Laplacian operator theories

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

Electrostatics is a 3-D theory, because it describes forces between particles that exist in a 3-D world. Nevertheless, it is often useful to consider versions of the Poisson and PB equations where the field depends on only one or two spatial variables, and the 3-D Laplacian operator is replaced by l-D or 2-D analogs. In the context of electrostatics, these reduced dimensional versions arise in situations with spatial symmetry such that the dependence of the field along one or two Cartesian directions vanishes. Since it is easier to notate the l-D than the 3-D case, we shall often consider the simpler versions when discussing real-space lattice methodology. In order to derive the l-D update equation (13) from the steepest descents principle embodied by equation (12), we should construct a l-D version of the action indicated in equation (8). That is, suppose the charge distribution depends only on x (and is uniform in the y. z directions). Then the electric potential

[Pg.2099]

Vol. 1862 B. Helffer, F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians (2005)... 

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