Soft potential model

A. Incoherent inelastic neutron scattering measurements [53] gave some evidences that the boson peak is a localized mode on 6 and 11 monomers for PS and PB, respectively. Such localization was predicted in molecular dynamics simulations [54-56]. In the framework of the soft potential model, which will be mentioned later, the number of atoms participating in the boson peak mode was evaluated, showing the localization of the boson peak mode on several tens to hundreds of atoms for three inorganic glasses [28] and on several monomers for three polymers [57]. 

In a recently published paper Parshin and Sahling [8] showed the complexity of the interpretation of the heat release data when thermally activated relaxation is taken into account within the framework of the soft potential model. Further theoretical work on the residual properties of two-level systems and its dependence on the cooling procedure has been published by Brey and Prados [9]. 

At temperatures above 1 K tunneling transitions and localized vibrations influence the physical properties [9]. Both types of mechanisms have recently been incorporated in the soft potential model [10] which contains the well-known tunneling model as a special case. 

The soft docking model represents the target and docking molecules as a collection of cubes rather than spheres. This method combines aspects of surface complementarity, grid search, and soft potential modeling. The cubic representation along with a grid search makes the translational and rotational searches much more efficient. In addition, the cubes implicitly allow for some volume overlap, which can be used in combination with surface complementarity to screen docked complexes [208]. 

Both models were simulated with the same crosslinked configurations. In both cases the relaxation of the systems was extremely slow. Considering tat w 180r for N — (Ns) and 3700r for N= 50 the primary chain length, it was observed that the characteristic relaxation time increased by about a factor of 10 compared to rj r=5o or more than 100 compared to the Rouse time of the average inner strand for the LJ case. The soft potential model displayed an even slower relaxation. 

The soft-core model may be more convenient in molecular dynamics simulation, since a continuously differentiable potential is available to calculate the force. In the case of a hardcore potential, collision times of all atom pairs have to be monitored and used to control the time step. 

Soft sensors can be used for closed-loop control, but caution must be used to ensure that the soft-sensor model is applicable under all operating conditions. Presumably one would need to test any potential process condition to validate a soft-sensor model in the pharmaceutical industry, making their use in closed-loop control impracticable due to the lengthy validation requirements. An important issue in the use of soft sensors is what to do if one or more of the input variables are not available due, for example, to sensor failure or maintenance needs. Under such circumstances, one must rely on multivariate models to reconstruct or infer the missing sensor variable.45 A discussion of validating soft sensors for closed-loop control is beyond the scope of this book. 

In spite of the fact that the decay after excitation of the hard-potential itinerant oscillator is similar to the experimental computer simulation result of Figs. 7 and 8, we do not believe that it is the reduced model equivalent to the one-dimensional many-particle model under study. As remarked above, indeed, the e(r) function is not correctly reproduced by this reduced model. The choice of a virtual potential softer than the linear one seems also to be in line with the point of view of Balucani et al. They used an itinerant oscillator with a sinusoidal potential, which is the simplest one (to be studied via the use of CFP) to deal with the soft-potential itinerant oscillator. Note that the choice... 

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

The n= 12 soft sphere model is the high-temperature limit of the 12-6 Lennard-Jones (LJ) potential. Agrawal and Kofke [182] used this limit as the starting point for another Gibbs-Duhem integration, which proceeded to lower temperatures until reaching the solid-liquid-vapor triple point. The complete solid-fluid coexistence line, from infinite temperature to the triple point, can be conveniently represented by the empirical formula [182]... 

DFT has been much less successful for the soft repulsive sphere models. The definitive study of DFT for such potentials is that of Laird and Kroll [186] who considered both the inverse power potentials and the Yukawa potential. They showed that none of the theories existing at that time could describe the fluid to bcc transitions correctly. As yet, there is no satisfactory explanation for the failure of the DFTs considered by Laird and Kroll for soft potentials. However, it appears that some progress with such systems can be made within the context of Rosenfeld s fundamental measures functionals [130]. 

---