Statistical Mechanics of Lattice Systems pp 67-91 | Cite as

# The Mean-Field Approximation, Scaling and Critical Exponents

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## Abstract

We consider the Ising model on a regular lattice (see Appendix A.1) where each interior site has the same number of nearest-neighbour sites. This is called the *coordination number* of the lattice and will be denoted by z. We shall denote the dimension of the lattice by the symbol *d*. It is assumed that, in the thermodynamic limit, boundary sites can be disregarded and that, with *N* sites, the number of nearest-neighbour site pairs is 1/2z*N*. Since the Ising model is restricted in the sense of Sect. 2.3.4, the Helmholtz free energy A depends on the two extensive variables *M* and *N*, the latter being equal to the number of spins and also to the reduced volume *V* if the volume μ_{0} per site is taken as the standard volume (see discussion after (2.55)). The configurational Helmholtz free energy per site a depends on *T* and the (relative) magnetization per site *m*, defined by (2.70), so that *n* = 2.

## Keywords

Ising Model Critical Exponent Helmholtz Free Energy Free Energy Density Canonical Partition Function## Preview

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