Thermodynamic limit of pressure (lattice)

The existence and properties of the pressure in the infinite-volume limit for lattice systems.

Thermodynamic limit of pressure (lattice)

For a lattice system on a finite region $\Lambda \subset \mathbb{Z}^d$ at inverse temperature $\beta$ , the pressure (or log-partition density) is

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

where $Z_\Lambda$ is the partition function .

Thermodynamic limit

Under standard conditions (translation-invariant interactions with suitable decay), the limit

p(\beta) = \lim_{\Lambda \nearrow \mathbb{Z}^d} p_\Lambda(\beta)

exists and is independent of the sequence of regions $\Lambda$ and of boundary conditions .

Properties

$p(\beta)$ is a convex function of $\beta^{-1}$ .
Derivatives of $p$ with respect to parameters yield thermodynamic densities (energy, magnetization, etc.).
Non-analyticity of $p(\beta)$ signals a phase transition .

The pressure is closely connected to the finite-volume Gibbs measure and its infinite-volume limit ; see also the general notion of thermodynamic limit .