  The Orthodox kinetic energy has, in fact, two hidden-variable components: one
linked to the current (or Bohmian) velocity, and another linked to the osmotic
velocity (or quantum potential), and which are respectively identified with
phase and amplitude of the wavefunction. Inspired by Bohmian and Stochastic
quantum mechanics, we address what happens to each of these two velocity
components when the Orthodox kinetic energy thermalizes in closed systems, and
how the pertinent weak values yield experimental information about them. We
show that, after thermalization, the expectation values of both the (squared)
current and osmotic velocities approach the same stationary value, that is,
each of the Bohmian kinetic and quantum potential energies approaches half of
the Orthodox kinetic energy. Such a `kinetic energy equipartition' is a novel
signature of quantum thermalization that can empirically be tested in the
laboratory, following a well-defined operational protocol as given by the
expectation values of (squared) real and imaginary parts of the
local-in-position weak value of the momentum, which are respectively related to
the current and osmotic velocities. Thus, the kinetic energy equipartion
presented here is independent on any ontological status given to these hidden
variables, and it could be used as a novel element to characterize quantum
thermalization in the laboratory, beyond the traditional use of expectation
values linked to Hermitian operators. Numerical results for the nonequilibrium
dynamics of a few-particle harmonic trap under random disorder are presented as
illustration. And the advantages in using the center-of-mass frame of reference
for dealing with systems with many indistinguishable particles are also
discussed.
