Ising model mean-field approximation

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

A four-state Ising-Potts model is applied to the [Mn(taa)] system to elucidate thermodynamic relations [21]. An [Mn(taa)] molecule is assumed to take four different microscopic states state 0 is the LS state and states 1-3 are the HS states with the elongation axis parallel to x, y, and z, respectively. Interactions are assumed only between nearest neighbor molecules. Under a mean-field approximation [22], the internal energy of the [Mn(taa)] system is expressed in terms of populations Pi (i = 0,1,2,3) of four microscopic states,... 

Table 3 Parameters of the Ising-like binuclear model, treated in the mean-field approximation.

Ising model and equilibrium properties based on the mean field approximation... 

Fisher and Wortis have shown that Tohnan s length is zero for symmetric fluid coexistence and non-zero for asymmetric fluid coexistence. " Symmetric fluids are represented by the lattice-gas (Ising) model in which the shape of the coexistence curve is perfectly symmetric with respect to the critical isochore. Real fluids always possess some degree of asymmetryAsymmetry in the vapour-liquid coexistence in helium, especially in He, is very small, but not zero. In the mean-field approximation, the asymmetry in the vapour-liquid coexistence is represented by the rectilinear diameter ... 

Equation 10.92 contains an asymmetric term ocM that is not present in the (symmetric) Ising model or lattice gas. We note that in the mean-field approximation is of order t. Equation 10.85 for the renormalized potential Or now becomes ... 

We will calculate the values of a and y for the Ising model in the mean field approximation. From Eqs. (D.89) and (D.90) we obtain the following expression for the value of the average magnetization at temperature T below E, in the limit H —> 0 ... 

The precise formulation of the interaction mechanism of the Ising model requires one to introduce separate probabilities p. governed by separate master equations of the type of (2.38) for each individual active center. At this point, we introduce an approximation which is known in statistical physics as mean field or Bragg-Williams approximation. It takes into account the interaction between active centers by making the rate constants k and k for the average probability p in (2.38) functions of p such that... 

---