Phase space structure dividing surface

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

Figure 9.18 Surfaces of section for the HCCH [Nj, l] = [4,0] and [8,0] polyads. The surface section for the [4,0] polyad shows that all of phase space is divided between cis-bend and trans-bend normal modes. The phase space structure for the [8, 0] polyad contains large scale chaos as well as at least two new qualitative behaviors (from Jacobson, et al., 1999).

Locally, the 2d — l)-dimensional energy surface E(E) has the structure of X R, i.e., fhe Cartesian product of a 2d — 2)-dimensional sphere and a line, in fhe 2rf-dimensional phase space. The energy surface S(E) is splif locally info two components, "reactants" and "products," by a (2d — 2)-dimensional "dividing surface" fhaf is diffeomorphic to and which we therefore denofe by S (E). The dividing surface that we construct has the following properties ... 

Amorphous silicas play an important role in many different fields, since siliceous materials are used as adsorbents, catalysts, nanomaterial supports, chromatographic stationary phases, in ultrafiltration membrane synthesis, and other large-surface, and porosity-related applications [16,150-156], The common factor linking the different forms of silica are the tetrahedral silicon-oxygen blocks if the tetrahedra are randomly packed, with a nonperiodic structure, various forms of amorphous silica result [16]. This random association of tetrahedra shapes the complexity of the nanoscale and mesoscale morphologies of amorphous silica pore systems. Any porous medium can be described as a three-dimensional arrangement of matter and empty space where matter and empty space are divided by an interface, which in the case of amorphous silica have a virtually unlimited complexity [158],... 

Similar to unstable periodic orbits, an NHIM has stable and unstable manifolds that are of dimension 2 — 2 and are also structurally stable. Note that a union of the segments of the stable and unstable manifolds is also of dimension 2n — 2, which is only of dimension one less than the energy surface. Hence, as far as dimensionality is concerned, it is possible for a combination of the stable and unstable manifolds of an NHIM to divide the many-dimensional energy surface so that reaction flux can be dehned. However, unlike the fewdimensional case in which a union of the stable and unstable manifolds necessarily encloses a phase space region, a combination of the stable and unstable manifolds of an NHIM may not do so in a many-dimensional system. This phenomenon is called homoclinic tangency, and it is extensively discussed in a recent review article by Toda [17]. 

Figure 1. A. Computer graphic portion of a periodic surface of constant mean curvature, having the same space group and topological type as the Schwarz D minimal surfhce. This surbce, together with an identical displaced copy, would represent the polar/apolar dividing surface in a cubic phase with space group 224 (Pn3m). The two graphs shown would thread the two aqueous subspaces. B. Computer graphic of a portion of the Schwarz D minimal sur ce (mean curvature identically zero). In the 224 cubic phase structure, this sur ce would bisect the surfactant bilayer.

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