Random fields

Bury, K. V. 1978 On Product Reliability under Random Field Loads. IEEE Transactions on Reliability, R-27(4), 258-260. 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

In the light of the above questions, it is tempting to refer to the results emerging from numerous theoretical and computer simulation studies [40,41,85-88,129-131] of the random field Ising model, and we shall do so, but only after completing the present discussion. 

From Eq. (33) it follows that, in the case of very large homogeneous domains, even very small heterogeneity effects should completely destroy any phase transition connected with the adsorbate condensation. This result is quite consistent with the theoretical predictions stemming from the random field Ising model [40,41]. 

In the next paper [160], Villain discussed the model in which the local impurities are to some extent treated in the same fashion as in the random field Ising model, and concluded, in agreement with earlier predictions for RFIM [165], that the commensurate, ordered phase is always unstable, so that the C-IC transition is destroyed by impurities as well. The argument of Villain, though presented only for the special case of 7 = 0, suggests that at finite temperatures the effects of impurities should be even stronger, due to the presence of strong statistical fluctuations in two-dimensional systems which further destabilize the commensurate phase. 

Record—The elements (fields) may be of different types and may be accessed at random fields and their types are assigned at declaration and may not be changed field values are assigned as are variable values. 

T. Nattermann, in Spin Glasses and Random Fields, A. P Young (ed.). World Scientific, Singapore, 1998, p. 277. 

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t the function U(x, D varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x lJ(x. t) is a random process with respect to t. This behavior is illustrated in Fig. 2.2. The meaning of random in the context of turbulent flows is simply that a variable may have a different value each time an experiment is repeated under the same set of flow conditions (Pope 2000). It does not imply, for example, that the velocity field evolves erratically in time and space in an unpredictable fashion. Indeed, due to the fact that it must satisfy the Navier-Stokes equation, (1.27), U(x, t) is differentiable in both time and space and thus is relatively smooth. ... 

Figure 2.1. Three components of the random field U(x, t ) as a function of x = xi with fixed t =t. The velocity was extracted from DNS of isotropic turbulence (Rk = 140) with (U> = 0. (Courtesy of P. K. Yeung.)...

Because the random velocity field U(x, t) appears in (1.28), p. 16, a passive scalar field in a turbulent flow will be a random field that depends strongly on the velocity field (Warhaft 2000). Thus, turbulent scalar mixing can be described by a one-point joint velocity, composition PDF /u,< (V, i/r,x, t) defined by... 

Conditional moments of this type cannot be evaluated using the one-point PDF of the mixture fraction alone (O Brien and Jiang 1991). In order to understand better the underlying closure problem, it is sometimes helpful to introduce a new random field, i.e.,15... 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

Note that hv operates on the random field U(r, f) and (for fixed parameters V, x, and t) produces a real number. Thus, unlike the LES velocity PDF described above, the FDF is in fact a random variable (i.e., its value is different for each realization of the random field) defined on the ensemble of all realizations of the turbulent flow. In contrast, the LES velocity PDF is a true conditional PDF defined on the sub-ensemble of all realizations of the turbulent flow that have the same filtered velocity field. Hence, the filtering function enters into the definition of /u u(V U ) only through the specification of the members of the sub-ensemble. 

The joint velocity, composition PDF is defined in terms of the probability of observing the event where the velocity and composition random fields at point x and time t fall in the differential neighborhood of the fixed values V and ip ... 

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13... 

The expected value on the left-hand side is taken with respect to the entire ensemble of random fields. However, as shown for the velocity derivative starting from (2.82) on p. 45, only two-point information is required to estimate a derivative.14 The first equality then follows from the fact that the expected value and derivative operators commute. In the two integrals after the second equality, only /u,[Pg.264]

In summary, due to the linear nature of the derivative operator, it is possible to express the expected value of a convected derivative of Q in terms of temporal and spatial derivatives of the one-point joint velocity, composition PDF. Two-point information about the random fields U and

expected value and derivative operators commute, and does not appear in the final expression (i.e., (6.9)). 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

Conceptually, since a random field can be represented by a Taylor expansion about the point (x, f), the random... 

Note that, even though Y and to are fixed, the initial velocity and composition will be random variables, since U(x, 0 and (. f) are random fields. 

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

The coherence of a random field can be evaluated by its correlation function, i.e.. 

Conversely, a rf field is totally correlated because it is represented by a sine (or cosine) function and, as a consequence, its value at any time t can be predicted from its value at time zero. The efficiency of a random field at a given frequency co can be appreciated by the Fourier transform of the above correlation function... 

We now intend to derive the Bloch equations in order to express Ti and T2 according to spectral densities at appropriate frequencies. The starting point is the evolution equation of an elementary magnetic moment p subjected to a random field b... 

Fig. 8. Evolution of the longitudinal and transverse relaxation times (Ti and T2, respectively) as a function of (for a fixed measurement frequency Vo = 400 MHz) assuming that the considered spin is subjected to random fields whose correlation function is proportional to being the correlation time. Notice the continuous...

Because of fast S relaxation (which usually prevents the observation of J splittings), JjsSx can be considered as a random field jbx acting on I. Consequently, correlation functions of the type Sx(t)Sx(0) have to be evaluated (in this case T plays the role of a correlation time). The forthcoming calculations make use of the following identity... 

Classical relaxors [22,23] are perovskite soUd solutions like PbMgi/3Nb2/303 (PMN), which exhibit both site and charge disorder resulting in random fields in addition to random bonds. In contrast to dipolar glasses where the elementary dipole moments exist on the atomic scale, the relaxor state is characterized by the presence of polar clusters of nanometric size. The dynamical properties of relaxor ferroelectrics are determined by the presence of these polar nanoclusters [24]. PMN remains cubic to the lowest temperatures measured. One expects that the disorder -type dynamics found in the cubic phase of BaTiOs, characterized by two timescales, is somehow translated into the... 

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