[Paper Review] Chaos in de Broglie - Bohm quantum mechanics and the dynamics of quantum relaxation

Christos Efthymiopoulos, G. Contopoulos+1 more|arXiv (Cornell University)|Mar 28, 2017

Quantum Mechanics and Applications9 citations

TL;DR

This paper identifies the nodal point–X-point complex as a universal mechanism generating chaos in de Broglie–Bohm quantum trajectories in 2D and 3D systems, demonstrating that chaos is necessary for quantum relaxation—the dynamical approach to Born’s rule. It shows that chaotic trajectories drive efficient quantum relaxation, while regular trajectories fail to achieve full relaxation due to non-penetration effects, quantifying the rate via Lyapunov scaling laws and entropy dynamics.

ABSTRACT

We discuss the main mechanisms generating chaotic behavior of the quantum trajectories in the de Broglie - Bohm picture of quantum mechanics, in systems of two and three degrees of freedom. In the 2D case, chaos is generated via multiple scatterings of the trajectories with one or more `nodal point - X-point complexes'. In the 3D case, these complexes form foliations along `nodal lines' accompanied by `X-lines'. We also identify cases of integrable or partially integrable quantum trajectories. The role of chaos is important in interpreting the dynamical origin of the `quantum relaxation' effect, i.e. the dynamical emergence of Born's rule for the quantum probabilities, which has been proposed as an extension of the Bohmian picture of quantum mechanics. In particular, the local scaling laws characterizing the chaotic scattering phenomena near X-points, or X-lines, are related to the global rate at which the quantum relaxation is observed to proceed. Also, the degree of chaos determines the rate at which nearly-coherent initial wavepacket states lose their spatial coherence in the course of time.

Motivation & Objective

• To identify the dynamical mechanisms responsible for chaos in Bohmian trajectories within quantum systems of two and three degrees of freedom.
• To establish the necessity of chaos for the dynamical emergence of quantum equilibrium (i.e., quantum relaxation toward Born’s rule).
• To quantify the rate of quantum relaxation in relation to the degree of chaos, particularly through Lyapunov scaling laws and entropy dynamics.
• To classify trajectories as regular, chaotic, or partially integrable, and to analyze their impact on the relaxation process.
• To extend the nodal point–X-point complex framework from 2D to 3D systems, identifying nodal lines and X-line structures as sources of 3D chaos.

Proposed method

• Analyzing the local structure of the quantum flow around moving nodal points using expansions of the wavefunction and its associated Bohmian equations of motion.
• Deriving scaling laws that relate the local Lyapunov exponent of a trajectory to the size and velocity of nodal point–X-point complexes.
• Using numerical simulations to track ensembles of Bohmian trajectories initialized in coherent wavepackets, comparing regular and chaotic initial conditions.
• Computing time evolution of statistical measures: the Kullback–Leibler divergence $D(t)$ and the entropy $H(t)$, to quantify quantum relaxation rates.
• Applying the concept of invariant 2D surfaces in 3D systems to identify partially integrable cases with explicit integrals of motion.
• Employing the sub-quantum H-theorem framework to interpret the long-term approach to quantum equilibrium as a consequence of chaotic mixing.

Experimental results

Research questions

• RQ1 What dynamical mechanism generates chaos in Bohmian trajectories in systems with two and three degrees of freedom?
• RQ2 How does the local Lyapunov exponent of a trajectory depend on the kinematics of nodal point–X-point complexes?
• RQ3 To what extent does the presence of regular (non-chaotic) trajectories prevent complete quantum relaxation in isolated systems?
• RQ4 Can the rate of quantum relaxation be quantitatively linked to the degree of chaos in the system’s trajectory dynamics?
• RQ5 How do nodal lines and X-lines in 3D configuration space give rise to chaotic scattering and global mixing in Bohmian trajectories?

Key findings

• The nodal point–X-point complex is a universal mechanism generating chaos in 2D quantum systems, with trajectories exhibiting sensitive dependence on initial conditions and positive Lyapunov exponents.
• Scaling laws were derived linking the local Lyapunov exponent to the size and speed of nodal point–X-point complexes, enabling quantitative prediction of chaotic scattering behavior.
• In 3D systems, nodal points form continuous nodal lines, and X-points organize into cylindrical structures around them, enabling chaotic scattering via close encounters with X-lines.
• Regular Bohmian trajectories, which avoid nodal point–X-point complexes, remain spatially coherent and fail to exhibit quantum relaxation, as shown by $D(t)$ and $H(t)$ remaining bounded away from zero.
• Chaotic ensembles exhibit faster and more complete quantum relaxation, with $D(t) \simeq 12.3$ and $H_s \simeq 6.4$ approaching asymptotic values, though never reaching zero due to non-penetration of regular domains.
• Partially integrable 3D systems feature invariant 2D surfaces defined by explicit integrals of motion, restricting all trajectories to these surfaces and limiting chaotic mixing.