Spatial evolution

Figure 3. Temporal and Spatial Evolution of Reaction Rates in the Liquid Phase Reaction Zone. Rates were calculated as a function of time and distance from the bubble surface assuming only conductive heat transport from a sphere with radius 150ym at 5200K, embedded in an infinite matrix at 300K.

The strategy in a molecular dynamics simulation is conceptually fairly simple. The first step is to consider a set of molecules. Then it is necessary to choose initial positions of all atoms, such that they do not physically overlap, and that all bonds between the atoms have a reasonable length. Subsequently, it is necessary to specify the initial velocities of all the atoms. The velocities must preferably be consistent with the temperature in the system. Finally, and most importantly, it is necessary to define the force-field parameters. In effect the force field defines the potential energy of each atom. This value is a complicated sum of many contributions that can be computed when the distances of a given atom to all other atoms in the system are known. In the simulation, the spatial evolution as well as the velocity evolution of all molecules is found by solving the classical Newton equations of mechanics. The basic outcome of the simulation comprises the coordinates and velocities of all atoms as a function of the time. Thus, structural information, such as lipid conformations or membrane thickness, is readily available. Thermodynamic information is more expensive to obtain, but in principle this can be extracted from a long simulation trajectory. 

Initially, Oz diffuses through the bentonite and granitic domains, controlling the redox state of the system. Once 02 is exhausted, granitic groundwater controls the redox state of the system. The results of these calculations performed with the PHREEQC geochemical code (Parkhust Appelo 1999) clearly indicate that there is a substantial variability in pH/pe space along the temporal and spatial evolution of the near field of a repositoiy. This has clear consequences for the subsequent interactions with the Fe canister material and finally with the spent fuel matrix. 

Concentration grating Due to the Ludwig-Soret effect, the temperature grating is the driving force for a secondary concentration grating, which starts to build up and is superimposed upon the thermal one. Its temporal and spatial evolution is obtained from the one-dimensional form of the extended diffusion equation... 

Undoubtedly, the most promising modehng of the cardiac dynamics is associated with the study of the spatial evolution of the cardiac electrical activity. The cardiac tissue is considered to be an excitable medium whose the electrical activity is described both in time and space by reaction-diffusion partial differential equations [519]. This kind of system is able to produce spiral waves, which are the precursors of chaotic behavior. This consideration explains the transition from normal heart rate to tachycardia, which corresponds to the appearance of spiral waves, and the fohowing transition to fibrillation, which corresponds to the chaotic regime after the breaking up of the spiral waves, Figure 11.17. The transition from the spiral waves to chaos is often characterized as electrical turbulence due to its resemblance to the equivalent hydrodynamic phenomenon. 

Basu A. R., Poreda R. J., Renne P. R., Teichmann F., Vasiliev Y. R., Sobolev N. V., and Turrin B. D. (1995) High-He-3 plume origin and temporal spatial evolution of the Siberian flood basalts. Science 269, 822—825. 

Horan M. F., Walker R. J., Fedorenko V. A., and Czamanske G. K. (1995) Osmium and neodymium isotopic constraints on the temporal and spatial evolution of Siberian flood basalt sources. Geochem. Cosmochim. Acta 59, 5159—5168. 

An independent and very important sphere for the application of operator-network approaches is represented by evolutionary stereochemistry, where the structural-electronic level of mapping chemical properties and their mutual transmutations are considered in terms of their spatial evolution, i.e. the variation of their symmetry and configurational characteristics (the latter for chiral structures) is considered. 

Feistel, R., Nausch, G., Matthaus, W., Hagen, E., 2003a. Temporal and spatial evolution of the Baltic deep water renewal in spring 2003. Oceanologia, 45(4), 623-642. 

The MBI in January 2003 and the temporal and spatial evolution of the Baltic deepwater renewal have been investigated in detail by Feistel et al. (2003b). The propagation of the saline and oxygen-rich water into the Baltic Sea and the ventilation of anoxic water between Bomhohn Basin and central Baltic were recorded by the Darss Sill measuring mast, the Arkona Basin buoy, a subsurface mooring in the Eastern Gotland Basin (cf. Chapter 3), and several research cruises. 

Thomas GB Jr, Finney RL (1996) Calculus and Analytic Geometry. Addison-Wesley Publishing Company, 9th Edition, Reading, Massachusetts Tomiyama A, Miyoshi K, Tamai H, Zun 1, Sakaguchi T (1998) A bubble tracking method for the prediction of spatial evolution of bubble flow in a vertical pipe. Third International Conference on Multiphase Flow, Lyon, France Trapp JA (1986) The mean flow character of two-phase flow equations. Int J Multiphase Flow 12(2) 263-276... 

But by variation of the gradient strength G, the time axes for spectroscopic and for spatial evolution can be separated. By dividing the time axis t into two parts,... 

In previous work of the same group (72), the electrokinetic injection of DNA fragments was optimized as well by means of a simplex method. CGE-LIF was also used. In this case, BGE concentration, sample injection voltage, and time were the factors to be optimized. The optimum conditions were reached after only nine experiments. Figure 6.5 shows the spatial evolution of the simplex method used in this work (the initial tetrahedron (vertices 1-4) and the subsequent movements of reflection and contraction). Vertex 9 was considered as the optimum for injection of the Ikbp DNA ladder (l.OmM TTE buffer, 20s injection, 55V/cm electric field injection). 

FIGURE 6.5. Spatial evolution of the simplex optimization. The sohd bold lines link the initial conditions (vertices 1-4).The dashed lines show the simplex figure after the radical contraction (vertices 4,7-9) and the first reflection after contraction (vertex 10, dotted lines). The arrow shows the best result. Reprinted from Reference 72 with permission from Wiley-VCH Verlag. 

We investigate the spatial evolution of the optical fields propagating in the Raman coupler depicted in Fig. 28. Working in the Heisenberg picture, we solve a set of Heisenberg-Langevin equations for the z-dependent field operators. It is convenient to employ momentum operator G rather than Hamiltonian H to describe the operation of the coupler because this approach allows us to take dispersion into account. 

Abundant spins with a high gyromagnetic ratio, e.g., protons, lead in the absence of line-narrowing methods to unresolved spectra described by the Hamiltonian of Equations (4.1) and (4.2). The only relevant observable is the spatial distribution of the polarization, P(r t). Using the master-equation approach (Equation (4.8)), the spatial evolution of the polarization is described by a random walk on a grid that can be approximated by a diffusion equation... 

In this chapter, we use the results of numerical infiltration experiments in dual porosity media performed with a three-dimensional lattice-gas model to characterize preferential flow as response to rainfall intensity. From the temporal and spatial evolution of the water content during infiltration and drainage, we evaluate the adequacy of a kinematic wave approximation to describe the flow. We also discuss the conceptual basis of the asymptotic kinematic approach to Richards equation in comparison with the macropore kinematic equation. 

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