Complex systems periodic potentials

Erom a general experience with wave-packet motion in periodic potentials [237], it may be expected that the complexity of the dynamics is partially caused by the symmetric excitation of the system (i.e., at wave function right from the beginning. To simplify the analysis, it is therefore helpful to invoke an initial preparation that results in a preferred direction of motion of the system. With this end in mind, we next assume that the initial wave packet contains a dimensionless average momentum of po = 23.24, corresponding to an... 

In the third period, which ended in 1999 after the book VIG was published, various fluids had been studied strongly polar nonassociated liquids, liquid water, aqueous solutions of electrolytes, and a solution of a nonelectrolyte (dimethyl sulfoxide). Dielectric behavior of water bound by proteins was also studied. The latter studies concern hemoglobin in aqueous solution and humidified collagen, which could also serve as a model of human skin. In these investigations a simplified but effective approach was used, in which the susceptibility % (m) of a complex system was represented as a superposition of the contributions due to several quasi-independent subensembles of molecules moving in different potential wells (VIG, p. 210). (The same approximation is used also in this chapter.) On the basis of a small-amplitude libration approximation used in terms of the cone-confined rotator model (GT, p. 238), the hybrid model was suggested in Refs. 32-34 and in VIG, p. 305. This model was successfully employed in most of our interpretations of the experimental results. Many citations of our works appeared in the literature. 

The upshot is that the measured voltage necessary to drive a cell with the minimum two electrodes is a complex mixture of potentials at both electrodes together with various voltage losses in the system, and we do not know, if the required voltage should suddenly be seen to rise after a period of electrolysis, whether this is due to some time-dependent effect at the anode, the cathode or elsewhere in the system. 

What cannot be seen in the abstract are the several themes of this short (seven-page) paper. These themes set standards both for the quality of simulation and the style of attack on complex systems. The paper discusses the methodology of numerical finite-difference integration of the dynamical equations of motion in detail in an appendix, with proper attention to numerical accuracy. The limitations of the simple pair-additive interatomic potential, of the cutoff range of the interaction, and of the periodic boundary conditions are also all noted. Validation of the underlying potential model is seriously considered via direct comparison with available experimental data for the atomic diffusion constant and for the interatomic-pair distribution function, from X-ray scattering. 

On the micro-level of the brain, there are massively many-body-problems which need a reduction strategy to handle with the complexity. In the case of EEG-pictures, a complex system of electrodes measures local states (electric potentials) of the brain. The whole state of a patient s brain on the micro-level is represented by local time series. In the case of, e.g., petit mal epilepsy, they are characterized by typical cyclic peaks. The microscopic states determine macroscopic electric field patterns during a cyclic period. Mathematically, the macroscopic patterns can be determined by spatial modes and order parameters, i.e., the amplitude of the field waves. In the corresponding phase space, they determine a chaotic attractor characterizing petit mal epilepsy. 

The processes of cathodic protection can be scientifically explained far more concisely than many other protective systems. Corrosion of metals in aqueous solutions or in the soil is principally an electrolytic process controlled by an electric tension, i.e., the potential of a metal in an electrolytic solution. According to the laws of electrochemistry, the reaction tendency and the rate of reaction will decrease with reducing potential. Although these relationships have been known for more than a century and although cathodic protection has been practiced in isolated cases for a long time, it required an extended period for its technical application on a wider scale. This may have been because cathodic protection used to appear curious and strange, and the electrical engineering requirements hindered its practical application. The practice of cathodic protection is indeed more complex than its theoretical base. 

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