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Existence of the thermodynamic limit of the pressure

For translation-invariant short-range lattice systems, the finite-volume pressure (1/|Λ|) log Z_Λ has a limit as Λ grows, typically via subadditivity and Fekete’s lemma.

Existence of the thermodynamic limit of the pressure

Statement (pressure limit exists)

Let $\Lambda\subset\mathbb Z^d$ be finite volumes, with a translation-invariant finite-range interaction defining a lattice Hamiltonian $H_\Lambda$ and finite-volume partition function (see lattice partition function )

Z_\Lambda(\beta)=\sum_{\sigma_\Lambda}\exp\big(-\beta H_\Lambda(\sigma_\Lambda)\big).

Define the finite-volume pressure

p_\Lambda(\beta)=\frac{1}{|\Lambda|}\log Z_\Lambda(\beta)

(see pressure (lattice) ).

Assume standard short-range conditions (finite range or sufficiently fast decay, plus stability/regularity) so that boundary effects are at most of order $|\partial\Lambda|$ . Then for any sequence of boxes $\Lambda_n\nearrow\mathbb Z^d$ with vanishing boundary-to-volume ratio,

p(\beta)=\lim_{n\to\infty} p_{\Lambda_n}(\beta)

exists and is independent of the chosen sequence. The limit $p(\beta)$ is the thermodynamic pressure.

Key hypotheses

Translation invariance and short-range interaction (finite range is the cleanest case).
Control of boundary terms: $\log Z_{\Lambda\cup\Lambda'}$ differs from $\log Z_\Lambda+\log Z_{\Lambda'}$ by at most $O(|\partial|)$ when $\Lambda,\Lambda'$ are separated/combined appropriately.
Temperedness/stability ensuring $Z_\Lambda(\beta)<\infty$ for all finite $\Lambda$ and relevant $\beta$ .

Conclusions

Existence: $p(\beta)$ is well-defined as a thermodynamic limit.
Shape independence: the limit does not depend on the particular exhausting sequence of volumes (under vanishing boundary/volume ratio).
Foundation for infinite-volume theory: existence of $p(\beta)$ underlies compactness arguments for infinite-volume Gibbs measures (see infinite-volume Gibbs measure ).

Cross-links to definitions

Finite-volume objects: lattice Hamiltonian , finite-volume Gibbs measure , partition function , pressure .
Core tools: subadditivity of log partition functions , Fekete’s lemma .

Proof idea / significance

One shows an “almost subadditivity” inequality for $\log Z_\Lambda$ when tiling large boxes by smaller boxes: the bulk term adds while the interaction across tile boundaries contributes only a boundary correction. After dividing by volume and taking box sizes to infinity (so boundary/volume vanishes), subadditivity methods plus Fekete’s lemma yield existence of the limit. This is the standard route to defining thermodynamic potentials rigorously in lattice statistical mechanics.