Bose-Einstein condensation

Cool a gas of bosons far enough and something startling happens: instead of spreading thinly across many low-lying states, a macroscopic fraction of all the particles — perhaps trillions of atoms — abruptly crowds into the single lowest quantum state and behaves as one giant coherent matter wave. This is Bose-Einstein condensation, a phase transition driven not by any force between particles but purely by quantum statistics. It is, in Einstein's phrase, condensation in momentum space.

The transition happens when the thermal de Broglie wavelength lambda grows to about the spacing between particles, so their wave packets overlap and indistinguishability becomes unavoidable. For an ideal Bose gas the condition is n lambda^3 = 2.612..., which defines the critical temperature T_c = (2 pi hbar^2 / m k)(n / 2.612)^(2/3). Below T_c the chemical potential is pinned at the ground-state energy (essentially mu = 0) and the excited states can hold only a limited number of particles; every particle beyond that capacity must go into the ground state. The condensate fraction grows as N_0/N = 1 - (T/T_c)^(3/2) for the uniform ideal gas, reaching 1 at T = 0. The macroscopic ground-state occupation serves as the order parameter of the transition.

Bose-Einstein condensation underlies superfluid helium-4 (approximately, since interactions matter there) and was achieved cleanly in dilute atomic gases of rubidium and sodium in 1995, cooled to billionths of a kelvin (Nobel Prize 2001). Two honest points: BEC occurs even for NON-interacting bosons, which sharply distinguishes it from ordinary gas-to-liquid condensation that requires attraction; and for a uniform ideal Bose gas in one or two dimensions there is no condensation at any finite temperature — the phenomenon depends on dimensionality and on the trap geometry.