Localized Near Field Theory

Corresponding to the partition P = Pp -H P of the transition probability, the concentration c can be broken into two parts c = Cp -I- C)., . Since the far-field part Cp obeys a gradient-diffusion 

The near-field part of the concentralion accounts for the nondiffusive character of near-field dispersion. It is given by 

These equations provide a complete solution to the forward problem of calculating c(z) from a given (f z) in a uniform canopy. The rec [uired turbulence properties are canopy profiles of the standard deviation J,dz) and the Lagrangian time scale Tiiz) for vertical velocity. 

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The variation of 5(7) near the N-I phase transition will be measured in this experiment and will be compared with the behavior predicted by Landau theory, " " which is a variant of the mean-field theory first introduced for magnetic order-disorder systems. In this theory, local variations in the environment of each molecule are ignored and interactions with neighbors are represented by an average. This type of theory for order-disorder phase transitions is a very useful approximate treatment that retains the essential features of the transition behavior. Its simplicity arises from the suppression of many complex details that make the statistical mechanical solution of 3-D order-disorder problems impossible to solve exactly. 

Because fluctuations become large at the critical point, the simple, mean-field theory used here breaks down. Large fluctuations mean that the approximation, JijSiSj -> JijSi( ), used to simplify the partition function, Eq. 0-65), is no longer valid since the local value of the concentration is no longer approximately given by the average concentration. Even if these fluctuations are included as corrections to the mean-field approximation, the theory becomes quantitatively inaccurate near the critical point. A detailed theoretical treatment of these critical phenomena is outside the scope of this book (see for example Ref. 24). However, analysis of both simple mean-field theories plus their fluctuation corrections includes most of the important physics and provides a guide to when one must include more sophisticated treatments very close to the critical point. 

Wu s explanations for this fundamental observation were couched in field theory, such as overlapping of concentrated stress fields around particles or a transition in local stress state from plane strain to plane stress, and could not furnish a specific material dimension, which, as we present below, depends on the type of the polymer and its crystalline state. The required fully consistent explanation for the discovery was provided by the studies of Muratoglu et al. (1995a), who proposed that the material-specific level of A is a consequence of a preferred form of crystallization of polyamide lamellae near particle interfaces, extending to a certain distance I away from the interface. This results in an anisotropic plastic resistance in this layer, which upon percolation through the matrix and in an... 

It is known that the value of dielectric constant is a function of the local electric field. Since the electric field near the interface is not a constant, but a function of distance from it, the further refinement of EDL theory and potential profile at the BLM interface should include this positional dependency of dielectric coefficient. The local value of this coefficient is of course a measure of the influence of uncharged species such as water or lipid molecules on the interaction between charges in their vicinity. The dielectric coefficient of bulk aqueous electrolyte at... 

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