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In comparison with superconductivity what is the microscopic theory of superfluidity? Who has done it?

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Superfluids come in several types. The most theoretically accessible case are superfluids that are Bose-Einstein condensates, just like in the case of superconductors. However, the bosons that get condensed in superconductors are Cooper pairs (of electrons). It's the whole atoms, including nuclei (e.g. helium-4), that get Bose-Einstein condensed in the superfluid case.

See e.g. this old paper

http://www.sciencedirect.com/science/article/pii/0003491665902800

for a microscopic theory of superfluidity in helium. Other superfluids include superfluid Fermi gases whose dynamics is quite different but also mostly understood. The description of all the possible behaviors and transitions between them is a major part of condensed matter physics so one can't really summarize all of them in one answer.

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A microscopic theory of superfluidity of Helium-4 was developed by Bogoliubov around 1947 (please see a short review and a reference in, e.g., http://www.lptl.jussieu.fr/files/Dupuis09a.pdf (PRL 102, 190401 (2009)). It is a bit difficult to summarize the theory here.

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