Disordered systems, theory

Nov. 21, 1931, Tbilisi, Georgia, USSR - May 13, 1985) Dogonadze was one of the founders of the new science - electrochemical physics [i]. The main scientific interests of Dogonadze were focused on condensed-phase reactions. His pioneering works of 1958-59 have laid the foundations of the modern quantum-mechanical theory of elementary chemical processes in electrolyte solutions. He developed a comprehensive quantum-mechanical theory of the elementary act of electrochemical reactions of -> electron and -> proton transfer at metal and - semiconductor electrodes [ii—v]. He was the first to obtain, by a quantum-mechanical calculation, the expression for the electron transfer probability, which was published in 1959 in his work with -> Levich. He conducted a number of studies on the theory of low-velocity electrons in disordered systems, theory of solvated electrons, and theory of photochemical processes in solutions. He made an impressive contribution to the theory of elementary biochemical processes [vi]. His work in this area has led to the foundation of the theory of low-temperature -> charge-transfer processes cov-... 

Gravitis, J. Lignin structure and properties from the viewpoint of general disordered systems theory. In Kennedy, J.F., Phillips, G.O., Williams, P.A. (eds.) Lignocellulosic Science, Technology, Development and Use, pp. 613-627. Ellis Horwood, New York (1992)... 

In the framework of the mobile order and disorder (MOD) theory five components contribute most to the Gibbs free energy of partitioning of a solute in a biphasic system of two essentially immiscible solvents [23] ... 

The effect can be applied, for example, to estimate a bond length or atomic spacing, to observe valence electron spin distribution around a specific atom and to derive information of the nearest neighbor atom distribution in a disordered system such as amorphous, under an expansion of the theory. 

This branch of polymer physics is closely connected to another of I.M. Lif-shitz favorite directions in physics, namely, the theory of disordered systems [73]. The situation when different samples have only statistical similarity is typical for the physics of chaos, and many concepts I.M. Lifshitz developed are also quite naturally applied in the physics of disordered polymers. The idea of self-averaging in general and self-averaging of free energy [74], in particular, are examples of such concepts. 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Another perspective comes from famiiy systems theory, which characterizes the schizophrenic patient s family as having disordered communication, with various members playing unusual or aberrant roles. According to this theory, patients experience double binds when faced with contradictory expectations ( 7). Related controversial hypotheses held that the schizophrenogenic mother was the critical factor and then later that the schizophrenic s father also played a significant role (8, 9 and 10). Intensive therapy, in the context of in-hospital separation from the family, was considered the treatment of choice. In contrast to classic psychodynamic therapy, which focuses on the individual patient, this approach attempts to resolve conflicts in the family system, as well as in the patient s psyche. Typically, this involves sessions that include all or as many members as possible. Thus, even though one member is identified as the patient, it is the disturbed communication and interactions among all members that is the focus of therapy. This and subsequent therapeutic approaches may be most effective when medication is used concurrently. 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

I.M. Lifshits, S.A. Gradeskul and L.A. Pastur, Introduction to the Theory of Disordered Systems, Nauka, Moscow, 1982 (in Russian). 

The theory predicts a strong dependence of photogeneration efficiencies on the field and it approaches unity at high field. The temperature sensitivity decreases with the increase in field. The theory has found satisfactory explanations in the photogeneration process in many organic disordered systems, such as PVK (Scheme la) [25], and triphenylamine doped in polycarbonate [26], Figure 4 shows an example of the field dependence of c() calculated from Eq. (22) (the solid lines) to fit the quantum efficiency data at room temperature for hole and electron generation in an amorphous material. The material consists of a sexithiophene covalently linked with a methine dye molecule (compound 1) (Scheme 2). 

See, for instance, M. Lax, in Multiple Scattering and Waves in Random Media, P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Eds., North Holland, 1981, and references therein J. Klafter and M. S. Schlesinger, Proc. Nat. Acad. Sci. U.S.A. 83, 848 (February 1986), and references therein R. Brown, Thesis, University of Bordeaux 1,1987 R. Brown et al. J. Phys. C20, L649 (1987) 21 (1988) in press. (This last work thoroughly discusses the applicability of fractal theory to isotopically mixed crystals as disordered system. Serious criticism is presented both of the analysis of the experimental data and of the fundementals of their description in the present theory of fractals. For this reason we omit all works treated there. 

The relaxation process may be accompanied by diffusion. Consequently, the mean relaxation time for such kinds of disordered systems is the time during which the relaxing microscopic structural unit would move a distance R. The Einstein-Smoluchowski theory [226,235] gives the relationship between x and R as... 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

In 1991, Bonny and Leuenberger [40] explained the changes in dissolution kinetics of a matrix controlled-release system over the whole range of drug loadings on the basis of percolation theory. For this purpose, the tablet was considered a disordered system whose particles are distributed at random. These authors derived a model for the estimation of the drug percolation thresholds from the diffusion behavior. 

The Effective Medium Approximation (EMA), based in some assumptions, allows us to employ linear regressions as an approximation of the behavior of a disordered system outside the critical range. Based on EMA theory, two linear regressions have been performed as an approximation for estimating the percolation threshold as the point of intersection between both regression lines (see Figures 43 15). The values of the excipient percolation thresholds estimated for all the batches studied, based on the behavior of the kinetic parameters, ranged from 25.99 to 26.77%. 

The variation of 5(7) near the N-I phase transition will be measured in this experiment and will be compared with the behavior predicted by Landau theory, " " which is a variant of the mean-field theory first introduced for magnetic order-disorder systems. In this theory, local variations in the environment of each molecule are ignored and interactions with neighbors are represented by an average. This type of theory for order-disorder phase transitions is a very useful approximate treatment that retains the essential features of the transition behavior. Its simplicity arises from the suppression of many complex details that make the statistical mechanical solution of 3-D order-disorder problems impossible to solve exactly. 

The use of the percolation model to analyze the d.c. conductivity in hydrated lysozyme powders (Careri et al., 1986, 1988) and in purple membrane (Rupley et al, 1988) introduces a viewpoint from statistical physics that is relevant to a wide range of problems originating in disordered systems. Percolation theory is described in the appendix to this article, for readers unfamiliar with it. Here, we discuss the significance of percolation specihcally for protein hydration and function. 

The power of percolation theory is its offering of a unitary treatment for widely different phenomena occurring in disordered systems. The central feature of percolation theory is the existence of a threshold at which long-range connectivity suddenly emerges. 

The mean field approximate theories, ATA and CPA, are final simplifications of multiple scattering approaches to simulate a real disordered system by an equivalent effective system. 

Lattice methods are well suited to treatment of the intermolecular, or steric , part of the configuration partition function for a system comprising particles or molecules in which volume exclusion plays its usual, dominant role. In principle, these well established methods are no less applicable to systems consisting of species of asymmetric shape, although certain modifications of the conventional procedures are required in order to accommodate highly asymmetric molecules on a lattice when they maintain a degree of orientational disorder. Lattice theory has been adapted to this purpose and the practicability of this approach has been demonstrated... 

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. 

II. Statistical Theory of EELS of the Simple Disordered Systems... 

The direct reconstruction of a disordered system by EELS is an important technique by virtue of its wide applicability. The theory for this problem requires only simple expressions that connect the inelastic cross section with the structural characteristics of the system. The second goal of this chapter will be to develop this theory. Note that the modern source of information about distribution functions is the Fourier transform of the static structure factors for a system. 

II. STATISTICAL THEORY OF EELS OF THE SIMPLE DISORDERED SYSTEMS... 

J. Klafter and M. F. Shlesinger, On the Relationship Among Three Theories of Relaxation in Disordered Systems, Proc. Natl. Acad. Sci., USA, 83 (1986) 848. 

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