Rigid constraint model

Thus, it is argued that tte rigid constraint model can provide exact results for important structural properties [49-51]. This assumption has been adopted by workers in the area of off-lattice polymer simulations [45,47,51], and will be retained in this work. However, we wish to point out that the validity of this assumption has been tested mostly on short chain molecules [55, 56, 58-60], and on a small protein [57]. As our computational facilities and methods improve, and it becomes possible to work with denser systons of longer chains, this assumption must be scrutinized closely and carefully, since the approximation might deteriorate with chain length. 

Bead-spring models without explicit solvent have also been used to simulate bilayers [40,145,146] and Langmuir monolayers [148-152]. The amphi-philes are then forced into sheets by tethering the head groups to two-dimensional surfaces, either via a harmonic potential or via a rigid constraint. 

In the enumeration of chirality elements of flexible molecules all arrangements are taken into account which are permitted by the given constraints under the observation conditions. Here, one must always assume a rigid skeletal model and freely rotating ligandsF That arrangement for which the lowest number of chirality elements is found equal zero determines the number of chirality elements for the whole ensemble. 

As is apparent from Figure 10.1, an a-helical structure imposes fairly rigid constraints on the relative positions of successive residues in a peptide chain. Thus there is a loss of entropy that must be overcome energetically in order for an a-helix to form. To explain the underlying biophysics of this system, John Schellman introduced a theory of helix-coil transitions that is motivated by the Ising model for one-dimensional spin system in physics [180, 170],... 

In contrast, random displacements of individual sites of a chain (or a few neighboring sites), when feasible, can be a valuable tool (see Fig. 1). For this approach to be applicable, the chain backbone cannot have rigid constraints (e.g., rigid bonds and bond angles). It is particularly effective for coarse-grained models that allow wide fluctuations of individual sites around their bonded neighbors. Relevant examples are the bond-fluctuation model and certain bead-spring models such as that employed by Binder... 

We now consider the case where the beads are subject to rigid constraints. This is necessary to deal with the problems of suspensions of a rigid body, or polymers with rigid constraints (such as the rodlike polymer, or the freely jointed model), but the reader who is interested only in flexible polymers can omit this section. 

Note that if the constraint would be elastic, the gyristors (GR) would stay in the model and represent the fictitious forces like the centrifugal force (in case of a rigid constraint, the corresponding velocity and thus the corresponding power is zero, such that the contribution becomes irrelevant for the behavior). 

This was not the case. Marsh turned to Hill s analysis of a spherical cavity expanding under an internal pressure and not forcing the material to the surface as the rigid die model does because experimental observation shows the plastic zone to be hemispherical for a great range of materials. He had to introduce constraints to account for the inclusion of a flat surface to produce a hemisphere. For ceramic and glass materials with high values of o-y/E Marsh derived semi-empirically the relationship... 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

To illustrate this theory, we consider a one-component fluid with the interaction between the same species given by Eq. (36). Obviously, the model differs from that described in Sec. (II Bl). In particular, the geometrical constraints, which determine the type of association products in the case of a two-component model, are no longer valid. If we restrict ourselves to the case L < cr/2, only dimers and -mers built up of rigid, regular polygons are possible. 

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