A Proof that the Free Energy of a Spin System is Extensive

The free energy obtained from the canonical partition function for a finite spin system possesses a certain convexity property, of which theorems by Peierls and Bogoliubov are particular applications. This property is used in proving the following result: Consider a regular lattice of spins in the form of a parallelepiped (in two dimensions a parallelogram, in one dimension a linear chain). The free energy of the system divided by the number of spins approaches a definite limit as the linear dimensions of the system become infinite. The limit is not influenced by certain common types of boundary conditions. A similar result, but with convergence understood in a weaker sense, holds for derivatives of the free energy such as entropy, magnetization, and specific heat. In the proof it is necessary to assume that the Hamiltonian has the translational symmetry of the spin system, and that long-range interactions decrease sufficiently rapidly with the distance r between spins. (For example, as r−3−ε with ε 0 for interactions between pairs of spins in 3 dimensions.)