At ultra-cold temperatures, a gas of bosons undergoes a phase transition where there is a macroscopic occupation of the ground state. The top plot shows how the density profile of bosons in a 3D harmonic trap changes with temperature, .

For temperatures above this transition, the bosons occupy many energy states, so the density profile has a broad spatial distribution. Below the transition, there is a macroscopic occupation of the ground state, and particles accumulate at the bottom of the trap (near the origin); in a harmonic potential, the ground state wavefunction is Gaussian. At the transition, there is a jump in the specific heat, , (bottom plot) indicating a first-order phase transition.

The temperature is in units of ℏω, and the position is in units of the trap length, , where is the trap frequency. The example uses only 1000 particles.