Chemical equilibrium of a classical spin gas ($n_{\pm }=n_Q{e}^{\beta \mu }{e}^{\pm \beta {\mu }_{B}B}$)

For a nondegenerate spin-one-half ideal gas with energies $\mp {\mu }_{B}B(\mathbf{r})$, the two component chemical potentials are $k_{B}T\ln (n_{\pm }/n_Q)\mp {\mu }_{B}B$. Diffusive and spin-conversion equilibrium require one spatially constant common chemical potential, provided spin conversion is allowed. Hence the displayed densities give total number density $n=2n_Q{e}^{\beta \mu }\cosh (\beta {\mu }_{B}B)$ and magnetization $M=2{\mu }_B{n}_Q{e}^{\beta \mu }\sinh (\beta {\mu }_{B}B)=n{\mu }_B\tanh (\beta {\mu }_{B}B)$. The high-temperature fixed-density limit is $M\simeq n{\mu }_B^{2}B/(k_{B}T)$, a Curie law. At fixed chemical potential the density itself varies with temperature, so the unnormalized magnetization need not have a pure $1/T$ dependence.