Generalized diffusion equation,

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

Eq. (2.39) may be considered as the generalized diffusion equation in angular space with a diffusion coefficient that varies in time as... 

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. 

Over the last four decades or so, transport phenomena research has benefited from the substantial efforts made to replace empiricism by fundamental knowledge based on computer simulations and theoretical modeling of transport phenomena. These efforts were spurred on by the publication in 1960 by Bird et al. (6) of the first edition of their quintessential monograph on the interrelationships among the three fundamental types of transport phenomena mass transport, energy transport, and momentum transport. All transport phenomena follow the same pattern in accordance with the generalized diffusion equation (GDE). The unidimensional flux, or overall transport rate per unit area in one direction, is expressed as a system property multiplied by a gradient (5)... 

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

As a basic model of a set of equation derived above, the generalized diffusion equation with non-linear source could be used... 

Strictly speaking, equations for the joint densities of similar particles have to be solved with the boundary condition (5.1.40) imposed at the coordinate origin. However, the singular terms S(r — r ) with r — 0 have to modify it. To illustrate this point, let us consider the generalized diffusion equation with the singular term... 

With regard to Eq. (4.2), the general diffusion equation describing three standard geometries (slab sheet, cylinder, and sphere) is... 

In this subsection we illustrate the attempt made in Ref. 59 to derive a generalized diffusion equation from the Liouville approach described in Section III. We are addressing the apparently simple problem of establishing a density equation corresponding to the simple diffusion equation... 

Notice that this leads us to the generalized diffusion equation... 

In the case where the correlation function <3> (f) has the form of Eq. (148), with p fitting the condition 2 < p < 3, the generalized diffusion equation is irreducibly non-Markovian, thereby precluding any procedure to establish a Markov condition, which would be foreign to its nature. The source of this fundamental difficulty is that the density method converts the infinite memory of a non-Poisson renewal process into a different type of memory. The former type of memory is compatible with the occurrence of critical events resetting to zero the systems memory. The second type of memory, on the contrary, implies that the single trajectories, if they exist, are determined by their initial conditions. 

It is now well understood that this is not an approximation rather it is a way to force an equation with infinite memory to become compatible with Levy diffusion. The assumption (152) makes it possible for us to get rid of the time convolution nature of the generalized diffusion equation (133). At the same time, this key relation replaces the correlation function (t) with its second-order derivative and, as a consequence of Eq. (147), with the waiting time distribution /( ). This fact is very important. In fact, any Liouville-like approach makes the correlation function 3F(f) enter into play. The CTRW is a perspective resting on trajectories and consequently on /(f). Establishing a connection between the two pictures implies the conversion of 4> (f) into j/(r), or vice versa. Here, this conversion has been realized paying the price of altering the physics of the generalized diffusion equation (133). 

This is a plausible way to prove that Eq. (162) is the diffusion equation that applies to the gaussian condition. It is important to point out that a more satisfactory derivation of this exact result can be obtained by using the Zwanzig projection approach of Section III [67,68]. Thus, Eq. (162) as well as Eq. (133) must be considered as generalized diffusion equations compatible with a Liouville origin. 

In Section IV we have shown that the CTRW can be thought of as being the solution of an equation of motion with the same time convoluted structure as the GME derived from a Liouville or Liouville-like approach by means of a contraction on the irrelevant degrees of freedom. However, this formal equivalence may not imply an equivalence of physical meanings. This review affords the information necessary to establish this important point. Let us consider, for instance, the diffusion process dx/dt = widely discussed in this chapter. From within the density method treatment, this equation has been proven to generate two possible generalized diffusion equations, which are written here again for the reader s convenience. 

The first generalized diffusion equation is described by Eq. (133), which reads... 

The CTRW approach generates a third generalized diffusion equation, whose form is... 

Let us consider now the case p < 2. In this case the third generalized diffusion equation, derived from a CTRW, is well established [43] as a paradigmatic case of subdiffusion. The correlation function of the... 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

Equations (6.305) and (6.306) are different because of the drift term V (PV D), which is sometimes called a spurious drift term. These diffusion equations have different equilibrium distributions and are two special cases of a more general diffusion equation. 

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