction with the surrounding environment and thereby induced dynamics dictates the complex properties of the living organism. Modeling of biofilms began with a low dimensional continuum description first based on kinematics and translational diffusions; later, more sophisticated microscopic dynamical mechanisms are introduced leading to the anomalous diffusion and dissipation encountered by various components in biofilms. Recently, biofilm and bulk fluid (or solvent) coupling has been investigated using discretecontinuum, multifluid and single fluid multicomponent models to treat the entire biofilm-bulk-fluid system either as a system consisting of various components whose dynamics exhibits different time scale or as a whole. We classify the models into roughly four classes: low-dimensional continuum models, diffusion limited aggregation models, continuum-discrete models, and fully coupled biofilm-fluid models. We will address some hybrid models that combine the ideas from the above categories and new computational protocols combining the existing computational tools for cell dynamics coupled with the discrete-continuum biofilm model.

We have studied the self-diffusion properties of butyl-methyl-imidazolium bis(trifluoromethylsulfonyl)-imide ([BMIM][TFSI]) + water system. The self-diffusion coefficients of cations, anions, and water molecules were determined by pulsed field gradient NMR. These measures were performed with increased water quantity up to saturation (from 0.3 to 30 mol %). Unexpected variations have been observed. The self-diffusion coefficient of every species increases with the quantity of water but not in the same order of magnitude. Whereas very similar evolutions are observed for the anion and cation, the increase is 25 times greater for water molecules. We interpret our data by the existence of phase separation at microscopic scale.