Spatial randomness

Although the deposition of radiation energy is spatially random on the molecular scale, the chemical changes are not random. 

The soliton conductivity model for rrans-(CH) was put forward by Kivelson [115]. It was shown that at low temperature phonon assisted electron hopping between soliton-bound states may be the dominant conduction process in a lightly doped one - dimensional Peierls system such as polyacetylene. The presence of disorder, as represented by a spatially random distribution of charged dopant molecules causes the hopping conduction pathway to be essentially three dimensional. At the photoexitation stage, mainly neutral solitons have to be formed. These solitons maintain the soliton bands. The transport processes have to be hopping ones with a highly expressed dispersive... 

Figure 8. Production of an image current signal from spatially random but phase coherent ion population.

Most of theories of spin dynamics in semiconductors, as discussed above, consider the SO coupling as uniform, that is electron-coordinate independent. The reality is, however, different the coupling produced by randomly distributed charged impurities is spatially random as well. 

Note that this result is independent of the residual stress, because the terms containing (crb + crT) in Eqns. (26) and (33) cancel when inserted into Eqn. (40a). The coefficient x depends on the spatial aspects of crack evolution periodic, random, etc. Simulations for spatial randomness indicate that X = 1.6.29 Moreover, these same simulations indicate that the saturation stress should scale with amc, such that... 

The general physical model of percolation theory applies to a broad range of phenomena that are rooted in spatially random events and topological disorder and that reflect long-range connectivity. One can picture the protonic process as being a transfer along threads of hydro-... 

Percolation theory provides a well-defined model applicable to a wide variety of spatially random phenomena, both macroscopic and microscopic (Table X). The characteristic length scales for these phenomena... 

The difficulty of having to evaluate or measure the matrix elements density distribution of the atomic system, due to random site occupancy (Fig. 1, inset (a)) and short-range interaction with the probe (e.g., a neutron or thermal atom). The elements structure factor over all relevant k ... 

Ries B. and Bassler H., Monte Carlo study of dispersive charge-carrier transport in spatially random systems with and without energetic disorder Phys. Rev. B, 35, (1987) 2295-2302, DOI 10.1103/... 

O. Fudym, H.R.B. Orlande, M. Bamford, and J.C. Batsale, Bayesian Approach for Thermal Diffusivity Mapping from Infrared Images Processing with Spatially Random Heat Pulse Heating, Journal of Physics. Conference Series (Online),... 

This case is of interest for the MPD process, since a previous investigation"" has indicated that this dissociation type will normally lead to spatially randomized fragments. Therefore, spherical symmetry has been assumed in the MPD experiments reported here, though the limited... 

Much of the information in a smooth image is highly correlated by virtue of the fact that, for example, pixel values are not spatially random and that the value of one pixel indicates the likelihood of its neighbours values. Several types of correlation exist in an image ... 

One layer of the multi-layer model is modeled by randomly placed fibers which are dilated with respect to 3D as described in Eq. (24.1), where the starting points of the polygonal tracks follow the principle of complete spatial randomness. Note that due to the multi-layer approach, only a 2D polygonal track has to be modeled, which reduces the complexity dramatically compared vsdth a (real) 3D model of curved fiber trajectories. 

We model the 3D set of vertices by a multi-layer approach, whereby all points of one thin section are projected on to their bases, that is, we have to find a 2D model for the projected points. Therefore, we use a generalized Thomas process with elliptically shaped clusters, which has the following structure see also, for example, [27]. The parent points form a stationary Poisson point process with intensity Xp > 0, that is, they follow the principle of complete spatial randomness. The random number of child points per cluster is Poisson distributed with expectation c, and the random deviations of child points from their parent points are given via a 2D normal distribution N(0, C), with expectation vector 0 and covariance matrix... 

SHINOZUKA, M., Response Variability Due to Spatial Randomness, Proceedings of the 2nd International Workshop on Stochastic Methods in Structural Mechanics, (Pavia SEAG), 1985b. 

M. Shinozuka and T, Nomoto, Response Variability Due to Spatial Randomness, Report No. CU-CEEM-MS-80-1, Department of Civil Engineering and Engineering Mechanics, Columbia University, August 1980. 

Kinetic analysis of the propagation and termination rate constants in the polymerization of MMA-EGDMA mixtures is more difficult than for MM A. An understanding of the polymerization depends on knowledge of the individual concentrations of C=C bonds that are (1) in monomer molecules and (2) attached to polymer molecules. This information may be obtainable for NMR spectra utilizing differences in the relaxation times of the two types of C=C environments. However, it is likely that the polymerization is heterogeneous with a non spatially random distribution of crosslinks. 

The entropy change is negative and quite large mainly due to the fact that the random O2 gas becomes locahzed in the red solid HgO, which is a drastic reduction in spatial randomness. We note that if A5 is negative, the process is going toward a more ordered state. That is quite against the natural tendency of entropy to increase, but if we compute the heat of the reaction, we see that the reaction is also very exothermic, so we could say in this case the energy released in the reaction makes it possible for the reaction to proceed to a more ordered state (Table 5.1) ... 

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