= 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 $g=1$.
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