A Bose–Einstein condensate (BEC) is a state of matter that occurs at extremely low temperatures, close to absolute zero (0 K or -273.15 °C). It was predicted by the physicists Satyendra Nath Bose and Albert Einstein in the early 20th century.
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A Bose-Einstein condensate is a state with macroscopic occupation of a one-particle mode. A phase-selected mean-field description can use , while a fixed-number finite-volume state need not have a nonzero field expectation. Bose-Einstein condensation and superfluidity are related but distinct notions; long-wavelength phase fluctuations can invalidate a uniform condensate saddle in low dimensions.
Inward Bound by Abraham Pais (1988) page 282 shows how this can be generalized from the Maxwell-Boltzmann distribution