Non-Gaussian behavior

Now, we consider this phenomenon from the viewpoint of the non-Gaussian behavior of the network chain. As is well known, when we assume the idealized molecular network consisting... 

We shall now discuss the non-Gaussian behavior of our self-correlation functions. Rahman32 pointed out that it is convenient to do this by introducing the coefficients a N(t) which for Gs(r, t) are defined as... 

These coefficients are strongly dependent on the number of molecules used in the simulations. For example, Figures 37 and 38 present the coefficients from the Stockmayer simulation using 216 and 512 molecules, respectively. The corresponding coefficients from the 216 and 512 molecule systems differ substantially from each other. Therefore, we feel that these coefficients from our simulations are only qualitative indications of the non-Gaussian behavior of our self-correlation functions. Figure 41 presents the coefficients from the modified Stockmayer simulation. Comparing the results for the two simulations we see ... 

This appendix gives some of the properties of the Hermite polynomials, HeN(jc). These polynomials form a basis set for Rahman s32 expansion of C s(v)(r, t) and play a fundamental role in the discussion of the non-Gaussian behavior of this latter function. Brief sketches of this expansion and of the... 

Fig. 9.2.5. Elution of a components (peak) showing non-Gaussian behavior (tailing)...

Moreover, the Gaussian prediction for is accurate in the very short and long time limits but fails at intermediate times. Thus, the non-Gaussian behavior observed here is quite similar, qualitatively, to that reported by... 

Values of the correlational contribution are listed in Table I. The correlational contributions decrease with an increase in the mole percent of short chains in the network. This trend is also consistent with the conclusion that intermolecular interactions have little to do with the non-Gaussian behavior exhibited by bimodal PDMS networks. 

Hurley, M. M., and Harrowell, P., Non-gaussian behavior and the dynamical complexity of particle motion in a dense two-dimensional liquid. J. Chem. Phys. 105,10521 (1996). Hyde, P. D., Evert, T. E., and Ediger, M. D., Nanosecond and microsecond study of probe... 

The use of Stirling s approximation assumes a sufficiently large value of Njj. This assumption fails at high degrees of crosslinking and high extensions. The failure leads to non-Gaussian behavior. Note the similarity of the last term of this sum and that evaluated in the... 

Birefringence can be used to characterize non-Gaussian behavior in PDMS bimodal elastomers. - A large decrease the stress-optical coefficient (ratio of birefringence to stress) was observed over a relatively small range in elongation, presumably due to limited extensibility of the short chains. 

In one application, a filled PDMS network was modeled as a composite of cross-linked polymer chains and spherical filler particles arranged on a cubic lattice. The filler particles increase the non-Gaussian behavior of the chains and increase the moduli. It is interesting to note that composites with such structural regularity have actually been produced and mechanical properties have been reported. -... 

A method for accurately estimating the second-order statistical responses required for robust design optimization in case of nonlinear response. This is accomplished by a modified method of stochastic equivalent linearization, especially purported to take into account the actual non Gaussian behavior of hysteretic oscillators proposed by the author in (Hurtado and Barbat 1996 Hurtado and Barbat 2000). 

To accotmt for these effects, several theoretical approaches were proposed. Statistical description taking into account a finite extensibility of subchains and their non-Gaussian behavior at high deformations was put forward by Kuhn and Griin. In the framework of this model, the elastic deformation can be described at any extension by using the inverse Eangevin function. This direction was developed further by... 

Specifically, at least three typical features are related with nonequilibrium granular flow systems (i) feasibility of one granular temperature (ii) non-Gaussian behavior of velocity distribution, and (iii) strong correlated density fluctuations in forms of bubbling or clustering. In the following section, we detail the above three points. 

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