On Entropy Production in the Madelung Fluid and the Role of Bohm's Potential in Classical Diffusion
Abstract
The Madelung equations map the nonrelativistic timedependent Schrödinger equation into hydrodynamic equations of a virtual fluid. While the von Neumann entropy remains constant, we demonstrate that an increase of the Shannon entropy, associated with this Madelung fluid, is proportional to the expectation value of its velocity divergence. Hence, the Shannon entropy may grow (or decrease) due to an expansion (or compression) of the Madelung fluid. These effects result from the interference between solutions of the Schrödinger equation. Growth of the Shannon entropy due to expansion is common in diffusive processes. However, in the latter the process is irreversible while the processes in the Madelung fluid are always reversible. The relations between interference, compressibility and variation of the Shannon entropy are then examined in several simple examples. Furthermore, we demonstrate that for classical diffusive processes, the "force" accelerating diffusion has the form of the positive gradient of the quantum Bohm potential. Expressing then the diffusion coefficient in terms of the Planck constant reveals the lower bound given by the Heisenberg uncertainty principle in terms of the product between the gas mean free path and the Brownian momentum.
 Publication:

Foundations of Physics
 Pub Date:
 July 2016
 DOI:
 10.1007/s1070101600031
 arXiv:
 arXiv:1509.01265
 Bibcode:
 2016FoPh...46..815H
 Keywords:

 Madelung equations;
 Entropy;
 Hydrodynamics;
 Quantum Physics;
 Condensed Matter  Other Condensed Matter
 EPrint:
 9 pages. The second version contains more examples and explanations