Diffusion equation, interpretation

Torkar et al. [702,706—708] identified nucleation as an autocatalytic process at the (hk0) planes of hexagonal platelets of NaN3. The decelera-tory reaction fitted the first-order equation [eqn. (15)]. Values of E tended to be irreproducible for the pure salt E was about 180 kJ mole 1 but this was reduced to about half by doping. This influence of an additive and the observed similarities in magnitudes of E for decomposition and for diffusion were interpreted as indicating that growth of nuclei was controlled by a diffusion process. 

With the advent of picosecond-pulse radiolysis and laser technologies, it has been possible to study geminate-ion recombination (Jonah et al, 1979 Sauer and Jonah, 1980 Tagawa et al 1982a, b) and subsequently electron-ion recombination (Katsumura et al, 1982 Tagawa et al, 1983 Jonah, 1983) in hydrocarbon liquids. Using cyclohexane solutions of 9,10-diphenylanthracene (DPA) and p-terphenyl (PT), Jonah et al. (1979) observed light emission from the first excited state of the solutes, interpreted in terms of solute cation-anion recombination. In the early work of Sauer and Jonah (1980), the kinetics of solute excited state formation was studied in cyclohexane solutions of DPA and PT, and some inconsistency with respect to the solution of the diffusion equation was noted.1... 

A mathematically simple case, that occurs frequently in solvent extraction systems, in which the extracting reagent exhibits very low water solubility and is strongly adsorbed at the liquid interface, is illustrated. Even here, the interpretation of experimental extraction kinetic data occurring in a mixed extraction regime usually requires detailed information on the boundary conditions of the diffusion equations (i.e., on the rate at which the chemical species appear and disappear at the interface). 

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

Neither the Ito nor the Stratonovich interpretation of an SDE leads naturally to a term of this form. The Ito interpretation yields a diffusion equation of the form given in Eq. (2.222), in which the diffusivity instead appears inside two derivatives, while the Stratonovich interpretation yields Eq. (2.255), in which is decomposed into two factors of B, one of which appears inside both derivatives and the other between them. 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

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. 

Exercise. The equation (2.4) for the probability of one particle may be interpreted as the diffusion equation for independent particles. Show that the equilibrium distribution and the diffusion constant agree with the expressions that could have been deduced directly from the model, without the intervening calculations. 

Geometrical Interpretation of the Diffusion Equation when Diffusivity is Constant... 

Because of the assumed dual sorption mechanism present in glassy polymers, the explicit form of the time dependent diffusion equation in these polymers is much more complex than that for rubbery polymers (82-86). As a result exact analytical solutions for this equation can be found only in limiting cases (84,85,87). In all other cases numerical methods must be used to correlate the experimental results with theoretical estimates. Often the numerical procedures require a set of starting values for the parameters of the model. Usually these values are shroud guessed in a range where they are expected to lie for the particular penetrant polymer system. Starting from this set of arbitrary parameters, the numerical procedure adjusts the values until the best fit with the experimental data is obtained. The problem which may arise in such a procedure (88), is that the numerical procedures may lead to excellent fits with the experimental data for quite different starting sets of parameters. Of course the physical interpretation of such a result is difficult. 

The conditions basic to the Fickian sorption were that (1) D is a function of ct only and (2) a constant surface concentration is maintained during sorption. So long as we wishes to retain the Fick diffusion equation as the basis of the discussion, any attempt for the theoretical interpretation of non-Fickian characteristics must abandon either or both of these conditions. In this section we give a brief account of a theory which involves an alternation of condition (1). It is due originally to Crank and Park (1951). 

This particular solution of the rotational diffusion equation can be interpreted as the transition probability that is, the probability density for a rod to have orientation u at time t, given that it had an orientation Uo initially. 

Aller (1984) created a mechanistic model for the multi-dimensional transport of dissolved pore-water species by animals. He observed that ammonia profiles caused by sulfate reduction in the top-ten-centimeter layer of Long Island Sound sediments could not be interpreted by onedimensional diffusion (Equation (3)). The multidimensional effects of irrigation were reproduced mathematically by characterizing the top layer of... 

Reeburgh (1980) suggested that the pore-water distributions of SO and CH4 indicate that CH4 is being oxidized anaerobically with SO4 being the electron acceptor. This suggestion, which is virtually unavoidable based on the metabolite distributions and interpretation by diffusion equations (see also Murray et al., 1978), was not... 

Soonawala (1976) used a computer-adapted finite difference method to solve the diffusion equation for Rn in two and three dimensions. Using theoretical and laboratory studies he was able to explain satisfactorily field emanation data from the Eldorado area of Saskatchewan and the Kaipokok Bay area of Labrador with the diffusion model of Rn transport. The results shown in Table ll-Xlll are taken from Novikov and Kapkov (1965). To interpret them the reader must visualise an inactive soil layer of thickness h... 

Finally, we present an interpretation of our observations in terms of diffusion paths. Basically, the diffusion equations are solved for the case of two semi-infinite phases brought into contact under conditions where there is no convection and no interfacial resistance to mass transfer. Other simplifying assumptions such as uniform density and diffusion coefficients in each phase are usually made to simplify the mathematics. The analysis shows that the set of compositions in the system is independent of time although the location of a particular composition is time-dependent. The composition set can be plotted on the equilibrium phase diagram, thus showing the existence of intermediate phases and, as explained below, providing a method for predicting the occurrence of spontaneous emulsification. 

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). 

This relation is referred to as the Maxwell-Stefan model equations, since Maxwell [65] [67] was the first to derive diffusion equations in a form analogous to (2.302) for dilute binary gas mixtures using kinetic theory arguments (i.e., Maxwell s seminal idea was that concentration gradients result from the friction between the molecules of different species, hence the proportionality coefficients, Csk, were interpreted as inverse friction or drag coefficients), and Stefan [92] [93] extended the approach to ternary dilute gas systems. It is emphasized that the original model equations were valid for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. 

As noted above, one of the most successful strategies for dealing with complex problems involving many different scales is to find a way to separate scales. For example, in the context of diffusion, fhe use of fhe diffusion equation (see eqn (7.18)) is an example of this approach in which the physics of the omitted temporal scales appears in the diffusion constant. This interpretation of the... 

The latter interpretation was successfully used to analyze natural convection at vertical plate electrodes by including a magnetic field-related term in the classical convective diffusion equation. " The beneficial effect of the magnetic field on mass transport may be estimated from the ratio of the limiting current density in a magnetic field to that in its absence, called the augmentation factor ... 

Today, spin diffusion is often used in a quite general way to describe multispin polarization-transfer processes, whether or not the process can actually be described by a diffusion equation. In this overview, we will interpret spin diffusion in this broad sense. If applied in the context of two-dimensional homonuclear experiments, it becomes synonymous with total through-space correlation spectroscopy (TOSSY) [2], the dipolar equivalent of the liquid-state TOCSY [3] experiment. 

---