Anomalous Diffusion

Background and the purpose of the study: This study reports the laboratory optimization for the preparation of salbutamol sulphate-ethylcellulose microparticles by a non-solvent addition coacervation technique through adjustment of the ratio of salbutamol sulphate to ethylcellulose. The variation of drug release between the microparticles and tabletted microparticles was also investigated. Methods: In vitro release profiles of developed microparticles and tabletted microparticles were studied using USP XXIV dissolution apparatus I and II, respectively, in 450 ml double distilled water at 50 rpm maintained at 37°C. Results: White microparticles with no definite shape having good entrapment efficiency (96.68 to 97.83%) and production yield (97.48 ± 1.21 to 98.35 ± 1.08%) were obtained. In this investigation, initial burst effect was observed in the drug release behavior. The rate of drug release from microparticles decreased as the concentration of polyisobutylene was increased from 6...

The objective of the present study was to develop a once-daily sustained-release (SR) matrix tablet of famotidine. Nine different formulations (F1–F9) were prepared by direct compression method using Avicel PH101 as filler/binder in the range of 41–27% in F1–F3, 18–22% in F4–F7, and 16–18% in F8–F9 and hydroxypropyl methylcellulose (4,000 cps) as hydrophilic matrix was used in F1–F3 from 19% to 30%, around 40% in F4–F7, and 42–45% in F8–F9. Talc and Aerosil were added in the ratio of 0.7–1.2%. The tablets were subjected to various physical parameters including weight variation test, hardness, thickness, diameter, friability, and in vitro release studies. Assay was also performed according to the USP 30 NF 25 procedure. The results of the physical parameters and assay were found to be within the acceptable range. In vitro dissolution results indicated that formulation F4–F7, having around 40% of rate control polymer, produced a SR pattern throughout 24 h. F1–F3 showed drug release at a faster rate, while F8–F9 released much slower, i.e., r 2 = 0.984) and F6 in Korsmeyer and Higuchi (r 2 = 0.992 and 0.988). The parameter n indicated anomalous diffusion, while β in Weibull showed a parabolic curve with higher initial slope. The f 2 similarity test was performed taking F4 as a reference formulation. Only the F5–F7 formulations were similar to the reference formulation F4. The mean dissolution time was around 10 h for the successful formulation.

This paper examines the properties of a fractional diffusion equation defined by the composition of the inverses of the Riesz potential and the Bessel potential. The first part determines the conditions under which the Green function of this equation is the transition probability density function of a Lévy motion. This Lévy motion is obtained by the subordination of Brownian motion, and the Lévy representation of the subordinator is determined. The second part studies the semigroup formed by the Green function of the fractional diffusion equation. Applications of these results to certain evolution equations is considered. Some results on the numerical solution of the fractional diffusion equation are also provided.

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker-Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov-Feller equation for discontinuous Markovian processes, and the fractional Fokker-Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

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.

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.

Boron is implanted in crystalline silicon through oxide layers with different thicknesses. The implantation is carried out at various doses and energies of interest in ultra large scale integration (ULSI) application. Rapid thermal annealings (RTA) are used to obtain shallow junctions and electrical activation of the B atoms. However, transient enhanced diffusion induced by implantation damage can be observed. The boron concentration profiles before and after annealing are obtained with secondary ion mass spectrometry (SIMS). It is found that the diffusion transient in the tail region of the boron profile increases with decreasing oxide thickness. Even more, if the implantation damage concerns mostly the oxide, i.e. when the concentration peak is located in this oxide, the oxygen knocked into the silicon substrate could play this way an important role in restricting the boron diffusion, which is good to obtain very shallow junctions. On the other hand, for thinner oxide, boron enhanced diffusion is attributed to the implantation induced damage into silicon at high doses. The diffusion process of boron in oxide and monocrystalline silicon during rapid thermal annealing is investigated. The boron diffusion profiles obtained by computer simulation are compared with the measured results. It is shown by this comparison that the intrinsic coefficient cannot be considered as constant along all the silicon depth.

The experimental investigation reported in this paper focuses on the effect of induced implantation damage on the boron diffusion process. Boron is implanted at various fluences and energies in Cz-(100) silicon through different oxide layer thicknesses. Rapid thermal annealing (RTA) is used to activate shallow p + layers (0"1-0"15/am) following boron implantation. Concentrations versus depth boron profiles are measured using a secondary ion mass spectrometry (SIMS) analyser. An enhanced boron diffusion is detected in the tail region when the oxide thickness decreases. If the concentration peak is located at the oxide/silicon interface or in the substrate, further implantation damage is generated. This observed enhanced boron diffusion is thus attributed to the implantation-induced damage. The point defects, which act as a driving force for this enhanced boron diffusion at different annealing stages, are detected by the deep level transient spectroscopy (DLTS) technique. In particular, the effect of knocked-on oxygen during the implantation step on the generation of deep centres and defects is investigated. Finally, all the results reveal that DLTS coupled to SIMS ~,=lysis provides an eflident method with ~which to identify the origin and the nature of implanted and RTA related defects.

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.

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, x 2 (t) ∝ t, while anomalous behavior is expected to show a different time dependence, x 2 (t) ∝ t δ with δ < 1 for subdiffusive and δ > 1 for superdiffusive motions. Here we demonstrate that this kind of qualification, if applied straightforwardly, may be misleading: There are anomalous transport motions revealing perfectly "normal" diffusive character ( x 2 (t) ∝ t), yet being non-Markov and non-Gaussian in nature. We use recently developed framework [1, Phys. Rev. E 75, 056702 ] of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of paradoxical diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).

... 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 ...

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 characteristic even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytucal expressions for the velocity correlation functions. Knowledge of the results presented here are expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.

In this paper, we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation. We investigate properties of a coupled continuous-time random walk that follows from modeling the occurrence of jumps with compound counting processes. As a result, two different diffusion processes corresponding to over-and undershooting operational times, respectively, have been found. We show that within the proposed framework, all empirical two-power-law relaxation patterns may be derived. This work is motivated by the so-called "less typical" relaxation behavior observed, e.g., for gallium-doped Cd 0.99 Mn 0.01 Te mixed crystals.

To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present 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.

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 study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up off self-similar with stationary increments processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and beta=1, and time-fractional diffusion stochastic processes when alpha=beta in (0,1). The latters have marginal probability density function governed by time-fractional diffusion equations of order beta. The ggBm is defined through the explicit construction of the underline probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the M-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the $H$-{\bf sssi} nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differenital equation of fractional type.

We propose a method for determining the solution and source term of a generalized timefractional diffusion equation. The method is based on selecting a bi-orthogonal basis of L 2 space corresponding to a nonself-adjoint boundary value problem. Uniqueness is proven and an existence result is obtained for smooth initial and final conditions. The asymptotic behavior of the generalized Mittag-Leffler function is used to relax the smoothness requirement on these conditions.

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. Integro-Difference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;alpha&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/=2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.

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.

We propose diffusionlike 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, cannot 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 their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating sup...

Complex systems such as glasses, gels, granular materials, and systems far from equilibrium exhibit violation of the ergodic hypothesis (EH) and of the fluctuation-dissipation theorem (FDT). Recent investigations in systems with memory [1] have established a hierarchical connection between mixing, the EH and the FDT. They have shown that a failure of the mixing condition (MC) will lead to the subsequent failures of the EH and of the FDT. Another important point is that such violations are not limited to complex systems: simple systems may also display this feature. Results from such systems are analytical and obviously easier to understand than those obtained in complex structures, where a large number of competing phenomena are present. In this work, we review some important requirements for the validity of the FDT and its connection with mixing, the EH and anomalous diffusion in one-dimensional systems. We show that when the FDT fails, an out-ofequilibrium system relaxes to an effective temperature different from that of the heat reservoir. This effective temperature is a signature of metastability found in many complex systems such as spin-glasses and granular materials.

An intermittent nonlinear map generating subdiffusion is investigated. Computer simulations show that the generalized diffusion coefficient of this map has a fractal, discontinuous dependence on control parameters. An amended continuous time random walk theory well approximates the coarse behavior of this quantity in terms of a continuous function. This theory also reproduces a full suppression of the strength of diffusion, which occurs at the dynamical transition from normal to anomalous diffusion. Similarly, the probability density function of this map exhibits a nontrivial fine structure while its coarse functional form is governed by a time fractional diffusion equation. A more detailed understanding of the irregular structure of the generalized diffusion coefficient is provided by an anomalous Taylor-Green-Kubo formula establishing a relation to de Rham-type fractal functions.

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.

Analysing the quasi-elastic neutron scattering from 8 at. % hydrogen in the metallic glass NiUZq6, we find that the hydrogen motion can be described in terms of anomalous diffusion. This process is analogous to that which leads to a strong frequency dependence of the conductivity in fast ionic conducting glasses. The data can be well described by an effective medium calculation based on a model with a broad distribution of activation energies. The same model can also successfully be applied to describe the anomalous temperature dependence of proton spin relaxation in amorphous metals.

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.

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.

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] Accurate and efficient simulation of contaminant transport in fractured rock is practically important, yet current approaches suffer from numerical constraints that limit the full inclusion of fracture network properties at the regional scale. We propose a multidimensional transport model, nonlocal in space and time, which describes complex transport behavior in fractured media at regional scales, without introducing the computational burden of explicitly incorporating individual rock fractures. The model utilizes a direction-dependent tempered stable process to describe the transition of multiscaling anomalous diffusion to asymptotic diffusion limits. Applications show that the model can capture efficiently the transient superdiffusion found in two-dimensional synthetic fracture networks, and suggest that the truncation of leading plume edges is related to fracture network properties. Citation: Zhang, Y., B. Baeumer, and D. M. Reeves (2010), A tempered multiscaling stable model to simulate transport in regional-scale fractured media, Geophys.

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).

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.

The linear dielectric response of an assembly of noninteracting linear ͑needlelike͒ dipole molecules ͑each of which is free to rotate in space͒ is evaluated in the context of fractional dynamics. The infinite hierarchy of differential-recurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional Fokker-Planck ͑fractional Klein-Kramers͒ equation. On solving this hierarchy in terms of continued fractions ͑as in the normal rotational diffusion͒, the complex dynamic susceptibility is obtained and is calculated for typical values of the model parameters. It is shown that the model can reproduce nonexponential anomalous dielectric relaxation behavior at low frequencies ͑р1, where is the Debye relaxation time͒ and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies ͑in the far infrared region͒ so that Gordon's sum rule for integral dipolar absorption is satisfied.

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Oceanic turbulence has been considered for a while as one of the main sources of the heterogeneity of the phytoplankton field over a wide range of scales. However, it is only recently that the intermittency of turbulence has been taken into account, although it is rather indispensable in the explanation of the observed patchiness of the plankton field. In order to improve the understanding and characterization of the diffusion of copepods within an heterogeneous phytoplankton field, we developed a model based on particle diffusion in a multifractal field. After discussing this model with some detail, we used it and the corresponding numerical simulations in order to investigate the fundamental question: does the strong heterogeneity of the phytoplankton generate a anomalous diffusion of the copepods? Although a positive answer is obtained in a rather straightforward manner for a one-dimensional multifractal field of phytoplankton concentration, the answer is Ž. rather more involved for a greater topological dimension of the field, contrary to some previous claims.

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(&#39;)tau(gamma(&#39;))[ partial differential (gamma(&#39;))rho(x,t)]/ partial differential t(gamma(&#39;))= integral dmu(&#39;)dnu&#39;D(mu(&#39;),nu(&#39;))[ partial differential (mu(&#39;))[rho(x,t)](nu(&#39;))]/ partial differential x(mu(&#39;)), where tau(gamma(&#39;)) and D(mu(&#39;),nu(&#39;)) are kernels to be chosen; the choice tau(gamma(&#39;))=delta(gamma(&#39;)-1) and D(mu(&#39;),nu(&#39;))=delta(mu(&#39;)-2)delta(nu(&#39;)-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 mecha...

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(&#39;)tau(gamma(&#39;))[ partial differential (gamma(&#39;))rho(x,t)]/ partial differential t(gamma(&#39;))= integral dmu(&#39;)dnu&#39;D(mu(&#39;),nu(&#39;))[ partial differential (mu(&#39;))[rho(x,t)](nu(&#39;))]/ partial differential x(mu(&#39;)), where tau(gamma(&#39;)) and D(mu(&#39;),nu(&#39;)) are kernels to be chosen; the choice tau(gamma(&#39;))=delta(gamma(&#39;)-1) and D(mu(&#39;),nu(&#39;))=delta(mu(&#39;)-2)delta(nu(&#39;)-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 mecha...

Experimental data are reported on moisture diffusion and the elastoplastic response of an intercalated nanocomposite with vinyl ester resin matrix and montmorillonite clay filler at room temperature. Observations in diffusion tests show that water transport in the neat resin is Fickian, whereas it becomes anomalous (non-Fickian) with the growth of the clay content. This transition is attributed to immobilization of penetrant molecules on the surfaces of hydrophilic clay layers. Observations in uniaxial tensile tests demonstrate that the response of vinyl ester resin is strongly elastoplastic, whereas an increase in the clay content results in a severe decrease of plastic strains observed as a noticeable reduction of curvatures of the stress-strain diagrams. This is explained by slowing down of molecular mobility in the host matrix driven by confinement of chains in galleries between platelets. Constitutive equations are developed for the anomalous moisture diffusion through and the elastoplastic behavior of a nanocomposite. Adjustable parameters in these relations are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. A striking similarity is revealed between changes in diffusivity, ultimate water uptake and the rate of plastic flow with an increase in the clay content.

Nonmagnetic particles in suspension in a ferrofluid act as magnetic holes when an external magnetic field is exerted: They acquire an effective dipolar moment opposing the surrounding one, which induces dipolar magnetic interactions. For a large enough imposed field and particle size, the induced interactions dominate the thermal forces, dipolar chains form. At equilibrium, these chains fluctuate under the effect of brownian noise. When the imposed magnetic field is suddenly decreased, these chains roughen dynamically. We study the time and size scaling of these fluctuations and roughening, and the relationships between the equilibrium and out of equilibrium behavior. We compare the experimental data both to Brownian dynamics simulations, and to a simple theory of semiflexible polymer chains, a generalization of the Rouse model. The scaling behavior of the experiments agree with the predictions of both theory and simulations over 5 orders of magnitudes. The roughening follows three successive regimes: The root mean square width of the chain initially evolves as W ϳ t 1/2 , then it enters a subdiffusive regime where W ϳ t 1/4 and eventually it saturates to a level W ϳ N, where N is the number of particles in the chain. The exact prefactors as a function of the applied field, particle diameter, and temperature are also derived analytically. We also show that this phenomenon can be described equivalently as a non-Markovian diffusion process for a particle in an environment with memory effects. Within this framework, our system is shown to confirm the predictions of theories for anomalous diffusion in systems with memory.

The foundations of the fractional diffusion equation are investigated based on coupled and decoupled continuous time random walks (CTRW). For this aim we find an exact solution of the decoupled CTRW, in terms of an infinite sum of stable probability densities. This exact solution is then used to understand the meaning and domain of validity of the fractional diffusion equation. An interesting behavior is discussed for coupled memories (i.e., L\'evy walks). The moments of the random walk exhibit strong anomalous diffusion, indicating (in a naive way) the breakdown of simple scaling behavior and hence of the fractional approximation. Still the Green function $P(x,t)$ is described well by the fractional diffusion equation, in the long time limit.