Phase-space transition states dimensions

GEOMETRY OF PHASE-SPACE TRANSITION STATES MANY DIMENSIONS, ANGULAR MOMENTUM... 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

This paper by Ya.B. helped lay the foundation for the study of the kinetics of phase transitions of the first kind. It considers the fluctuational formation and subsequent growth of vapor bubbles in a fluid at negative pressures. It is assumed that the fluid state is far from the boundary of metastability and that the volume of the bubbles formed is still small in comparison with the overall volume of the fluid. The first assumption ensures slowness of the process the time of transition to another phase is large compared to the relaxation times of the fluid per se. This allows the application of the Fokker-Planck equation in the space of embryo dimensions to describe the growth of the embryos. 

The location of the saddle point in phase space is specified by and Pi = 0, where qi is the reaction coordinate. On top of the saddle point, the reaction coordinate is completely separated from the rest of the degrees of freedom. Therefore, a set of orbits where Pi) is fixed on the saddle point while the rest are arbitrary is invariant under dynamical evolution. Its dimension in phase space is 2n — 2. Such invariant manifolds are considered as the phase-space structure corresponding to transition states, and will play a crucial role in the following discussion. 

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