Timescales for dynamical relaxation to the Born rule

M. D. Towler, N. J. Russell, A. Valentini

Published 2011-03-08, updated 2011-09-27Version 2

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.