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The thermodynamics of non-interacting 1d Bose gas with finite number of particle in a trap has been discussed by N. J. van Druten and W. Ketterle, Phys. Rev. A 54, 656 (1996).
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Although true phase transitions can not occur in a finite size system, the temperature range over which the off-diagonal order is turned on decreases so rapidly with N so that for all practical purposes, the BEC appears to be "sharp." See N. J. van Druten and W. Ketterle, ibid.
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