Analytic theory

Studies of double carrier injection and transport in insulators and semiconductors (the so called bipolar current problem) date all the way back to the 1950s. A solution that relates to the operation of OLEDs was provided recently by Scott et al. [142], who extended the work of Parmenter and Ruppel [143] to include Lan-gevin recombination. In order to obtain an analytic solution, diffusion was ignored and the electron and hole mobilities were taken to be electric field-independent. The current-voltage relation was derived and expressed in terms of two independent boundary conditions, the relative electron contributions to the current at the anode, JgU, and at the cathode, 

Clearly, the optimal injection condition is for electrons and holes to be in balance, but this does not necessarily guarantee that b = l. For example, when both contacts are injection limited and the injected charge densities are small, the 

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An early analytic theory by Hoeve accurately predicts the number of loops ni(s) and trains n,(s) having s segments... 

Stell G 1977 Fluids with long-range forces towards a simple analytic theory Statistical Mechanics part A, Equilibrium Techniques ed B Berne (New York Plenum)... 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

As we have seen, all the analytic coexistence curves are quadratic in the limit, so for all these analytic theories, tire exponent (3=1/2. 

For analytic theories, y is simply 1, and we have seen that for the van der Waals fluid F / F equals 2. Divergences with exponents of the order of magnitude of unity are called strong . 

It is curious that he never conuuented on the failure to fit the analytic theory even though that treatment—with the quadratic fonn of the coexistence curve—was presented in great detail in it Statistical Thermodynamics (Fowler and Guggenlieim, 1939). The paper does not discuss any of the other critical exponents, except to fit the vanishing of the surface tension a at the critical point to an equation... 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

The coexistence curve is nearly flat at its top, with an exponent p = 1/8, instead of the mean-field value of 1/2. The critical isothemi is also nearly flat at the exponent 8 (detemiined later) is 15 rather than the 3 of the analytic theories. The susceptibility diverges with an exponent y = 7/4, a much stronger divergence than that predicted by the mean-field value of 1. 

Garel T, Orland H and Thirumalai D 1996 Analytical theories of protein folding New Developments in Theoretical Studies of Protein Folding e6 R Elber (Singapore World Scientific) pp 197-268... 

Phase transitions in two-dimensional layers often have very interesting and surprising features. The phase diagram of the multicomponent Widom-Rowhnson model with purely repulsive interactions contains a nontrivial phase where only one of the sublattices is preferentially occupied. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are molecular layers of H2, D2, N2 and CO molecules on graphite substrates. We review the path integral Monte Carlo (PIMC) approach to such phenomena, clarify certain experimentally observed anomalies in H2 and D2 layers, and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are analyzed via PIMC as well. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions where quantum effects play a role. 

Complementing these very well established approaches for the study of any scientific field, namely experiments and analytical theory, very recently, computer simulations have become a powerful tool for the study of a great variety of processes occurring in nature in general [4-6], as well as surface chemical reactions in particular [7]. Within this context, the aim of this chapter is not only to offer a critical overview of recent progress in the area of computer simulations of surface reaction processes, but also to provide an outlook of promising trends in most of the treated topics. 

Experimentally, these functions are usually determined only indirectly via the scattering functions of the whole system or the scattering functions of marked chains (see, e.g., [34]). This is one of the advantages of computer simulations over to experiments. However, in order to make significant statements for experimental systems it is always very important to directly compare computer simulations with experimental investigations as well as analytic theories. 

A. Valance, C. Misbah, D. Temkin, K. Kassner. Analytic theory for parity breaking in lamellar eutectic growth. Phys Rev E 48 92A, 1993. 

The analytic theory outlined above provides valuable insight into the factors that determine the efficiency of OI.EDs. However, there is no completely analytical solution that includes diffusive transport of carriers, field-dependent mobilities, and specific injection mechanisms. Therefore, numerical simulations have been undertaken in order to provide quantitative solutions to the general case of the bipolar current problem for typical parameters of OLED materials [144—1481. Emphasis was given to the influence of charge injection and transport on OLED performance. 1. Campbell et at. [I47 found that, for Richardson-Dushman thermionic emission from a barrier height lower than 0.4 eV, the contact is able to supply... 

Finally, interest in these questions came from a new line of research, namely, the theory of automatic control systems. Here, however, contrary to the analytical theories that are summarized in this review, these new piecewise analytic or piecewise linear phenomena in control systems are nonanalytic by their very essence. They open an entirely new field, which is still in an early stage of development. 

Now Lienard developed a purely geometrical argument that enabled him to bypass, as it were, the analytical theory, which is discussed in Part II of this chapter. His equation is 8... 

Although Li nard s method is purely qualitative, it has the advantage that one can carry out the argument without worrying about the smallness of the parameter p in Eq. (6-17) as we shall see later, this is an essential limitation of the analytical theory. The work of Li6nard has been continued by numerous mathematicians and, although we are unable to go into these details,8 we shall give a few conclusions. 

The foregoing discussion shows that complicated facts require a continuous readjustment of the analytical theory. Historically, the theory of nonlinear oscillations progressed precisely in this manner in the hands of the early pioneers—Lord Rayleigh, van der Pol, Appleton, and others. The following sections give a brief account of some of these investigations. 

From the standpoint of the classical (analytical) theory with which we were concerned in this review, the situation is obviously absurd since each of these two equations is linear and of a dissipative type (since h > 0) trajectories of both of these equations are convergent spirals tending to approach a stable focus. However, if one carries out a simple analysis (see Reference 6, p. 608), one finds that change of equations for = 0, results in the change of the focus in a quasi-discontinuous manner, so that the trajectory can still be closed owing to the existence of two nonanalytic points on the -axis. If, however, the trajectory is closed, this means that there exists a stationary oscillation and in such a case the system (6-197) is nonlinear, although, from the standpoint of the differential equations, it is linear everywhere except at the two points at which the analyticity is lost. 

The equations, especially in the case of ternary systems, are necessarily more complicated, but nothing fundamentally new appears. We shall therefore omit all the analytical theory of such cases. 

Chandra and his coworkers have developed analytical theories to predict and explain the interfacial solvation dynamics. For example, Chandra et al. [61] have developed a time-dependent density functional theory to predict polarization relaxation at the solid-liquid interface. They find that the interfacial molecules relax more slowly than does the bulk and that the rate of relaxation changes nonmonotonically with distance from the interface They attribute the changing relaxation rate to the presence of distinct solvent layers at the interface. Senapati and Chandra have applied theories of solvents at interfaces to a range of model systems [62-64]. 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

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