Convergence to Quantum Equilibrium: Deterministic vs Stochastic Pilot Wave Dynamics

We compare the onset of quantum equilibrium in the (deterministic) de Broglie-Bohm (dBB) formalism with its counterpart in the (stochastic) formalism of Bohm, Hiley, and Nelson (BHN). We bring to light the role played by microstructures in the position distributions regarding the approach to quantum equilibrium. We show that they provide a marker which makes it possible to distinguish deterministic chaos à la dBB from stochastic dynamics à la BHN. We also introduce a measure of the degree of mixing of two ensembles initially separated in space; this measure is inspired by Bernoulli processes.