Dynamical origin of quantum probabilities

We study the origin of the Born probability rule ρ = |ψ|2 in the de Broglie–Bohm pilot–wave formulation of quantum theory. It is argued that quantum probabilities arise dynamically, and have a status similar to thermal probabilities in ordinary statistical mechanics. This is illustrated by numerical simulations for a two–dimensional system. We show that a simple initial ensemble, with a non–standard distribution ρ ≠ |ψ|2 of particle positions, evolves towards the quantum distribution to high accuracy. The relaxation process ρ→|ψ|2 is quantified in terms of a coarse–grained H–function (equal to minus the relative entropy of ρ with respect to |ψ|2), which is found to decrease approximately exponentially over time, with a time constant that accords with a simple theoretical estimate.