Gaussian models

In a dense melt, the excluded volume of the monomeric units is screened and chains adopt Gaussian conformations on large length scales. In the following, we shall describe the conformations of a polymer as space curves r(r), where the contoiu parameter r runs from 0 to 1. The probability distribution P[r] of such a path r(r) is given by the Wiener measure 

Combining these with the chain conformations r(r), we can define a microscopic density a r) for segments of type a = A or B 

Here the sum ij rims over aU nj polymers of type /, and ry(r) denotes the conformation of the polymer. The segment density is normalized by the polymer number density, q = rij/V (17 being the volume of the system). 

As a result, the x-parameter depends on the temperature, pressure and composition, as it is often observed in experiments. 

Within the coarse-grained model the canonical partition function of a polymer mixture with tij chains of type / takes the form  

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Step 4 This PC A defines a multivariate Gaussian model... 

As dense clouds move downwind, they are diluted with air until they eventually become neutrally buoyant. Thus, the gaussian models presented earher are applicable for dense cloud releases at distances Far downwind from the release. 

Here,. Ai(X) is the partial SASA of atom i (which depends on the solute configuration X), and Yi is an atomic free energy per unit area associated with atom i. We refer to those models as full SASA. Because it is so simple, this approach is widely used in computations on biomolecules [96-98]. Variations of the solvent-exposed area models are the shell model of Scheraga [99,100], the excluded-volume model of Colonna-Cesari and Sander [101,102], and the Gaussian model of Lazaridis and Karplus [103]. Full SASA models have been used for investigating the thermal denaturation of proteins [103] and to examine protein-protein association [104]. 

In addition to the Gaussian modeling techniques already discussed, four other methods will be considered. 

H. Mtiller-Krumbhaar. Kinetic Gaussian model. Z Physik B 25 287, 1976. 

MPTER is a multiple point-source Gaussian model witli optional terrain adjustments. 

TLTPOS is a Gaussian model tluit estimates dispersion directly from fluctuation statistics at plume level. 

Similar results may be obtained not only for exponential R t) but for the Gaussian model of the kernel [56] and some others. It has recently been shown [57] that Kj(t) changes sign at any k when... 

Much of the initial development of Gaussian modeling and definition of dispersion paramenters was done during and after World War I in addressing the problem of poison gas dispersal. These studies involved the definition of risk factors, such as exposure and dose. The next intensive development effort came during and after World War II with the nuclear weapons program. 

The dielectric constant of the external medium is e. The most significant point of Eq. (16) is that 5/z is proportional to q2. On this basis, we change variables, writing 8U = q >, and consider the Gaussian model (Hummer etal, 1998c),... 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

Hummer, G. Pratt, L. Garcia, A. E., Multistate Gaussian model for electrostatic solvation free energies, J. Am. Chem. Soc. 1997,119, 8523-8527... 

The deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

Virtual sources As indicated above, the gaussian model was formulated for an idealized point source, and such an approach may be unnecessarily conservative (predict an unrealistically large concentration) for a real release. There are formulations for area sources, but such models are more cumbersome than the point source models above. For point source models, methods using a virtual source have been proposed in the past which essentially use the maximum concentration of the real source to determine the location of an equivalent upwind point source that would give the same maximum concentration at the real source. Such an approach will tend to overcompensate and unrealistically reduce the predicted concentration because a real source has lateral and along-wind extent (not a maximum concentration at a point). Consequently, the modeled concentration can be assumed to be bounded above, using the point source formulas in Eq. (23-78) or (23-79), and bounded below by concentrations predicted by using a virtual source approach. 

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