= Two-dimensional harmonic-trap Bose condensation
{title2=$k_BT_c=\hbar\omega\sqrt{6N}/\pi$}
For an ideal <Bose gas> in a two-dimensional isotropic harmonic trap, the leading <density of states> measured above its <ground state> is $D(E)=E/(\hbar\omega)^2$. At saturation of the <Bose-Einstein distribution>, $N_{\mathrm{ex}}=(k_BT/(\hbar\omega))^2\int_0^\infty u/(e^u-1)\,du=(\pi^2/6)(k_BT/(\hbar\omega))^2$. The finite excited-state capacity gives the displayed <critical temperature> and <condensate fraction> $N_0/N=1-(T/T_c)^2$ below it. The appropriate trapped-gas <thermodynamic limit> fixes $N\omega^2$ as $N\to\infty$ and $\omega\to0$; a finite trap exhibits <finite-size rounding>. The <number density> is inhomogeneous, and the physical <chemical potential> approaches the oscillator <ground state> <energy>, not an unshifted zero by convention.
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