Boundary Conditions
This ice model provide an important 'counterexample' in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions. The model was analytically solved for periodic boundary conditions, anti-periodic, ferromagnetic and domain wall boundary conditions. Six vertex model with domain wall boundary conditions on a square lattice has specific significance for algebraic combinatorics, it helps to enumerate Alternating sign matrix. In this case the partition function can be represented as a determinant of a matrix (dimension of the matrix is equal to the size of the lattice), but in the other cases the enumeration of does not come out in such a simple closed form.
Domain wall gives the smallest . Clearly, the largest is given by free boundary conditions (no constraint at all on the configurations on the boundary), but the same occurs, in the thermodynamic limit, for periodic boundary conditions, as used originally to derive .
Read more about this topic: Ice-type Model
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