Derivatives fractional diffusion equations

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

The fact that the temporal occurrence of the motion events performed by the random walker is so broadly distributed that no characteristic waiting time exists has been exploited by a number of investigators [7,19,31] in order to generalize the various diffusion equations of Brownian dynamics to explain anomalous relaxation phenomena. The resulting diffusion equations are called fractional diffusion equations because in general they will involve fractional derivatives of the probability density with respect to the time. For example, in fractional noninertial diffusion in a potential, the diffusion Eq. (5) becomes [7,31]... 

The idea of introducing fractional derivatives in diffusion equations to account for long-range memory effects has given rise to a large number of publications. In this context, the work by Metzler, Glockle, and Nonnenmacher is noteworthy [297]. They realized that (6.10) suffers from a pathological defect For d/ = 1 and dy, = 2 one should recover the standard diffusion equation however, this is not the case. To overcome this difficulty they propose... 

With the help of fractional calculus, Dassas and Duby123 have worked on the problem of diffusion towards the fractal interfaces. They have proposed the following generalized diffusion equation involving a fractional derivative operator ... 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

Before closing this chapter we would like to mention briefly a novel consideration of diffusion based on the recently developed concepts of fractional kinetics [29]. From our previous discussion it is apparent that if ds < 2, diffusion is recurrent. This means that diffusion follows an anomalous pattern described by (2.10) the mean squared displacement grows as (z2 (t)) oc t1 with the exponent 7 1. To deal with this, a consistent generalization of the diffusion equation (2.18) could have a fractional-order temporal derivative such as... 

This equation describes the change of the molar concentration of A with respect to time t at a fixed point inside the pellet pores, this change resulting from the motion of A. Since the pores occupy only a fraction of the pellet volume and the flux JA relates to the unit area of the pellet, the porosity ep of the pellet is accounted for in Equation 5.15. The derivation of the Equation 5.15 makes it essential that neither chemical reaction or adsorption of A occurs. When Equation 5.1 for the flux is inserted into Equation 5.15, we get, for constant diffusivity,... 

Diffusion Equation with Fractional Derivatives. In normal as well as in anomalous diffusion the quantity... 

Then the diffusion equation with fractional derivatives allowing for inertial effects is... 

The second model of Debye or the Debye-Frohlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole-Cole equation arises naturally from the solution of a fractional Fokker-Planck equation in the configuration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole-Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24]... 

These investigations have been almost entirely based on fractional diffusion and Fokker-Planck equations with a fractional Riesz derivative and have turned out to be a convenient basis for mathematical manipulations, while at the same time being easy to interpret in the context of a dynamical approach. 

In an analysis of the diffusion equations in magnetic fields, Dumarque et al. introduced a diffusivity tensor and an effectiveness parameter related to the difference between diffusivity in the presence and in the absence of a magnetic field. In a less general, but somewhat more amenable approach to numerical estimations, Lielmezs and Musbally derived a magnetic correction factor called the arithmetic mean average fractional integral diffusion coefficient... 

The apparent diffusion coefficient, Da in Eq. (11) is a mole fraction-weighted average of the probe diffusion coefficient in the continuous phase and the microemulsion (or micelle) diffusion coefficient. It replaces D in the current-concentration relationships where total probe concentration is used. Both the zero-kinetics and fast-kinetics expressions require knowledge of the partition coefficient and the continuous-phase diffusion coefficient for the probe. Texter et al. [57] showed that finite exchange kinetics for electroactive probes results in zero-kinetics estimates of partitioning equilibrium constants that are lower bounds to the actual equilibrium constants. The fast-kinetics limit and Eq. (11) have generally been considered as a consequence of a local equilibrium assumption. This use is more or less axiomatic, since existing analytical derivations of effective diffusion coefficients from reaction-diffusion equations are approximate. 

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