The geometry of reaction dynamics

Abstract

The geometrical structures which regulate transformations in dynamical systems with three or more degrees of freedom (DOFs) form the subject of this paper. Our treatment focuses on the (2n − 3)-dimensional normally hyperbolic invariant manifold (NHIM) in nDOF systems associated with a centre × • • • × centre × saddle in the phase space flow in the (2n − 1)dimensional energy surface. The NHIM bounds a (2n − 2)-dimensional surface, called a 'transition state' (TS) in chemical reaction dynamics, which partitions the energy surface into volumes characterized as 'before' and 'after' the transformation. This surface is the long-sought momentum-dependent TS beyond two DOFs. The (2n − 2)-dimensional stable and unstable manifolds associated with the (2n − 3)-dimensional NHIM are impenetrable barriers with the topology of multidimensional spherical cylinders. The phase flow interior to these spherical cylinders passes through the TS as the system undergoes its transformation. The phase flow exterior to these spherical cylinders is directed away from the TS and, consequently, will never undergo the transition. The explicit forms of these phase space barriers can be evaluated using normal form theory. Our treatment has the advantage of supplying a practical algorithm, and we demonstrate its use on a strongly coupled nonlinear Hamiltonian, the hydrogen atom in crossed electric and magnetic fields.

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

“…First, expand the canonical transformation Eqs. (23) and (24) with respect to q and p, and then, truncated it at the order 21-st, which is the conventional prescription used in [29]. These actions are close with each other within O…”

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

“…This excludes possibly interesting applications. Settings where a noncompact, general geometric version of normal hyperbolicity may be useful include chemical reaction dynamics [Uze+02] and problems in classical and celestial mechanics [DLS06].…”

Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

“…Chemical reaction is one of the most important subjects of natural science through a wide range of fields not only just for chemistry but also for celestial mechanics, [1][2][3] atomic physics, 4,5 cluster physics, 6 environmental science, 7 and biology. 8 In the studies of chemical reactions, one of the central concepts is the rate constant, the proportion of the number of the species (molecules) in the reactant state reacting to form the products per unit time.…”

“…45 The validity of the usage of perturbation theory to take into account such nonlinearity in the region of rank-one saddle has been ensured by several studies in experiments 46,47 and theories [48][49][50][51][52][53][54][55][56][57] on the regularity of crossing dynamics over the saddle and the corresponding phase space geometrical structure (e.g., a no-return TS) in a wide class of Hamiltonian systems. [4][5][6][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] One can naturally adopt this perturbation theory without loss of generality as far as the total energy of the system is not so very high that any perturbation treatment is invalidated. These developments, however, are all based on the Hamiltonian formalism, which corresponds to isolated systems (i.e., gas phase).…”