Lattice pressure (finite volume)

Let $\Lambda$ be a finite region (see boundaries of finite regions ) and let $Z_\Lambda(\beta,\tau)$ be the finite-volume lattice partition function at inverse temperature $\beta$ with boundary condition $\tau$.

The finite-volume (dimensionless) pressure is

where $|\Lambda|$ is the number of lattice sites in $\Lambda$.

Equivalently, the finite-volume Helmholtz free energy is

so that $p_\Lambda(\beta,\tau) = -\beta\,\frac{F_\Lambda(\beta,\tau)}{|\Lambda|}$, matching the statistical-mechanics notion of free energy and the thermodynamic Helmholtz free energy (up to conventions and units).

Key properties

• Intensive quantity. $p_\Lambda$ is normalized per site and is therefore the natural quantity to send to the thermodynamic limit .

• Boundary effects are subextensive. For many short-range models (e.g. finite-range interactions ), changing $\tau$ changes $\log Z_\Lambda$ by at most order $|\partial \Lambda|$, so $p_\Lambda(\beta,\tau)$ is often asymptotically independent of $\tau$ as $|\Lambda|\to\infty$.

• Convexity in parameters. Because it is a log-partition function density, $p_\Lambda$ is typically convex in couplings and fields appearing linearly in the Hamiltonian (a finite-volume version of standard convexity for $\log Z$).

• Derivatives give observables. If the Hamiltonian includes an external field term (see external field coupling ), then derivatives of $p_\Lambda$ with respect to that field give magnetization density and susceptibilities (see susceptibility ).

Physical interpretation

$p_\Lambda$ is the free-energy density in units of $k_B T$ (see Boltzmann constant and temperature ): it encodes the competition between energy minimization and entropy maximization in a finite region. Non-smooth behavior of its infinite-volume limit is a key diagnostic of phase transitions .