Non-Gaussian case

For the non-Gaussian case, the distribution in angle of the individual chain segments is calculated first. Then, the probability of a given conformation is determined. Finally, the most probable distribution is derived by differentiation, as in the Gaussian case. 

It should be pointed out that, in the non-Gaussian case, P(R) going to zero at R leads to the force going to infinity (Figure 2.20). 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Models which also describe the molecular weight between crosslinks for neutral polymer networks but use a non-Gaussian chain distribution have also been derived. These models would be useful in cases of highly crosslinked polymer networks. Examples of these types of models include those of Peppas and Lucht [7], Kovac [8], and Galli and Brummage [9]. 

Although paraxial approximation becomes imsuitable for higher-WA optics and for non-Gaussian beams, the above insights should remain qualitatively valid in these cases as well. Since only the above-the-threshold intensity part of the spatio-temporal envelope of the beam is important for photomodification, usually this part can be reasonably well approximated by a Gaussian. 

Kuhn for statistically coiled molecules. The two dotted lines denoted by F and N stand for the free-draining and the non-draining case of Zimm s theoty for Gaussian coils. The hatched area indicates the area where the experimental points obtained on solutions of anionic polystyrenes are located (See Fig. 3.1). 

An example of a relevant optical property is the birefringence of a deformed polymer network.256 This strain-induced birefringence can be used to characterize segmental orientation, and both Gaussian and non-Gaussian elasticity.92,296-302 Infrared dichroism has also been particularly helpful in this regard.82,303 In the case of the crystallizable polysiloxane elastomers, this orientation is of critical importance with regard to strain-induced crystallization, and the tremendous reinforcement it provides.82... 

Section II is devoted to reviewing the basic ideas of the RMT, having in mind a linear chain of particles coupled with each other via linear interactions. Section III is devoted to illustrating the results of computer experiments that we did to supplement those of the interesting paper of Bishop et al. with additional information as to whether or not the non-Gaussian features of the velocity variable are the same as those of the real three-dimensional fluids. We shall show that these are qualitatively similar, although in the one-dimensional case the non-Gaussian character is more intense and much more persistent. 

In the case of simple liquids these deviations can be accounted for by usii techniques based on kinetic and mode-mode coupling theories - and are related to the non-Gaussian properties of the fluid, as observed in computer simulation experiments (see Chapter VI). 

The picture of the translation closely parallels die rotational picture. In this case the counterpart of the Anderson model is the random jump model (see Section VIII). An important theoretical prediction of the reduced model of Section FV is the deviation from Pick s law, which is firmly supported by experiment. Note that this deviation precisely depends on the fluctuating nature of the reduced model. On the other hand, the theoretical analysis of Chapter VI shows that this multiplicative fluctuation must be traced back to the nonlinear nature of the microscopic interaction, therby rendering plausible the appearance of non-Gaussian microscopic prop es. 

Lutz [47] also supposed that the random force X(t) is Gaussian. Equation (335) may also describe non-Gaussian processes. However, in that case, the higher-order moments X(t ) (o) X(tn) may not be expressed in terms of X(t) and... 

We now return to the simulated curves, in order to show how to extract the area, position, and width from a chromatographic peak. Several simple methods are available for symmetrical peaks (and even more for a special subset of these, Gaussian peaks), but since chromatographic peaks are often visibly asymmetric (and in that case obviously non-Gaussian), we will here use a method that is independent of the particular shape of the peak. It is a standard method that, in a chromatographic context, is described, e.g., by Kevra etal. in J. Chem. Educ. 71 (1994) 1023. 

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