Molecular trajectories

On purely kinetic grounds, however, the term random must be used carefully in describing a MaxweUian gas. The probabUity of a MaxweUian gas entering a duct is not a random function. This probabUity is proportional to the cosine of the angle between the molecular trajectory and the normal to the entrance plane of the duct. The latter assumption is consistent with the second law of thermodynamics, whereas assuming a random distribution entry is not. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

Finally, we were led to the last stage of research where we treated the crystallization from the melt in multiple chain systems [22-24]. In most cases, we considered relatively short chains made of 100 beads they were designed to be mobile and slightly stiff to accelerate crystallization. We could then observe the steady-state growth of chain-folded lamellae, and we discussed the growth rate vs. crystallization temperature. We also examined the molecular trajectories at the growth front. In addition, we also studied the spontaneous formation of fiber structures from an oriented amorphous state [25]. In this chapter of the book, we review our researches, which have been performed over the last seven years. We want to emphasize the potential power of the molecular simulation in the studies of polymer crystallization. 

Fig. 16 Molecular trajectory in the early stage of chain adsorption onto the thin substrate. The chain atoms on the crystalline substrate and the amorphous substrate are depicted in dark gray and in white, respectively. The presence of longloops in the amorphous region is quite pronounced...

Figure 21.6. Schematic representation of the relative phase-space volumes available to reactant, transition state, and product. A plane located at the most constricted place has the highest prohahility of being crossed only once by a molecular trajectory, which is the location of the transition state.

Figure 7. SiH4 molecular trajectories in free molecular flow in a very low pressure (1 Pa) hot-wall CVD reactor for Si epitaxy.

If the tracer is composed of the same species as that of the solid host, then the diffusion coefficient is named the tracer self-diffusion coefficient, where DA is the tracer self-diffusion coefficient. It is necessary to clarify that self-diffusion is a particle transport process that takes place in the absence of a chemical potential gradient [13]. This process is described, as explained later, by following the molecular trajectories of a large number of molecules, and determining their mean square displacement (MSD). 

Fig. 38 Molecular trajectories of a crystallizing chain (thick line) selected at random from 64 chains during crystallization at 370 K a at 0.128 ns, b at 0.64 ns, c at 1.28 ns, d at 2.56 ns, e at 6.40 ns, f 12.8 ns. Also shown are the growing crystalline domains of Fig. 12 (thin parallel lines). Pictures are all side view along the x-axis...

Figure 8.2. Simplified block diagram of the apparatus used to study the magnetic resonance spectrum of H2. The molecular beam source was cooled to liquid nitrogen temperature. M denotes fractionating pumps, whilst S and F are diffusion pumps. The appropriate molecular trajectories are not shown in this diagram the reader is referred back to figure 8.1.

With rate processes the situation is less clear-cut, In principle, ab initio calculations permit a more sophisticated approach to rate processes than that based on the transition state theory (TST) and represented by eqns. (5.14) and (5.l6)-(5.21). One may calculate the whole energy hypersurface and apply to it molecular trajectory calcu-... 

There is no sensitivity to individual molecular trajectories or dipole orientations, but one ends up directly with global figures to characterize the emitted fluorescence. Besides, distinguishing between the contributicms of the radiative rate and the collection efficiency remains a challenge, mainly because of the intrinsic difficulty to reliably measure collection efficiency. Lastly, the fluorescence enhancement factors are spectrally averaged within the fluorescence bandpass detection window. However, further investigations can provide some additional knowledge on these last two points, as we will discuss hereafter. 

The advent of methods of single molecule manipulation [6,7] and single molecule detection [8-10] have made it possible for the first time to follow the molecular trajectories of these motors and describe in increasing detail their dynamics. The variables that are more easily detected by these methods are force, displacement, and time. These quantities are also the ones of greatest functional value to understand the mechanochemical operation of these motors. 

Mechanisms and Molecular Trajectories for Hydroxylation by Cytochrome P450... 

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