  The Madelung equations map the non-relativistic time-dependent Schrodinger
equation into hydrodynamic equations of a virtual fluid. Here we show that an
increase of the Boltzmann entropy of this Madelung fluid is proportional to the
expectation value of its velocity divergence. Hence, entropy growth is
accompanied by expansion resulting from the ability of the Madelung fluid to be
compressible. The compressibility itself reflects superposition of solutions of
the Schrodinger equation. Thus, in unitary processes where the Madelung fluid
expands and then shrinks, the Boltzmann entropy may, correspondingly, grow and
then decrease. The notion of entropy growth due to expansion is common in
diffusive processes, however in the latter the process is irreversible. Much
unlike the Boltzmann entropy, the von Neumann entropy, does not vary with time.
To elucidate the physical underpinning of the Boltzmann entropy, we examine
several specific examples. We demonstrate that, for classical diffusive
processes, the "force" accelerating diffusion has the form of the positive
gradient of the quantum Bohm potential. In the Madelung fluid, the advective
and the diffusive velocities correspond respectively to the the real and
imaginary parts of the complex momentum. We find that the diffusion coefficient
provides a lower bound of Heisenberg uncertainty type product between the gas
mean free path and the Brownian momentum.
