= Chemical equilibrium of a classical spin gas
{title2=$n_\pm=n_Qe^{\beta\mu}e^{\pm\beta\mu_BB}$}
For a nondegenerate spin-one-half ideal gas with energies $\mp\mu_BB(\mathbf r)$, the two component <chemical potentials> are $k_BT\ln(n_\pm/n_Q)\mp\mu_BB$. 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> $n=2n_Qe^{\beta\mu}\cosh(\beta\mu_BB)$ and <magnetization> $M=2\mu_Bn_Qe^{\beta\mu}\sinh(\beta\mu_BB)=n\mu_B\tanh(\beta\mu_BB)$. The high-temperature fixed-density limit is $M\simeq n\mu_B^2B/(k_BT)$, 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.
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