Ising lattice theory

Moreover, some uncertainty was expressed about the applicability to fluids of exponents obtained for the Ising lattice. Here there seemed to be a serious discrepancy between theory and experiment, only cleared up by later and better experiments. By hindsight one should have realized that long-range fluctuations should be independent of the presence or absence of a lattice. 

The lattice gas (Ising model), the simplest model that describes condensation of fluids, has played an important role in the theory of critical phenomena [1] providing crucial tests for most basic theoretical concepts. Recently, accurate numerical results for the crossover from asymptotic (Ising-like) critical behavior to classical (mean-field) behavior have been reported both for two-dimensional [29, 30] and three-dimensional [31] Ising lattices in zero field with a variety of interaction ranges. The Ginzburg number, as defined by Eq. (36), depends on the normalized interaction range R = as... 

Another model of wide applicability is the Ising lattice model. The RIS theory is an Ising model in one dimension the model has been applied to other types of problems in spaces of two and higher dimensions. A recent 2-D model was developed to gain an understanding of the distribution of surfactant molecules in immiscible fluid systems (180). 

One approximation used in theories is to consider that adsorbed atoms are completely localized in registry with the surface lattice of adsorption sites [8], so that 2D Ising lattice model calculations are applicable [181]. However, such a complete localization seems to be more appropriate to chemisorbed systems than to physisorp-tion. The lattice gas theories can thus be considered as realistic only for small adsorbate atoms at low temperatures on an atomically rough solid surface or for problems near critical points, where the long-range nature of the correlations means that assumption of localization on a surface lattice is a less significant approximation [182],... 

The expressions whose limits give s and <7 are found to converge rapidly and the estimates for s and a so obtained are close to the values obtained experimentally. While improvement of the calculated values of s and <7 will undoubtedly be possible in the future, these results, together with the results of the theory of the one-dimensional Ising lattice, place the description of helix-coil transitions of polyfa-amino acids) on a firm combined molecular and statistical mechanical basis. 

In considering the cooperative behavior of linear chain molecules consisting of identical monomeric units, a particularly simple situation arises if cooperative interactions occur only between immediately adjacent monomeric units. This corresponds to the linear Ising lattice model [18] which is the basis for the Zimm-Bragg thermodynamic theory of the cooperative helix coil conformational transition of linear polymers [3]. Considering the processes in a simple chain of = 4... 

Lee T D and Yang C N 1952 Statistical theory of equations of state and phase transitions II. Lattice gas and Ising models Phys. Rev. 87 410... 

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

It is not surprising that attempts have been made to derive equations of state along purely theoretical lines. This was done by Flory, Orwoll and Vrij (1964) using a lattice model, Simha and Somcynsky (1969) (hole model) and Sanchez and Lacombe (1976) (Ising fluid lattice model). These theories have a statistical-mechanical nature they all express the state parameters in a reduced dimensionless form. The reducing parameters contain the molecular characteristics of the system, but these have to be partly adapted in order to be in agreement with the experimental data. The final equations of state are accurate, but their usefulness is limited because of their mathematical complexity. 

Figure 4.26 The log of the average relaxation time versus Adam-Gibbs exponent U jlTS for the facilitated Ising model on the square lattice. The points are from Monte Carlo simulations and the straight line through the points is the prediction of the Adam-Gibbs theory. (From Fredrickson 1988, with permission from the Annual Review of Physical Chemistry, Volume 39, 1988, by Annual Reviews, Inc.)...

Certainly the most important models for the development of modem scaling theory of critical phenomena have been the discrete Ising model of ferromagnetism and its antipode - the continuum van der Waals model of fluid. The widespread belief is that real fluids and the lattice-gas 3D-model belong to the same universality class but the absence of any particle-hole-type symmetry in fluids requires the revised scaling EOS. The mixed variables were introduced to modify the original Widom EOS and account the possible singularity of the rectilinear diameter. 

In order to understand why Landau s theory is inaccurate, let ns recall the justification of eq. (14) in terms of the coarse-graining eq. (13), where short wavelength fluctuations of a microscopic model [such as the Ising model, eq. (1)] are eliminated. In fact, if L in eq. (13) would be the lattice spacing a,... 

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