= Nonrelativistic entropy at zero chemical potential
{title2=$s\sim g m^3(2\pi)^{-3/2}x^{-1/2}e^{-x}$}
For a dilute nonrelativistic gas with $x=m/T\gg1$ and zero <chemical potential>, the <Bose-Einstein distribution> approaches the <Maxwell-Boltzmann distribution>. Expanding $E=m+p^2/(2m)+\cdots$ in the <number density> integral gives $n\sim g(mT/(2\pi))^{3/2}e^{-x}$. Since $\rho=mn+3nT/2+\cdots$ and $P=nT$, the <entropy density at zero chemical potential> gives $s=n(x+5/2+\cdots)$ and hence the displayed leading expression. This includes the internal multiplicity $g$; a single real <scalar field> has $g=1$.
The displayed asymptotic uses <natural units> $k_B=c=\hbar=1$. Thus $m$ and $T$ are both <energy> scales, and $x=m/T$ is dimensionless. Restoring SI units replaces this ratio by $mc^2/(k_BT)$ and includes $k_B$ in the physical <entropy> per particle.
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