Origin of chaos near critical points of quantum flow

Abstract

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.

Applied Bohmian mechanics

“…In a series of works published later on by Wisniacki and coworkers [282,283] it was shown that the origin of Bohmian chaos is in the evolution in time of those vortices. An analogous conclusion was also found by Efthymiopoulos and coworkers [197,[284][285][286][287], who showed that chaos is due to the presence of moving quantum vortices forming nodal point-X-point complexes. In particular, in reference [197] a theoretical analysis of the dependence of Lyapunov exponents of Bohmian trajectories on the size and speed of the quantum vortices is presented, which explains their earlier numerical findings [284,285].…”

Wave packet dynamics for a non-linear Schrödinger equation describing continuous position measurements

“…The de Broglie -Bohm theory, also called the pilot-wave theory or the quantum theory of motion, is an interpretation of quantum theory which has attracted the attention of researchers in recent years [1,2,15,[17][18][19][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. In their review of 100 years of quantum physics when celebrating the first hundred years of quantum physics, Tegmark and Wheeler considered Bohmian mechanics one of the most significant elements in its history [27].…”

“…In their review of 100 years of quantum physics when celebrating the first hundred years of quantum physics, Tegmark and Wheeler considered Bohmian mechanics one of the most significant elements in its history [27]. Among the fascinating facets of Bohm's theory one can mention its relevance for the study of quantum chaos [32] and for the quantum measurement problem [1,2,19], and its applications to quantum cosmology [28][29][30][31]. Thermal equilibrium of quantum systems [34], entanglement [33], and the dynamics of quantum particles with position-dependent effective mass [35] have also been recently considered within Bohm's approach.…”

“…The stability criterion (32) ensures that the potential W has a minimum, which means the real constant κ has to be negative, κ < 0. Due to the "drag force" the particle looses "energy" and tends to end up at δ s , the bottom of the potential well, given by dW dδ δ=δs = 0, yielding…”

“…Again, due to the drag force, the "particle" moves towards the potential minimum at δ s provided that condition (32) holds, which ensures a potential well and dissipative dynamics.…”