We initiate the study of relaxation to quantum equilibrium over long
timescales in pilot-wave theory. We simulate the time evolution of the
coarse-grained H-function Hbar(t) for a two-dimensional harmonic oscillator.
For a (periodic) wave function that is a superposition of the first 25 energy
states we confirm an approximately exponential decay of Hbar over five periods.
For a superposition of only the first four energy states we are able to
calculate Hbar(t) over 50 periods. We find that, depending on the set of phases
in the initial wave function, Hbar can decay to a large nonequilibrium residue
exceeding 10% of its initial value or it can become indistinguishable from zero
(the equilibrium value). We show that a large residue in Hbar is caused by a
tendency for the trajectories to be confined to sub-regions of configuration
space for some wave functions, and that this is less likely to occur for larger
numbers of energy states (if the initial phases are chosen randomly). Possible
cosmological implications are briefly discussed.Comment: 23 pages, 11 figures. Significant improvements in v2; new section on
confinement of trajectories. Accepted by J. Phys. A: Math. Theo
