Thermodynamic limit

Remarks

The main example is $A_L = \ln Z_L$, with $Z_L$ the canonical or grand canonical partition function; the limit is the free energy or the pressure per unit volume, up to a factor. Example: $L^d$ independent spins $\sigma_i = \pm 1$ in a field $h$ have $Z_L = (2 \cosh \beta h)^{L^d}$. So $\ln Z_L / L^d = \ln (2 \cosh \beta h)$ for every $L$, and the limit exists. Non-example: for a system of $N$ particles in which every pair attracts with the same energy $-J < 0$, the energy of the ground state is $-J N (N - 1) / 2$. This grows like $N^2$, so the energy per volume has no finite thermodynamic limit at fixed density. More general sequences of regions, the van Hove sequences, are often allowed; they grow in all directions so that the boundary is small compared with the volume.