Trajectory mapping

Figure 1.146, Stre.ss trajectory map.s of southern Northeast Honshu in the late Cenozoic period, after Tsunakawa and Takeuchi (1986) with a slight addition. oh, . trajectory is drawn by smoothing the inferred stress orientations from the selected dike-swarms with K-Ar dates. Selected major faults with age estimation are also shown for indicating types of stress fields. T Extensional stress field, where ay > a 2>cth , and normal or gravity faulting is preferable. P Compre.ssional, oh > ay, reverse or thrust faulting...

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

Fig. 2.9. The T-values ((a) experimental 4>-values, and (b) 4>md obtained from MD trajectories) mapped onto the three-dimensional structure of goat a-lactalbumin [25]. Reproduced with permission from [25]...

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

Figure 1. Gradient trajectory map of the crystalline urea in the molecular plane. Bond-paths are indicated by heavy lines, those corresponding to the interaction surface crossing the plane are drawn by weaker, while those originating from a (3,-l-3)CP and terminating at a (3,-3)CP by the weakest lines. Reprinted with permission from Ref. [78].

This is a remarkable result it is showing that a certain type of continuous column can be modeled exactly using a simple batch experiment. In other words, the exact trajectory mapped out when boiling a liquid of a certain mixture is exactly the same path followed by the liquid down a continuously operated total reflux column. 

Figure 7.33 A fed-batch trajectory map for complex kinetics containing multiple nodes (multiple CSTR solutions). This map is generated by plotting many constant a trajectories for different feed point and the same sidestream composition C°. Adapted from Ming et al. (2013).

This methodology was successfully applied in this context focusing on the texture of the products. It is also possible to use it with flavour or aroma attributes but the patterns were less obvious than for texture, making the trajectory map less easy to read. 

An isolated system is characterized by a time-independent, translationally invariant and rotationally invariant Hamiltonian. Integration of the classical equations of motion for such a system leads, in the limit of infinite sampling, to a trajectory mapping a microcanonical (Nyf ) ensemble of microstates Assuming an infinite numerical precision, this is also what a standard MD simulation will deliver. 

---