= Magnetic entropy
{title2=$S_{\rm mag}$}
<Magnetic entropy> counts the uncertainty in the occupied magnetic <microstates>. For independent localized <magnetic moments> with one-site <probabilities> $p_j$,
$$
S_{\rm mag}=-Nk_B\sum_j p_j\ln p_j,
$$
where $N$ counts sites and $k_B$ is the <Boltzmann constant>. This is the <Gibbs entropy> for the magnetic degrees of freedom. Equal occupation of two states gives $Nk_B\ln2$; certain occupation of one state gives zero. Calling this disorder is useful only if the relevant states and their <probabilities> are specified. In an interacting material, correlations generally prevent replacing the many-body <Gibbs entropy> by this sum of independent one-site contributions.
Increasing the <magnetic field> at fixed positive <temperature> favors lower magnetic <energy levels>, reducing <magnetic entropy>. During ideal isolated demagnetization, both the level separation and <temperature> fall together, leaving the occupation <probabilities> unchanged. Thus the independent-spin idealization cools without increasing spin disorder along the isolated leg. When a real crystal's vibrating lattice also cools, its lost <entropy> can instead be taken up by the magnetic degrees of freedom. The conserved quantity is the total <entropy> of the thermally isolated stage, including any attached load in <thermal equilibrium>.
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