Phase-space transition states temperature

At a higher temperature T = 11.0, for flow rates near the transition rate c, the free-energy barrier between the coiled and stretched conformation is much lower than that for T = 9.0. The chain can therefore explore the phase space and jump back and forth from the coiled to the stretched state. Similar behavior has already been observed in [59] and [60]. Figure 27 illustrates this feature. 

The Allan variance analyses of energy fluctuations can tell us about the nature of the configurational energy landscapes and the existence of (multidimensional) cooperativity among individual DOFs, enhanced for an intermediate time scale at the transition temperature. However, what can we learn or deduce from an (observed) scalar time series about the geometrical aspects of the underlying multidimensional state (or phase) space buried in the observations The so-called embedding theorems attributed to Whitney [75] and Takens [76] provide us with a clue to the answer of such a question (see also Section VI.A and Refs. 77-80). 

One of the problems with VMC is that it favors simple states over more complicated states. As an example, consider the liquid-solid transition in helium at zero temperature. The solid wave function is simpler than the liquid wave function because in the solid the particles are localized so that the phase space that the atoms explore is much reduced. This biases the difference between the liquid and solid variational energies for the same type of trial function, (e.g. a pair product form, see below) since the solid energy will be closer to the exact result than the liquid. Hence, the transition density will be systematically lower than the experimental value. Another illustration is the calculation of the polarization energy of liquid He. The wave function for fully polarized helium is simpler than for unpolarized helium because antisymmetry requirements are higher in the polarized phase so that the spin susceptibility computed at the pair product level has the wrong sign ... 

In the previous section we demonstrated how variational transition state theory may be usefully applied to systems described by a space- and time-dependent generalized Lan-gevin equation. The harmonic nature of the bath implicit in the STGLE led to a compact analytical expression for the optimized planar dividing surface result. Except for very low temperatures, most reactive systems cannot be described in terms of a harmonic bath. In this section we demonstrate how the VTST formalism may be applied to general condensed phase reactive systems. For a recent review, see Ref. 80. 

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