A figure of merit for Bose-Einstein condensation is the phase space density which can be defined as
$$\rho=n\lambda_T^3,$$
where $n$ is the number density of atoms and $\lambda_T$ the thermal de Boglie wave length.
My question is simply, why is this called a phase space density? From the definition, it is just the number of atoms occupying a volume with $\lambda_T$ being the characteristic length of the volume.
Best Answer
Because $n$ tells you how the particles are spread out in real space, while $T$ tells you how the particles are spread in out in momentum space (temperature is the only free parameter in the equilibrium thermodynamic distributions).
Even from elementary quantum mechanics with 3D square wells of size $L$, you might remember that the volume of momentum space goes as $\propto 1/L^3$, so you can see how that is now represented by the $\propto 1/\lambda_T^3$.
"Phase space" is usually used to refer to the space where each unique point corresponds to an allowed state of the particle. It is usually the (outer) product of direct space and reciprocal space, which is this case would be position and momentum. And indeed, $n$ tells you how peaks the atoms are in real space.