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
Famous quotes containing the words boundary conditions, boundary and/or conditions:
“The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience.”
—Willard Van Orman Quine (b. 1908)
“In the west, Apollo and Dionysus strive for victory. Apollo makes the boundary lines that are civilization but that lead to convention, constraint, oppression. Dionysus is energy unbound, mad, callous, destructive, wasteful. Apollo is law, history, tradition, the dignity and safety of custom and form. Dionysus is the new, exhilarating but rude, sweeping all away to begin again. Apollo is a tyrant, Dionysus is a vandal.”
—Camille Paglia (b. 1947)
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)