= Solution
<Bose-Einstein condensation> is macroscopic occupation of the lowest <one-particle state>. It is a change in occupation statistics, not ordinary liquid formation and not a requirement that all atoms occupy one spatial point. For ideal <bosons>,
$$
\overline n_{\mathbf k}=\frac1{\exp[(\epsilon_{\mathbf k}-\mu)/(k_BT)]-1}.
$$
Set the lowest one-particle <energy> to zero. Then $\mu\leq0$, and the population of excited states in a uniform three-dimensional box is
$$
N_{\rm ex}=\frac V{\lambda^3}g_{3/2}(z),\qquad
z=e^{\mu/(k_BT)}\leq1,\qquad
\lambda=\sqrt{\frac{2\pi\hbar^2}{mk_BT}}.
$$
Here $g_s(z)=\sum_{j\geq1}z^j/j^s$. As $\mu$ approaches zero, the excited-state population saturates at $V\zeta(3/2)/\lambda^3$. Any excess particles occupy the <ground state>. In the <thermodynamic limit>,
$$
\boxed{T_c=\frac{2\pi\hbar^2}{mk_B}
\left[\frac{n}{\zeta(3/2)}\right]^{2/3}},\qquad
\boxed{\frac{N_0}{N}=1-\left(\frac T{T_c}\right)^{3/2}\quad(T<T_c)}.
$$
Thus the onset occurs when the <thermal de Broglie wavelength> is comparable to the mean interparticle spacing. At finite $N$, the <chemical potential> remains slightly below the ground-state <energy>; setting it equal to zero is a thermodynamic-limit statement.
The 1995 <rubidium> experiment used magnetic confinement and <evaporative cooling of atoms>. Absorption images after trap manipulation revealed a narrow, anisotropically expanding component superposed on the broader thermal distribution; the new component grew as cooling proceeded. The thermal component supplied an internal comparison through the same imaging optics. Trap decompression and release geometry mattered, and finite imaging resolution and interactions limited direct comparison with an ideal-gas <ground state>.<Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor by M. H. Anderson and others>[Science 269, pp. 198–201, experimental preparation and velocity distributions]{ref}
A narrow cloud alone is not sufficient evidence. One should check its emergence across a controlled cooling sequence, atom-number and <temperature> calibration, the thermal background and the predicted expansion anisotropy. These are tests of an occupation transition rather than a simple focusing artifact. The box formula above is not the exact critical-temperature or condensate-fraction formula for the trapped experiment: the trap changes the <density of states>, and interactions also matter. Condensation and superfluid response are related but distinct properties; the former definition is ground-state occupation.
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