Origin of chaos near critical points of quantum flow.
The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie-Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function psi there is an unstable point (called the X point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the "nodal-point- X -point complex." We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point- X -point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point- X -point complex. It is also an inverse power of the distance of a trajectory from the X point's stable manifold far from the complex. This distance plays the role of an effective "impact parameter." The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point- X -point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.
Efthymiopoulos
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Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4, GR-115 27 Athens, Greece. cefthim@academyofathens.gr
Kalapotharakos
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Contopoulos
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