The Mean-Field Approximation, Scaling and Critical Exponents
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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 1/2zN. 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 μ0 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.
KeywordsIsing Model Critical Exponent Helmholtz Free Energy Free Energy Density Canonical Partition Function
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