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Annealed disorder allows the variables representing disorder to equilibrate along with the spins. With a specified normalized prior measure over couplings, its partition function is the average ${\mathbb{E}}_J{Z}_J$. This differs from preparing samples with fixed couplings and averaging their measured thermodynamics, as in quenched disorder.^[ref]

Quenched disorder is randomness held fixed while the other degrees of freedom equilibrate. For a sample with couplings $J$, first compute its partition function $Z_J$ and Helmholtz free energy $-k_{B}T\log Z_J$, then average over samples. One must not silently replace ${\mathbb{E}}_J\log Z_J$ by $\log {\mathbb{E}}_J{Z}_J$.^[ref]