Random fields method

In the current paper, we discuss some of the new approaches and results that have been developed and obtained recently within the context of such molecular modeling research, and in particular with the mean field and Monte Carlo studies of a lattice model. The next section describes the Gaussian random field method (Woo et al, 2001), which provides a computationally efficient route to generate realistic representations of the disordered mesoporous glasses. Application of the mean field theory, and Monte Carlo simulations are described in Secs. 3 and 4, respectively. 

In contrast to the approach of simulating the physical synthesis of the material, the computational cost involved in the Gaussian random field method is minimal. Statistically independent realizations with the desired porosity and spatial correlations can be repeatedly generated simply by changing the random number seed of the Gaussian random field. 

Figure 1 Two-dimensional cross-section of a Vycor glass realization generated by the Gaussitm random field method. The cubic grid size is 10 A.

In the present example the confl iration of the solid sites is build to model the mesoporous structure of a porous glass. Each sample of the glass material is obtained with the Gaussian random field method [30], During a calculation, we use periodic boundary conditions in all directions of space. An illustration of a Vycor glass sample obtained with the Gaussian random field is reported on Fig. 1. We use the same procedure for CPG. 

Fig. 1 An illustration of a mesoporous Vycor glass built on the lattice model with the Ganssiaii random field method. The parameters used are p=0.3, a=15A, T ,=Z, =ii=80. For CPG we use p=0.6, a=15A,ix= i,=180.

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

I3G. Winkler, Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, Springer-Verlag, New York, 2003. 

Geostatistical methods are also termed random field sampling as opposed to random sampling [BORGMAN and QUIMBY, 1988], The spatial dependence of data and their mutual correlation can be analyzed by use of semivariograms. Statements on the anisotropy of the spatial distribution are also possible. Kriging, a geostatistical method of... 

The shortcoming of the mean field method is that it admits no correlation between the motions of the individual particles. This correlation can be introduced by means of the random phase approximation (RPA) or time-dependent Hartree (TDH) method. In order to formulate this method, we introduce excitation operators (Ep) which replace f) p by when applied to the mean field ground state of the crystal when applied to any other state, they yield zero. Then, we write the Hamiltonian as a quadratic form in the excitation operators (Ep)+ and their Hermi-tean conjugates Ep... 

Taking mean bias and random bias into account, we obtain the following expression for an individual measurement of a given sample by a field method... 

If a specific value for X is not available and the two field methods that are compared are likely to be associated with random error levels of the same order of magnitude, X can be set to 1. The Deming procedure is generally relatively insensitive to a misspecification of the X value. The slope and intercept may now be derived as... 

As mentioned earlier, the matrix-related random interferences may not be independent. In this case, simple addition of the components is not correct, because a covariance term should be included. However, we can estimate the combined effect corresponding to the bracket term, which then strictly refers to the CV of the differences (CV b2-rb])- As in the case with constant standard deviations, information on the analytical components is usually available, either from duplicate sets of measurements or from quality control data, and the combined random bias term in the second bracket can then be derived by subtracting the analytical component from CV21. Systematic and random errors can then be determined, and it can be decided whether a new field method can replace an existing one. Figure 14-31 shows an example with proportional random errors around the regression line. 

As noted above, one interesting application of these ideas is to the motion of a dislocation through an array of obstacles. An alternative treatment of the field due to the disorder is to construct a particular realization of the random field by writing random forces at a series of nodes and using the finite element method to interpolate between these nodes. An example of this strategy is illustrated in fig. 12.27. With this random field in place we can then proceed to exploit the type of line tension dislocation dynamics described above in order to examine the response of a dislocation in this random field in the presence of an increasing stress. A series of snapshots in the presence of such a loading history is assembled in fig. 12.28. 

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