Transport and turnstiles in multidimensional Hamiltonian mappings for unimolecular fragmentation : application to van der Waals predissociation

A four-dimensional symplectic (Hamiltonian) mapping of the type studied by Gaspard and Rice is used to model the predissociation of the van der Waals complex He–I2. Phase space structure and unimolecular decay in this mapping are analyzed in terms of a general approach recently developed by Wiggins. The two-dimensional area preserving map obtained by restricting the 4D map to the T-shaped subspace is studied first. Both the Davis–Gray theory and the analog of the alternative RRKM theory of Gray, Rice, and Davis for discrete maps are applied to estimate short-time decay rates. A four-state Markov model involving three intramolecular bottlenecks (cantori) is found to give a very accurate description of decay in the 2D map at short to medium times. The simplest version of the statistical Davis–Gray theory, in which only a single intermolecular dividing surface is considered, is then generalized to calculate the fragmentation rate in the full 4D map as the ratio of the volume of a four-dimensional turnstile lobe and a four-dimensional complex region enclosed by a multidimensional separatrix. Good agreement with exact numerical results is found at short times. The alternative RRKM theory is also applied, and is found to give a level of agreement with the Davis–Gray theory comparable to the 2D case. When the height of the barrier to internal rotation in the van der Waals potential is increased, however, it is found that volume-enclosing turnstile no longer exist in the 4D phase space, due to the occurrence of homoclinic tangency. The implications of this finding for transport theories in multimode systems are briefly discussed.