Multi-dimensional diffusion

A further generalization is to write down a multi-dimensional GLE, in which the system is described in terms of a finite munber of degrees of freedom, each of which feels a frictional and random force. For example, an atom diffusing on a surface, moves in three degrees of freedom, two in the plane of the surface and a third which is perpendicular to the siuface. Each of these degrees of Ifeedom feels a phonon friction. Multi-dimensional generalizations and considerations may be foimd in Refs. 72-82. 

The equations derived above, describing the A + B —> B reaction kinetics in terms of the correlation functions g and g2, have the form of the nonlinear generalised multi-dimensional diffusion equation. Ignoring the multidimensionality of the operator terms in (5.2.11), these equations could be formally considered as similar to the basic non-linear equations for the A + B — 0 reaction (Section 5.1). Equations studied in both Sections 5.1 and 5.2 are derived with the help of the Kirkwood superposition approximation, the use of which leads to several equations for the correlation functions of similar and dissimilar reactants. 

It has been reported that rates of proton transfer from carbon acids to water or hydroxide ion can be predicted by application of multi-dimensional Marcus theory to a model whereby diffusion of the base to the carbon acid is followed by simple proton transfer to give a pyramidal anion, planarization of the carbon, and adjustment of the bond lengths to those found in the final anion.124 The intrinsic barriers can be estimated without input of kinetic information. The method has been illustrated by application to a range of carbon acids having considerable variation in apparent intrinsic barrier. 

The extension of the methods described so far to multi-dimensional diffusion problems is straightforward in principle. However, in such an attempt one is faced with a quite considerable increase in computational effort. 

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... 

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

In addition to the consideration of equilibrium problems, the treatment of thermal excitations is also important to the investigation of kinetics. An intriguing insight into kinetics is that in many cases, the tools for treating thermal excitations to be described in this chapter can be used to consider driven processes such as diffusion. As will be noted in chap. 7, models of kinetics are tied to the idea that as a result of the random excursion of atoms from their equilibrium positions, a system may occasionally pass from one well in the multi-dimensional potential energy surface to another. Indeed, this eventuality has already been rendered pictorially in fig. 3.22 where we saw an adatom hopping from one energy well to the next in its... 

Laskar, J. (1993). Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Physica D, 67 257-281. 

Berezhkovskii A M and Zitserman V Yu 1993 Multi-dimensional Kramers theory of the reaction rate with highly anisotropio friotion. Energy diffusion for the fast coordinate versus overdamped regime for the slow coordinate Chem. Phys. Lett. 212 413-19... 

The fliamelet model for diffusion flames has been developed at the RWTH Aachen describing in a simplified manner the flame front of a multi-dimensional flow. The flame front is assumed to be locally one-dimensional. By introducing an appropriate coordinate, e.g., fuel-mass ratio, the determination of flame structure and of flame front propagation can be separated [79]. 

The two-phase integrated finite difference flow code TOUGH2 was selected to perform the hydro-logical calculations for this I BEX analysis. TOUGH2 is a general-purpose program for the simulation of multi-dimensional, multiphase fluid flow in porous and fractured media. Fluid advec-tion is calculated by a multiphase extension of Darcy s law, and diffusion is included. Local thermodynamic equilibrium is assumed in all... 

In summary, with multi-dimensional data analysis reliable background models can be obtained from the source measurement itself in the case of diffuse gamma-ray line emissirm. Simulations and bootstrapping are employed to ensure proper imaging results. 

Data about chain alignment and reorientation must usually be obtained by modelling or simulation of the lineshape, and usually, data from several different experiments are required to yield an unambiguous conclusion. An exception involves the more recent multi-dimensional experiments, described in section 8.5 [53, 59], that are designed to yield the appropriate data graphically or with simple calculation. Even with 2-D spectroscopy, if multiple motions are present or if diffusion is also present, there remains a requirement to model the data. 

NUMERICAL METHODS FOR SOLVING MULTI-DIMENSIONAL MULTIGROUP DIFFUSION EQUATIONS... 

We shall concentrate on surveying the available numerical methods for solving the multi-dimensional multigroup diffusion equations. Since Dr. Ehrlich has already sketched the numerical methods available for treating the case of one space variable, we shall therefore concentrate on the cases of several space variables, although in general our theoretical discussions will be phrased independently of the number of space variables. In all cases we shall attempt to discuss both the rigorous mathematical features and the practical applications of these various numerical methods to both the time independent and time dependent diffusion equations. 

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