Gaussian random fields

Fig. 10. Random Stark broadening of a hydrogen atom in a Gaussian random field with different modulation speeds (y/A = 2, 1, 0.2, 0).

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. 

Kennedy, DP (1994) The Term Structure of Interest Rates as a Gaussian Random Field. Mathematical Finance 4 247-258. 

A Gaussian random field 0(r) is defined by a probability distribution exp(-.Fo[0]) with a quadratic free energy functional. T ol I ]- This applies in particular to the functional defined in Eq. (15). 

H) S)/2n(xE)-T lull lines are the results for Gaussian random fields the symbols represent Monte Carlo data for different system sizes N. (a) (xe) > 0, (b) (xe) < 0- (From Ref 85.)... 

Quintanilla J, Reidy RF, Gorman BP, Mueller DW (2003) Gaussian random field models of aerogels. J Appl Phys 93 4584-4589. 

Mills and Zhu (1999) developed the closed-cell and opened-cell foams using the BBC lattice tetrakaidecahedral cell model were as shown in Table 5.1 where 0s is the volume fraction of solids in the cell edges (Zhu et al. 1997a, b Mills and Zhu 1999). Roberts and Garboczi (2001, 2002) investigated the modulus for both opened-cell and closed-ceU foam stracture of random models based on Voronoi tessellations and level-cut Gaussian random fields for a closed-cell tetrakaidecahedral model shown in Fig. 5.2. Lu et al. (1999) proposed macro-mechanic model to evaluate the modulus for low porosity. 

Before discussing gaussian random fields in more detail and explicitly proving (6. 25)-(6.27), we examine the implications of these equations for the existence of a SCF approximation. 

In which I/CEqI) Is the expected value of the flexibity and f(x) Is a zero-mean Gaussian random field with autocorrelation function... 

If the random field is Gaussian, the joint distribution of Xi for various elements remains Gaussian for both the midpoint and spatial averaging methods. However, for non-Gaussian fields the distribution of X, defined by Eq. 21 is difhcult or impossible to obtain. An approximate description of the distribution for that case has been suggested elsewhere [8]. Here, attention is restricted to Gaussian random fields so that a fair comparison between the two methods of discretization can be made. 

The example structure is one quarter of a 1 in. thick, 32x32 in square composite plate with a 4-inch diameter circular core region under distributed edge loads, as shown in Fig. 5. The bulk and shear moduli K and G, respectively, of the outside material are modeled as homogeneous Gaussian random fields. The moduli and of the core material and the load intensity V/ are considered to be Gaussian random variables. The assumed means and coefficients of variation are listed in Table 2. The shear and bulk moduli for each material are assumed to be statistically independent, but correlation between the same moduli for the two materials, i.e, pG Gi is considered. 

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. 

In the random wave model of Berk [7], a Gaussian random field i/ (r) is constructed by superposition of a large number N of cosine waves with random phases ... 

This continuous random process mean-square value of unity, is then clipped and transformed into a two-state discrete random process. By clipping we mean assigning a constant value -(-1 to the function whenever the Gaussian random field at that point is above a certain level called a, and a constant 0 whenever its value is below a. This transformation can be defined as follows ... 

At a first glance, the two profiles shown in Fig. 5 are very similar, except for absolute length scales (about 100 A vs. 1 pim). Similarly, the two real-space structures constructed on the basis of the Gaussian random-field theory show a striking resemblance, having sponge-like characteristics. Quantitative comparisons of the two systems were made... 

In principle the earthquakes may be modeled as nonstationary non-Gaussian random fields, even if the null-mean Gaussian approximation is considered acceptable by most authors. 

As a consequence of the central limit theorem, it converges forM oo to a Gaussian random field with the same mean value and autocorrelation structure as the target Gaussian random field. 

This distribution can be used to approximately represent a homogeneous Gaussian random field by a superposition of harmonics. For a real-valued random field, one obtains the representation... 

For a prescribed, uniformly bounded random field a(x, random variables 1, (0) in Eq. 8 would be dependent non-Gaussian random variables whose joint distribution function is very difficult to identify. If, on the other hand, independent but bounded distributions are prescribed for the random field oc(x, co) is not necessarily bounded for —> 00. Thus, one is left with Gaussian distributions for i(co) and a(x, co), with transformations of Gaussian random fields or with some situations, where nonnegative distributions for i(w) lead to meaningful (e.g.. Erlang) distributions for oc(x, co). 

Transformation techniques for non-Gaussian random fields seek to represent the non-Gaussian random field as a nonlinear transformation of a Gaussian random field ... 

Phoon et al. (2002, 2005) used the KL expansion for the simulation together with an iterative mapping scheme to fit the target marginal distribution function of non-Gaussian random fields. The method allows to simulate homogeneous as well as nonhomogeneous random fields. 

Spanos 1991 and Pig. 1). The product of Young s modulus and the thickness of the plate is assumed to be an isotropic Gaussian random field with covariance function... 

Review of Some Basic Theory This subsection reviews some fundamental properties of Gaussian random fields in D-dimensions. The case D = 2 is needed for surface modelling, and the D = 3 case for property modelling. 

---