Chaos in Bohmian Quantum Mechanics: A short review

TL;DR

This review discusses how chaos arises in Bohmian Quantum Mechanics, its impact on trajectories, and how quantum entanglement influences ergodicity and the emergence of Born's rule, combining analytical and numerical insights.

Contribution

It introduces a generic mechanism for chaos in Bohmian systems and explores the role of entanglement in trajectory behavior and statistical distributions.

Findings

Chaotic trajectories are ergodic, leading to consistent long-term distributions.

Strong entanglement results in most trajectories being chaotic and ergodic, aligning with Born's rule.

Weak entanglement causes ordered trajectories to dominate, preventing convergence to Born's rule.

Abstract

This is a short review in the theory of chaos in Bohmian Quantum Mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i.e. they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tends to Born's rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born's rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born's rule, unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian Mechanics, namely whether there is a dynamical approximation of Born's rule by an arbitrary initial distribution of Bohmian particles.