Ising model simulations

Typical surfaces observed in Ising model simulations are illustrated in Fig. 2. The size and extent of adatom and vacancy clusters increases with the temperature. Above a transition temperature (T. 62 for the surface illustrated), the clusters percolate. That is, some of the clusters link up to produce a connected network over the entire surface. Above Tj, crystal growth can proceed without two-dimensional nucleation, since large clusters are an inherent part of the interface structure. Finite growth rates are expected at arbitrarily small values of the supersaturation. 

The essential influence of surface roughening is also present in this model. Grand canonical Monte Carlo calculations were used to generate adatom populations at various temperatures up to Chemical potentials corresponding to those in the bulk LJ crystal were used, and these produced adatom densities that increased with temperature and roughly approximated the values observed in Ising model simulations below T. ... 

A. Lukkarinen, K. Kaski, and D. B. Abraham, Mechanisms of fluid spreading Ising model simulations, Phys. Rev. E 51, 2199-2202 (1995). 

Figure 5. Plot of the actual error versus the internally estimated standard deviation of the energy error for Ising model simulations with the Metropolis algorithm on a 16 x 16 lattice with a different linear congruential sequence at each lattice site. The dashed line shows the results when all the linear congruential sequences were started with the same seeds but with different additive constants. The solid line shows the results when the sequences were started with different seeds. We expect around 95% of the points to be below the dotted line (which represents an error of two standard deviations) with a good generator.

P. Coddington, Tests of Random Number Generators Using Ising Model Simulations, Int. J. Mod. Phys. C 7(3), 295-303 (1996). 

Figure 4. The correlation time for the 2D Ising model simulated using the Wolff algorithm. The measurements deviate from a straight line for small system sizes L, but a fit to the larger sizes, indicated by the dashed line, gives a reasonable figure of z = 0.25 0.02 for the dynamic exponent of the algorithm.

Figure 6. The correlation time T,tepl of the 20 Ising model simulated using the Wolff algorithm, measured in units of Monte Carlo steps (i.e., cluster flips). The fit gives us a value of zstep, = 0.50 0.01 for the corresponding dynamic exponent.

For the novice in Ising model simulations, we present here a short program (Fig. 3) and its description, in old-fashioned Fortran it looks very similar in BASIC. It treats an interface by keeping the upper and the lower border of an LxLxL lattice all spin up and all spin down, respecively. It should convince the reader that the basics of Monte... 

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. 

In both of these cases, represents the energy of two noninteracting Ising spin systems, one system having spins s over one sublattice and the other spins s over the other sublattice. Indeed, the Q2R CA actually provides an efficient microcanonical algorithm for performing parallel simulation of the Ising model (see discussion in section 7.1.5). 

Fig. 2 Typical Ising model surfaces produced by computer simulations. In this system Tji=0S2, in terms of the dimensionless temperature shown on the figure.

The approximations of the superposition-type like equation (2.3.54), are used in those problems of theoreticals physics when other-kind expansions (e.g., in powers of a small parameter) cannot be employed. First of all, we mean physics of phase transitions and critical phenomena [4, 13-15] where there are no small parameters at all. Neglect of the higher correlation forms a(ml in (2.3.54) introduces into solution errors which cannot be, in fact, estimated within the framework of the method used. That is, accuracy of the superposition-like approximations could be obtained by a comparison with either simplest explicitly solvable models (like the Ising model in the theory of phase transitions) or with results of direct computer simulations. Note, first of all, several distinctive features of the superposition approximations. 

Here, we adopted a spin analogy/lattice gas model, or SRS model, as shown in Fig. 1.28(a), which represents an oversimplified molecular structure yet still captures the essence of the molecule-surface interactions for describing SME profiles. Similar techniques using the Ising model to study other physical systems have been investigated [148,149,160] however, none of the literature deals with the simulation of PFPE lubricant dynamics described here. 

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.)...

In many physically important cases of localized adsorption, each adatom of the compact monolayer covers effectively n > 1 adsorption sites [3.87-3.89, 3.98, 3.122, 3.191, 3.214, 3.261]. Such a multisite or 1/n adsorption can be caused by a crystallographic Me-S misfit, i.e., the adatom diameter exceeds the distance between two neighboring adsorption sites, and/or by a partial charge of adatoms (A < 1 in eq. (3.2)), i.e., a partly ionic character of the Meads-S bond. The theoretical treatment of a /n adsorption differs from the description of the 1/1 adsorption by a simple Ising model. It implies the so-called hard-core lattice gas models with different approximations [3.214, 3.262-3.266]. Generally, these theoretical approaches can only be applied far away from the critical conditions for a first order phase transition. In addition, Monte Carlo simulations are a reliable tool for obtaining valuable information on both the shape of isotherms and the critical conditions of a 1/n adsorption [3.214, 3.265-3.267]. 

Zunger A, Wang LG, Hart GLW, Sanati M (2002) Obtaining Ising-like expansions for binary alloys from first-principles. Modell Simul Mater Sci Eng 10 685... 

Figure 2 shows the optimized histogram for the two-dimensional ferromagnetic Ising model. The optimized histogram is no longer flat, but a peak evolves at the critical region around E —1.41 N of the transition. The feedback of the local diffusivity reallocates resources towards the bottlenecks of the simulation which have been identified by a suppressed local diffusivity. 

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