Diffusion entropy analysis functions

Unfortunately, in a typical NSE experiment the number of points obtained as a representation of this function is in most cases too small and the error of the individual points is too high to allow for a fit with more than two or three adjustable parameters or for an analysis using Laplace transformation - and maximum entropy methods [56]. Without information from additional experiments it is only possible to compute an effective diffusion coefficient from the data by using a first- or second-order cumulant analysis [57]. 

Empirically, Uc has often been found to be an exponential function of c or V2, both from viscoelastic - and from diffusion - measurements. This relation may be useful for interpolation purposes, but does not appear to provide any insight into molecular mechanisms. There have been many empirical correlations of plasticizing effectiveness with various physical properties of diluents. - An analysis of the concentration dependence of Uc based on changes in configurational entropy has been presented by Havlicek and collaborators, and provides good agreement with experiment. 

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

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