Abstract
The use of hydrodynamics in the description of quantum systems has a long and diverse history. Perhaps one of the earliest applications [1] was to the stopping power problem which concerns the excitation phenomena taking place when a charged particle passes through matter. At an atomistic level, the moving ion suffers a loss of energy as a result of collisions with atomic electrons. However, individual Rutherford scattering events cannot in fact take place due to the long-range nature of the Coulomb potential which implies that the ion interacts at once with many electrons. Furthermore, the electrons which respond to the passage of the ion, interact with each other, so that the field experienced by a given electron is comprised of the total field due to the ion and the dynamic screening charge of all other electrons. This situation is exceedingly complex and indicates that a quantitative understanding can only be achieved by a consideration of the collective response of the electrons. Presumably this is what Bohr [2] had in mind when he imagined electrons as forming a trailing wake behind the ion. This picture has a natural realization in Bloch’s hydrodynamic theory [1] which treats the electrons as a charged fluid, analogous to an ordinary fluid, whose dynamical state is specified in terms of local variables such as density, velocity and pressure. What is far from obvious is the extent to which this picture is meaningful for a degenerate quantum system.
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Zaremba, E., Tso, H.C. (1998). Hydrodynamics in the Thomas-Fermi-Dirac-von Weizsäcker Approximation. In: Dobson, J.F., Vignale, G., Das, M.P. (eds) Electronic Density Functional Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0316-7_16
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