Legendre transforms for the Ising model

I’m reading a review on the Ising model and came across a section where they discuss Legendre transforms of thermodynamic potentials. Now I’m familiar with the classical thermodynamic relations such as $F = U – TS$ (Helmholtz), $G = F + pV$ (Gibbs) and so on, but fail to see the connection with the ones proposed for spin models. In particular, the authors define (for $beta=1/T=1$)

$$ F(J, h) = – ln{Z(J, h)}, $$

$$ S(chi, m) = min_{J, h}left[{-sum_{i} h_i m_i -sum_{i < j} J_{ij} chi_{ij} – F(J, h) }right], $$

$$G(J, m) = max_{h}left[ { sum_{i} h_i m_i + F(J, h)} right] $$

Now I fail to see how this relates to thermodynamic potentials in terms $N, V, T$ and so on now that we are working with a spin model. Is there a logical derivation for why $S$ and $G$ should follow these functional forms?