Trajectory properties, direct molecular

A sequence of successive con figurations from a Mon te Carlo simulation constitutes a trajectory in phase space with IlypcrC hem. this trajectory in ay be saved and played back in the same way as a dynamics trajectory. With appropriate choices of setup parameters, the Mon te Carlo m ethod m ay ach leve ec nilibration more rapidly than molecular dynamics. Tor some systems, then. Monte C arlo provides a more direct route to equilibrium sinictural and thermodynamic properties. However, these calculations can be quite long, depentiing upon the system studied. 

The density is a maximum in all directions perpendicular to the bond path at the position of a bond CP, and it thus serves as the terminus for an infinite set of trajectories, as illustrated by arrows for the pair of such trajectories that lie in the symmetry plane shown in Fig. 7.2. The set of trajectories that terminate at a bond-critical point define the interatomic surface that separates the basins of the neighboring atoms. Because the surface is defined by trajectories of Vp that terminate at a point, and because trajectories never cross, an interatomic surface is endowed with the property of zero-flux - a surface that is not crossed by any trajectories of Vp, a property made clear in Fig. 7.2. The final set of diagrams in Fig. 7.1 depict contour maps of the electron density overlaid with trajectories that define the interatomic surfaces and the bond paths to obtain a display of the atomic boundaries and the molecular structure. 

The state estimation technique can also be incorporated into the design of optimal batch polymerization control system. For example, a batch reaction time is divided into several control intervals, and the optimal control trajectory is updated online using the molecular weight estimates generated by a model/state state estimator. Of course, if batch reaction time is short, such feedback control of polymer properties would be practically difficult to implement. Nevertheless, the online stochastic estimation techniques and the model predictive control techniques offer promising new directions for the improved control of batch polymerization reactors. 

The existence of Arnold diffusion is irrelevant to the properties of separatrix manifolds, which still mediate the transport of chaotic trajectories within the regions of phase space they control. However, if Arnold diffusion is present in a given multidimensional system, the possibility exists for chaotic motion initially trapped between two nonreactive (trapped) KAM layers to eventually become reactive. This would presumably manifest itself as an apparent bottleneck to the rate of population decay, as chaotic trajectories slowly leak out from the region occupied by regular KAM surfaces into the portion of phase space more directly accessible to the hypercylinders. However, transport via the Arnold diffusion mechanism typically manifests itself on time scales much larger than those that we observe in numerical simulations (Arnold diffusion usually occurs on the order of thousands of mappings, or vibrational periods), and so it seems improbable that this effect would be observed in a typical reaction dynamics simulation. It would be interesting to characterize the effect of Arnold diffusion in realistic molecular models. 

The molecular dynamics method is conceptually simpler than the Monte Carlo method. Here again, we can compute various averages of the form (2.107) and hence the RDF as well. The method consists in a direct solution of the equations of motion of a sample of N (j= 10 ) particles. In principle, the method amounts to computing time averages rather than ensemble averages, and was first employed for simple liquids by Alder and Wainwright (1957) [see review by Alder and Hoover (1968)]. The problem of surface effects is dealt with as in the Monte Carlo method. The sequence of events is now not random, but follows the trajectory which is dictated to the system by the equations of motion. In this respect, this method is of a more general scope, since it permits the computation of equilibrium as well as transport properties of the system. 

In contrast to the above shown ab initio molecular dynamics approach, quantum wavepacket methods can provide a rigorous description for any chemically important properties, thereby properly taking account of the quantum effects of nuclear dynamics. However, the direct applications of the full quantum theories are often prohibitively difficult for many dimensional systems. Also, the time evolution of quantum wavefunctions generally requires the knowledge of the global PES beforehand, in contrast to the trajectory-based methods such as ab initio molecular dynamics, which demands only local information along the paths used. The latter is often referred to as on-the-fiy method. 

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