Consider the Ising model on a subset $\Lambda\subset\mathbb{Z}^d$, with partition function $Z_{\Lambda; \beta , h}$ where $\beta$ is the inverse temperature and $h$ the external magnetic field.
The rescaled logarithm of the partition function
$$
\psi_{\Lambda}(\beta , h):=\frac{1}{|\Lambda|}\log Z_{\Lambda; \beta, h}\,\,
$$
is sometimes called the pressure of the system
(see e.g. the book of Friedli and Velenik, Chapter 3.2). One can then define the pressure in the thermodynamic limit as $\psi(\beta, h)=\lim_{\Lambda \Uparrow Z^d} \psi_{\Lambda}(\beta, h)$.
What is the physical meaning of the term "pressure" in this context? (And any particular reason to develop things here in terms of the pressure instead of, say, the free energy – which, if I understood right, might be defined as $-1/\beta$ times the pressure?)
Best Answer
The pressure is the natural term in the lattice gas interpretation of the model, in which $+$ spins are interpreted as particles and $-$ spins as vacancies (this is discussed in detail in Chapter 4). The standard Ising model in this language describes a gas in the grand canonical ensemble. The free energy is then naturally associated to the corresponding description in the canonical ensemble.
Going back to the magnetic language, this corresponds to a system with fixed magnetization. So, we reserve the word free energy for the latter situation (see, for instance, Definitions 2.4 and 2.8 in Chapter 2, or Definitions 4.5 and 4.11 in Chapter 4).
So, in our book this is done more for convenience: it would be annoying (and potentially confusing for the readers) if we changed terminology each time we change interpretation. Note that the only chapters in which we really care about thermodynamic potentials per se (as opposed as using them as tools) are Chapters 1 and 4. In all others, the central objects are the Gibbs measures that provide full information on the system (much more than the thermodynamic potentials).
[Addendum: if the question is why the pressure can be expressed in terms of the grand canonical partition function, then this is discussed in Chapter 1, see Section 1.3 (in particular the discussion just before (1.41)).]