Finite-volume Gibbs measure

The equilibrium probability distribution on spin configurations in a finite region, defined from the Hamiltonian and temperature with a chosen boundary condition.

Finite-volume Gibbs measure

Let $\Lambda \Subset \mathbb{Z}^d$ be finite, with spin space $\mathcal S$ (spin space ) and configuration space $\mathcal S^\Lambda$ (configuration space ). Fix an inverse temperature $\beta$ (see inverse temperature $\beta$ ) and a boundary condition $\eta$ (boundary condition ).

Given a finite-volume Hamiltonian $H_\Lambda(\sigma_\Lambda\mid \eta)$ (lattice Hamiltonian ), the finite-volume Gibbs measure on $\mathcal S^\Lambda$ is

\mu_{\Lambda,\beta}^{\eta}(\sigma_\Lambda) = \frac{1}{Z_{\Lambda,\beta}^{\eta}} \exp\!\bigl(-\beta\, H_\Lambda(\sigma_\Lambda\mid \eta)\bigr),

where the normalizing constant

Z_{\Lambda,\beta}^{\eta} =\sum_{\sigma_\Lambda\in \mathcal S^\Lambda} \exp\!\bigl(-\beta\, H_\Lambda(\sigma_\Lambda\mid \eta)\bigr)

is the lattice partition function (for discrete spins; replace the sum by an integral for continuous spins).

Cross-links

Built from an interaction potential $\Phi$ and a boundary condition via the Hamiltonian .
The collection $\{\mu_{\Lambda,\beta}^{\eta}\}$ (varying $\Lambda$ ) forms a Gibbs specification ; its infinite-volume consistency is expressed by the DLR equation .
Thermodynamic quantities are derived from $Z_{\Lambda,\beta}^{\eta}$ , including the pressure and its thermodynamic limit .
Model examples: Ising model , Potts model , XY model .

Key properties

Boltzmann form and normalization. $\mu_{\Lambda,\beta}^{\eta}$ is a probability measure: $\sum_{\sigma_\Lambda}\mu_{\Lambda,\beta}^{\eta}(\sigma_\Lambda)=1$ by definition of $Z_{\Lambda,\beta}^{\eta}$ .
Dependence on boundary conditions. For finite $\Lambda$ , changing $\eta$ changes $\mu_{\Lambda,\beta}^{\eta}$ through boundary interaction terms; this dependence typically decays into the bulk away from $\partial\Lambda$ when correlations are short-ranged.
Domain Markov / consistency property. If $\Delta \subset \Lambda$ , then conditioning $\mu_{\Lambda,\beta}^{\eta}$ on the spins in $\Lambda\setminus\Delta$ yields a Gibbs measure in $\Delta$ with boundary condition given by the conditioned exterior configuration. This is the finite-volume precursor of the DLR consistency.
Thermodynamic limit and phases. Along an increasing sequence of volumes $\Lambda_n \uparrow \mathbb{Z}^d$ , subsequential weak limits of $\mu_{\Lambda_n,\beta}^{\eta}$ (if they exist) are candidates for infinite-volume Gibbs measures . Different $\eta$ can lead to different limits in the presence of a phase transition .

Physical interpretation

$\mu_{\Lambda,\beta}^{\eta}$ is the equilibrium distribution of a finite sample in contact with a heat bath at temperature $T=1/\beta$ (up to constants), with the outside world modeled by the boundary condition $\eta$ . Observables are computed as ensemble averages under this measure (compare ensemble average ), and the sensitivity of bulk behavior to $\eta$ distinguishes single-phase regimes from coexistence of multiple phases.