Relative dispersion and quantum thermal equilibrium in de Broglie–Bohm mechanics

Numerical computations (Valentini and Westman 2005 Proc. R. Soc. A 461 253?72) demonstrate that an initially arbitrary particle density, stirred by the field of de Broglie velocity associated with the Schr?dinger wavefunction for a sum of energy eigenstates, relaxes to the quantum thermal equilibrium that is the Born probability density, provided the particle density is coarse grained. The results are explained here by a Lagrangian or trajectory analysis, in terms of the relative dispersion of passive particles in a turbulent fluid. The analysis assumes that the turbulence statistics are stationary and isotropic, although these assumptions may be weakened. The relaxation to equilibrium is not reversible, owing to the coarse graining of the particle density and to the statistical inevitability of particle separation. There is no effective stirring toward equilibrium in very simple quantum systems such as a Gaussian wave packet or an energy eigenstate. However, it is argued that relaxation takes place during the emission of the packet or the establishment of the eigenstate, owing to stirring by the transients in the wavefunction for the entire system. The Lagrangian analysis is readily extended to nonrelativistic many-particle systems and to relativistic single-particle systems.