Closed circuit theory

Steady State Migration Fluxes in Multicomponent Electrolytes and the Central Problem with Closed Circuit Theory... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The theoretical model that best describes regulation of transepithelial transport is derived from the Ussing-Zerahn equivalent electrical circuit model of ion transport theory [57] (Figure 15.1B). The model predicts that epithelia are organized as a layer(s) of confluent cells, where plasma membranes of neighboring cells come into close contact and functionally occlude the intercellular space. Accordingly, molecules can move across epithelia either through the cells... 

The reduction of Cd(II) ions on DME was also investigated in 1 M perchlorate, fluoride and chloride solutions using dc, ac admittance, and demodulation methods [27]. It was found that in the perchlorate supporting electrolyte, the reduction mechanism is also CEE, and that the rate constant of the chemical step is quite close to the value characteristic for fluoride solutions. The theories available at present could not be applied to the Cd(II) reduction in chloride solution because of the inapplicability of the Randles equivalent circuit. 

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero,... 

Thus, a plot of (Csc) 2 vs. B should be linear with an intercept at the flat-band potential. This linearity can be used as an immediate test of the theory and an example is shown in Fig. 20 for p-GaAs [62]. The capacitance derived from the two-component circuit is almost independent of frequency for potentials more than 0.5 V from the flat-band potential and the intercept is clearly defined. Similar results for both n-GaAs and p-GaAs are shown in Fig. 21 [63] and an important check on the theory is that the flatband potentials for the n- and p-type electrodes differ by ca. 1.4 V, corresponding to a bandgap of 1.4 eV for GaAs. This is expected as the Fermi level will be very close to the band edges in the bulk for these wide band materials. 

Group (1) in based on the tree-like model with uncorrelated circuit closing in the gel, while the theories of group (2) more or less rigorously simulate spatial correlations manifested particularly by cyclization. 

The behaviour of the resistance Is as expected. At low salt concentration R decreases with increasing surface charge on the latex, but at high there Is no difference between the latlces because now the bulk conductivity dominates (R R ). There Is a tendency of R to level off at very low electrolyte concentration then the particles are effectively so close to each other that they are "short-circuited", leading to a constant polarization. The theory (capillary model) applies semlquantltatlvely it becomes more defective at low because polarization Is not accounted for. 

Despite the fact that open-system behavior in mineral-isotopic systems is governed by a combination of volume and short-circuit diffusional processes, most thermochronologists make the simplifying assumption that volume diffusion alone controls the open- to closed-system transition that is so important to thermochronologic theory. There are a variety of reasons to believe that this assumption is reasonable. First, the stmcture of short-circuit pathways is such that they should be characterized by much faster diffusion than the intact crystal structure that surrounds them if so, then the rate of daughter isotope loss should be limited by the rate of diffusion out of intact domains and into short-circuit pathways. Second, the volumetric proportion of short-circuit pathways to intact domains is small in all but the most strongly deformed natural crystals, implying that their contribution to bulk... 

A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. 

---