Thermodynamic limit of the lattice pressure

Let $(\Lambda_n)_{n\ge 1}$ be an increasing sequence of finite regions exhausting the lattice (often boxes, as in finite boxes ), and fix boundary conditions $\tau_n$ (e.g. free, fixed, or periodic; see boundary conditions ).

With $p_{\Lambda_n}(\beta,\tau_n)$ the finite-volume pressure , the thermodynamic-limit pressure (or bulk pressure) is the limit

when this limit exists and is independent of the chosen exhausting sequence and boundary conditions.

This is the lattice counterpart of the thermodynamic limit for macroscopic thermodynamic quantities.

Key properties

• Existence under standard hypotheses. For many models defined by a translation-invariant , finite-range (or sufficiently fast decaying) interaction , the limit exists due to subadditivity-type arguments and boundary terms being negligible compared to volume.

• Boundary-condition independence (bulk limit). When the limit exists, dependence on $\tau_n$ typically disappears because boundary contributions scale like $|\partial\Lambda_n|$ while the normalization is by $|\Lambda_n|$.

• Thermodynamic information. The bulk Helmholtz free-energy density is

  $$f(\beta) = -\frac{1}{\beta}p(\beta),$$

  aligning with Helmholtz free energy per site.

• Non-analyticity signals phase transitions. Lack of differentiability or other singular behavior of $p(\beta)$ (as a function of parameters such as $\beta$ or an external field) is a standard signature of phase transitions ; it is closely tied to non-uniqueness of infinite-volume Gibbs measures .

Physical interpretation

$p(\beta)$ is the macroscopic (infinite-volume) free-energy density in units of $k_B T$. It determines equilibrium thermodynamic response: derivatives with respect to temperature-like or field-like parameters yield energy density, magnetization, and susceptibilities in the bulk. When multiple equilibrium phases coexist, $p(\beta)$ remains well-defined but can develop singularities reflecting competing phases.