Condensation of N Bosons and the Laser Phase Transition Analogy

A simple analytic expression for the ground state of a dilute gas of $N$ ideal bosons in a 3D harmonic potential at temperature $T$ is derived from the steady state solutions of nonequilibrium equations of motion. The $N$ particle constraint plays the important role of introducing the essential nonlinearity yielding a Ginzburg-Landau free energy. The present analysis has much in common with the quantum theory of the laser, and with the laser phase transition analogy.