TY - JOUR

T1 - On Entropy Production in the Madelung Fluid and the Role of Bohm’s Potential in Classical Diffusion

AU - Heifetz, Eyal

AU - Tsekov, Roumen

AU - Cohen, Eliahu

AU - Nussinov, Zohar

N1 - Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - The Madelung equations map the non-relativistic time-dependent 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.

AB - The Madelung equations map the non-relativistic time-dependent 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.

KW - Entropy

KW - Hydrodynamics

KW - Madelung equations

UR - http://www.scopus.com/inward/record.url?scp=84962176256&partnerID=8YFLogxK

U2 - 10.1007/s10701-016-0003-1

DO - 10.1007/s10701-016-0003-1

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AN - SCOPUS:84962176256

SN - 0015-9018

VL - 46

SP - 815

EP - 824

JO - Foundations of Physics

JF - Foundations of Physics

IS - 7

ER -