Nearest-neighbor interaction model

An approximate nearest-neighbor interaction model, in which the states of residues i-1 and / are correlated, is also introduced a 3 x 3 matrix is required In this model. Both the 4 x4 and 3x3 matrix formulations involve four parameters (uci, and vjg). 

By choosing the c state as a standard state, the number of statistical weights is reduced to three (relative to that of the c state), viz. vs,i > and vh,l - Hence, the 3x3 matrix for the nearest-neighbor interaction model can be written as ... 

Interaction between calf thymus DNA and dodecylpyridinium halide (DoPX) was studied through binding isotherms, varying concentrations of added sodium chloride, and temperature. The results are ana ed in term of a one-dimensional nearest neighbor interaction model. The... 

Because the binding process for our systems cannot be described by the independent single-site model as discussed above, we further examined our data by the expressions derived on the basis of the nearest-neighboring interaction model. 

Predicting these magnetic structures is extremely difficult since in each case there is a fine balance between different interactions. Probably the considerations discussed earlier will be necessary, together with some features of the anisnttrcrpic next-nearest-neighbor interaction model (ANNNI), as expounded by Bak and van Boehm (1980), Villain and Gordon (1980), and Fisher and Selke (1980). 

In their model they retained only the first- and second-nearest neighbor interactions, so that the Hamiltonian assumed the following form... 

The Larson model and Larson-type models have been widely used to study micelles [37,111,114-120], amphiphiles at oil/water interfaces [121,122] bilayers [117,123] and various other problems [125-128]. The models differ from each other in the range of the interactions and in the treatment of the amphiphile monomers. Other than in Larson s original model, most authors include only nearest-neighbor interactions, sometimes in combination with a... 

More recently suggested models for bulk systems treat oil, water and amphiphiles on equal footing and place them all on lattice sites. They are thus basically lattice models for ternary fluids, which are generalized to capture the essential properties of the amphiphiles. Oil, water, and amphiphiles are represented by Ising spins 5 = -1,0 and +1. If one considers all possible nearest-neighbor interactions between these three types of particle, one obtains a total number of three independent interaction parameters, and... 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

To examine replication of IPBs we made MFKEi-based simulations using the simplest 2D alloy model with the nearest-neighbor interaction. Some results are presented in Figs. 8-10. The lower row in Fig. 10 illustrates possible effects of thermal fluctuations, similar to those discussed in Sec. 3 for the replication of APBs. The figure shows that peculiar features of microstructural evolution are preserved even under rather strong thermal fluctuations used in this simulation. 

In Fig. 12 we present some results of MFKEbbased simulation of spinodal decomposition with the vacancy-mediated exchange mechanism. We use the same 2D model on a square lattice with the nearest-neighbor interaction and Fp = 0 as in Refs., ... 

Note that for = 2 both Eqs. (17), (18) essentially reduce again to the Ising Hamiltonian, Eq. (9), with nearest neighbor interaction only. The latter model is described by the following critical behavior for its order parameter if/, ordering susceptibility and specific heat C ... 

Choice of lattice linear dimensions Lx, Ly (usually Lx = Ly = L, apart from physically anisotropic situations, such as the ANWII model ). Only finite lattices can be simulated. Usually boundary effects are diminished by the choice of periodic boundary conditions, but occasionally studies with free boundaries are made. Note that Lx, Ly must be chosen such that there is no distortion of the expected orderings in the system e.g. for the model of where due to a third nearest neighbor interaction superstructures with unit cells as large as 4 x 4 did occur, L must be a multiple of 4. 

A model with both next-nearest-neighbor and next-next-nearest-neighbor interactions is a little, but clearly, better than a model without next-next-nearest-neighbor interaction. There is however a clear correlation between these two interactions. Variations in the next-nearest-neighbor and next-next-nearest-neighbor interaction change the spectra only little as long as the sum of these interactions is kept the same. 

Figure 15 shows how the leave-one-out error changes with the model of the lateral interactions. It is clear that the nearest-neighbor and next-nearest-neighbor interactions are the most important. These two interactions already provide a good description of all the adlayer structures. The next-next-nearest-neighbor interaction does not improve the model, but remarkably the linear... 

The root-mean-square error of the model with only nearest-neighbor and next-nearest-neighbor interactions is 3.9 kJ/mol. The one of the model that also includes the linear 3-particle interaction is 2.9 kJ/mol. Using these errors as estimates for the errors in DFT calculations gives the error estimates of the lateral interactions in table 2. These should be regarded as lower estimates. If we would use a larger estimate for the errors in DFT then the errors of the lateral interactions would increase proportionally. Note that, because the term is always the same in the expressions for independent of the adlayer structure, possible systematic errors in DFT (cr in Section 3.4.2) only affect E 2 and not the lateral interactions. 

HI. STOCHASTIC MODEL FOR RESTRICTED SELF-AVOIDING CHAINS WITH NEAREST-NEIGHBOR INTERACTIONS... 

We now introduce nearest-neighbor interactions in the restricted chain models. In our particular model these interactions are nonzero only for those nearest-neighbor contacts which can lead to excluded configurations upon addition of a single step. Thus for example, in a square lattice, nonzero interactions involve only contacts created by the three-step chain configuration denoted by q(i The new transition probabilities are... 

The partition function, given by Eqs. (15-17) for chains with no chain-end effects, and by Eqs. (20-23) for chains with end effects, is restricted to the chain models in which only the lowest-order polygons are excluded. We can extend the derivation of these partition functions to a more general case, in which we eliminate all polygons of sizes t or less and restrict nearest-neighbor interactions to contacts which are separated by t — 1 and fewer chain elements. 

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