Separatrix Crossings Research Articles

Phase change between separatrix crossings

The change of the crossing parameter (essentially the phase) between separatrix crossings is calculated for Hamiltonian systems with one degree of freedom and slow time dependence. This completes the calculation of the map for an arbitrary sequence of separatrix crossings. The change of the crossing parameter is used to calculate the retrapping probability. It is found that, even in the limit of infinitely slow time dependence, correlations persist between the separatrix crossings, and corrections to the usual zero-order adiabatic approximation must be included. Comparison with numerical experiments show that the result derived here is accurate for quite large values of the slowness parameter.

Uniform adiabatic invariance analysis of chemical reaction dynamics

It is shown that the usual primitive adiabatic theory of classical reaction dynamics is inconsistent when separatrix crossing occurs. In such cases, primitive theory yields errors in the reaction probabilities and other observables which do not scale to zero even when the time scale ratios become infinitely large, i.e., the adiabatic limit. This motivates a fundamental modification to the classical adiabatic theory of reactions to include the effects of separatrix crossing. The approach is explicitly formulated for direct heavy–light–heavy collinear reactions where two separatrix crossings must occur during the course of each reactive trajectory: once when the orbit untraps from the incoming reactant channel well and once again when it retraps in the final product channel well. The uniform adiabatic invariance analysis we propose reduces the classical reaction dynamics to the form of a simple measure preserving map. That is, the final conditions of the product trajectory are written as explicit analytic functions of the initial conditions. This eliminates the need to propagate any trajectories. The map is formulated in terms of the quantities from the adiabatic theory of reactions, i.e., vibrationally adiabatic potential curves, instantaneous frequency, etc., which are easily computed numerically. It is found that the imaginary frequency of the potential surface along the ridge separating reactants from products is a crucial parameter in the reaction dynamics. The uniform adiabatic analysis permits the calculation of vibrational inelasticity, complex lifetimes, the structure of reactivity bands, and other quantities inaccessible in usual adiabatic theory of reactions. Numerical result are presented for the I+HI reaction where it is found that the theory is quite accurate.

Reaction probability for sequential separatrix crossings.

The change of the crossing parameter (essentially the phase) between sequential slow separatrix crossings is calculated for Hamiltonian systems with one degree of freedom. Combined with the previous separatrix crossing analysis, these results reduce the dynamics of adiabatic systems with separatrices to a map. This map determines whether a trajectory leaving a given separatrix lobe is ultimately captured by the other lobe. Averaging these results over initial phase yields the reaction probability, which does not asymptote to the fully phase-mixed result even for arbitrarily long times between separatrix crossings.

Pitch-angle scattering driven by a single wave in tokamak plasma

The interaction of particles with a single wave in a tokamak plasma is investigated. It is pointed out that the stochastic pitch-angle scattering across the trapped/passing boundary may be driven by a single wave. The characteristics of such separatrix crossings are discussed. It is also found that the wave-driven separatrix crossings are accompanied by a radial flow of particles, which is composed of a directional flow and a diffusional flow. The resultant pitch-angle and radial fluxes are calculated.

THE EFFECTS OF TRANSITIONAL PARTICLES DRIVEN BY SINGLE WAVE

The unperturbed separatrix crossings driven by a single wave are discussed. The separa-trix crossings are followed by a mixing process, and a small-scale structure occurs in the distribution function in h-ψ plane. The separatrix crossings form a convective process in h-ψ plane, and the convective flux and the net flux are calculated. The separatrix crossings are accompanied by a radial flux, which is composed of a directional flux and a diffusional flux, both of them are calculated.