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

TL;DR

This paper explores the connection between entropy production in the Madelung fluid, derived from the Schrödinger equation, and the role of Bohm's potential in classical diffusion, highlighting the physical mechanisms behind entropy growth.

Contribution

It demonstrates that entropy increase in the Madelung fluid is linked to its compressibility and expansion, driven by the quantum Bohm potential, providing new insights into quantum-classical diffusion relations.

Findings

Entropy growth is proportional to velocity divergence in the Madelung fluid.

The quantum Bohm potential acts as a force in classical diffusion processes.

The diffusion coefficient bounds the Heisenberg uncertainty product.

Abstract

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.