Ising model random-field

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

In the light of the above questions, it is tempting to refer to the results emerging from numerous theoretical and computer simulation studies [40,41,85-88,129-131] of the random field Ising model, and we shall do so, but only after completing the present discussion. 

From Eq. (33) it follows that, in the case of very large homogeneous domains, even very small heterogeneity effects should completely destroy any phase transition connected with the adsorbate condensation. This result is quite consistent with the theoretical predictions stemming from the random field Ising model [40,41]. 

In the next paper [160], Villain discussed the model in which the local impurities are to some extent treated in the same fashion as in the random field Ising model, and concluded, in agreement with earlier predictions for RFIM [165], that the commensurate, ordered phase is always unstable, so that the C-IC transition is destroyed by impurities as well. The argument of Villain, though presented only for the special case of 7 = 0, suggests that at finite temperatures the effects of impurities should be even stronger, due to the presence of strong statistical fluctuations in two-dimensional systems which further destabilize the commensurate phase. 

Figure 3. Distributions of magnetic moments in one realization of the random-field Ising model, at a selection of the temperatures covered by a single TSMC run. (The curves are merely a visual aid).

It is not clear how two phases coexist in disordered pores as alternating domains or as two infinite networks. Disordered porous materials with low porosity are more reminiscent of interconnected cylindrical pores and therefore a domain structure seems to be more probable [299, 311-315]. In highly porous materials, such as highly porous aerogels, infinite networks of two coexisting phases may be assumed. The critical point of fluids in disordered pores is expected to belong to the universality class of the random-field Ising model [316-318]. 

At d = 2 the correlation length exponent z/ is z/ = 1/(2 —C) = 3/4. This value is in good agreement with the result of numerical simulations of Ji and Robbins for an interface in the random-field Ising model. Our results are also supported by recent numerical simulations of a lattice model which is expected to be in the same universality class... 

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