Long-time relaxation in pilot-wave theory

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 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 over five periods. For a superposition of only the first four energy states we are able to calculate over 50 periods. We find that, depending on the set of phases in the initial wave function, 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 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.