Disordered systems, theory 1-dimensional

The relationship between excitation transport and fluorescence depolarization in two and three dimensional disordered systems has been discussed by Anfinrud and Struve . In the usual discussion of excitation transport by dipole-interaction it is conventional to assume that excitation is completely depolarized after a single hop. This supposition has been critically examined and a theory formulated suitable for application to Langmuir Blodgett films and absorbed species. 

Percolation theory represents one of the simplest models of disordered systems. It was developed to mathematically deal with disordered media, in which the disorder is defined by a random variation in the degree of connectivity [59]. The main concept of percolation theory is the existence of a percolation threshold, above which the physical property of whole system dramatically changes. A typical example of a percolation problem is that of the site percolation on a simple two-dimensional square lattice, as shown in Figure 10. The relevant entities could be eitlier the squares determined by the gridlines or the points where these lines intersect. If the squares are chosen to be considered, this problem is called the site percolation, while if the points are chosen to be considered, it is called the bond percolation. For the example shown in Figure 10, the squares arc chosen to be the relevant... 

One of the most powerful theoretical tools to account for charge carrier transport in disordered systems is provided by the percolation theory as described in numerous monographs (see, for instance, [15, 18]). According to the percolation theory, one has to connect sites with fastest transition rates in order to fiilfil the condition that the average number Z of connected bonds per site is equal to the so-called percolation threshold Be. In the three-dimensional case this threshold is [15,41,42] ... 

Percolation theory is helpful for analyzing disorder-induced M-NM transitions (recall the classical percolation model that was used to describe grain-boundary transport phenomena in Chapter 2). In this model, the M-NM transition corresponds to the percolation threshold. Perhaps the most important result comes from the very influential work by Abrahams (Abrahams et al., 1979), based on scaling arguments from quantum percolation theory. This is the prediction that no percolation occurs in a one-dimensional or two-dimensional system with nonzero disorder concentration at 0 K in the absence of a magnetic field. It has been confirmed in a mathematically rigorous way that all states will be localized in the case of disordered one-dimensional transport systems (i.e. chain structures). 

Questions that had been of fundamental importance to quantum chemistry for many decades were addressed. When the existence of bond alternation in trans-polyacetylene was been demonstrated [14,15], a fundamental issue that dates to the beginnings of quantum chemistry was resolved. The relative importance of the electron-electron and electron-lattice interactions in Ti-electron macromolecules quickly emerged as an issue and continues to be vigorously debated even today. Aspects of the theory of one-dimensional electronic structures were applied to these real systems. The important role of disorder on the electronic structure and properties of these low dimensional metals and semiconductors was immediately evident. The importance of structural relaxation in the excited state (solitons, polarons and bipolarons) quickly emerged. 

On physical grounds (see Section 10.3 and (15)) one should expect that low-energy polaritons in 2D organic microcavities would also be rendered strongly localized by disorder, as it would also follow from the general ideas of the theory of localization (39). Further work on microscopic models of two-dimensional polariton systems is required to quantify their localization regimes. 

In the recent past, analytical research in Celestial Mechanics has centred on KAM theory and its applications to the dynamics of low dimensional Hamiltonian systems. Results were used to interpret observed solutions to three body problems. Order was expected and chaos or disorder the exception. Researchers turned to the curious exception, designing analytical models to study the chaotic behaviour at resonances and the effects of resonant overlaps. Numerical simulations were completed with ever longer integration times, in attempts to explore the manifestations of chaos. These methods improved our understanding but left much unexplained phenomena. 

The recent experimental confirmation of the existence of one-dimensional metallic systems has led to a rapid increase in the experimental and theoretical study of these conducting systems. The objective of this section is to acquaint the reader with the physical basis of the concepts currently being used to explain the experimental results. Emphasis is given to the development of one electron band theory because of its central importance in the description of metals and understanding the effects of lattice distortion (Peierls transition), electron correlation, disorder potentials, and interruptions in the strands. It... 

Buzza and Gates (102) also addressed the question whether disorder or the increased dimensionality from two to three dimensions is responsible for the observed experimental behavior of the shear modulus. In particular, they explored the lack of the sudden jump in G from zero to a finite value at 0 = 0Q that is predicted by the perfectly ordered 2-D model. We have seen above that disorder appears to remove that abrupt jump in two dimensions (90). For drops on a simple cubic lattice, Buzza and Cates analyzed the drop deformation in uniaxial strain close to 0 = 0q, first using the model of truncated spheres . (For reasons given above, we believe this to be a very poor model.) They showed that this model did not eliminate the discontinuous jump in G. An exact model, based on a theory by Morse and Witten (103) for weakly deformed drops, led to G a 1/ In (0 - 0q), which eliminates the discontinuity, but still shows an unrealistically sharp rise at 0 = 0q and is qualitatively very different from the experimentally observed linear dependence of G on (0 - 0q). Similar conclusions were reached by Lacasse and coworkers (49, 104). A simulation of a disordered 3-D model (104) indicated that the droplet coordination number increased from 6 at to 10 at 0 = 0.84, qualitatively similar to what is seen in disordered 2-D systems (90). Combined with a suitable (anharmonic) interdroplet force potential, the results of the simulation were in close agreement with experimental shear modulus and osmotic pressure data. It therefore appears again that disor-... 

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