= Classical ideal gas in a spherical linear trap
{title2=$Z_1=8\pi(k_BT/a)^3/\lambda_T^3$}
For a structureless particle with <Hamiltonian> $p^2/(2m)+ar$, $a>0$, the classical <phase space> integral gives the displayed one-particle <partition function>. Its spatial integral is $4\pi\int_0^\infty r^2e^{-ar/(k_BT)}dr=8\pi(k_BT/a)^3$, while the momentum integral gives $\lambda_T^{-3}$. For $N$ indistinguishable dilute particles, $Z_N=Z_1^N/N!$, $F\simeq-Nk_BT[\ln(Z_1/N)+1]$, $U=(9/2)Nk_BT$ and $S\simeq Nk_B[\ln(Z_1/N)+11/2]$. These thermodynamic formulas use leading Stirling accuracy; the classical phase-space integral additionally requires temperatures above the discrete trap level scale.
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