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The Madelung equations map the non-relativistic time-dependent Schrodingerequation into hydrodynamic equations of a virtual fluid. Here we show that anincrease of the Boltzmann entropy of this Madelung fluid is proportional to theexpectation value of its velocity divergence. Hence, entropy growth isaccompanied by expansion resulting from the ability of the Madelung fluid to becompressible. The compressibility itself reflects superposition of solutions ofthe Schrodinger equation. Thus, in unitary processes where the Madelung fluidexpands and then shrinks, the Boltzmann entropy may, correspondingly, grow andthen decrease. The notion of entropy growth due to expansion is common indiffusive processes, however in the latter the process is irreversible. Muchunlike the Boltzmann entropy, the von Neumann entropy, does not vary with time.To elucidate the physical underpinning of the Boltzmann entropy, we examineseveral specific examples. We demonstrate that, for classical diffusiveprocesses, the "force" accelerating diffusion has the form of the positivegradient of the quantum Bohm potential. In the Madelung fluid, the advectiveand the diffusive velocities correspond respectively to the the real andimaginary parts of the complex momentum. We find that the diffusion coefficientprovides a lower bound of Heisenberg uncertainty type product between the gasmean free path and the Brownian momentum.

On entropy production in the Madelung fluid and the role of Bohm's potential in classical diffusion