  We illustrate through explicit numerical calculations how the Born-rule
probability densities of non-relativistic quantum mechanics emerge naturally
from the particle dynamics of de Broglie-Bohm pilot-wave theory. The time
evolution of a particle distribution initially not equal to the absolute square
of the wave function is calculated for a particle in a two-dimensional infinite
potential square well. Under the de Broglie-Bohm ontology, the box contains an
objectively-existing 'pilot wave' which guides the electron trajectory, and
this is represented mathematically by a Schroedinger wave function composed of
a finite out-of-phase superposition of M energy eigenstates (with M ranging
from 4 to 64). The electron density distributions are found to evolve naturally
into the Born-rule ones and stay there; in analogy with the classical case this
represents a decay to 'quantum equilibrium'. The proximity to equilibrium is
characterized by the coarse-grained subquantum H-function which is found to
decrease roughly exponentially towards zero over the course of time. The
timescale tau for this relaxation is calculated for various values of M and the
coarse-graining length epsilon. Its dependence on M is found to disagree with
an earlier theoretical prediction. A power law - tau inversely proportional to
M - is found to be fairly robust for all coarse-graining lengths and, although
a weak dependence of tau on epsilon is observed, it does not appear to follow
any straightforward scaling. A theoretical analysis is presented to explain
these results. This improvement in our understanding of timescales for
relaxation to quantum equilibrium is likely to be of use in the development of
models of relaxation in the early universe, with a view to constraining
possible violations of the Born rule in inflationary cosmology.
