Abstract We considered a long-range Ising model under Glauber dynamics and calculated the difference from the mean-field approximation in a finite-size system using perturbation theory. To deal with the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, we assumed that certain types of extensive properties have a Gaussian distribution, which turned out to be equivalent to the Kirkwood superposition approximation within the range of first-order perturbation. After several calculations, ordinary differential equations that describe the time development of a two-body correlation were derived. This discussion is the generalization of our previous study which developed a similar consideration on the infinite-range Ising model. The results of the calculation fit those of the numerical simulations for the case in which the decay of the interaction was sufficiently slow; however, they exhibited different behaviors when the decay became rapid.
Abstract We considered a long-range Ising model under Glauber dynamics and calculated the difference from the mean-field approximation in a finite-size system using perturbation theory. To deal with the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, we assumed that certain types of extensive properties have a Gaussian distribution, which turned out to be equivalent to the Kirkwood superposition approximation within the range of first-order perturbation. After several calculations, ordinary differential equations that describe the time development of a two-body correlation were derived. This discussion is the generalization of our previous study which developed a similar consideration on the infinite-range Ising model. The results of the calculation fit those of the numerical simulations for the case in which the decay of the interaction was sufficiently slow; however, they exhibited different behaviors when the decay became rapid.
Abstract We consider an infinite-range Ising model under the Glauber dynamics and determine the finite-size effect on the distribution of two spin variables as a perturbation of O 1 / N . Based on several considerations, ordinary differential equations are derived for describing the time development of both a two-body correlation and the autocorrelation function of magnetization. The results of the calculation fit the simulation results, unless the perturbation theory breaks down because of critical phenomena or magnetization reversal.
A version of the Ising model is developed in which the spin variables can be treated accurately in the continuum approximation. The perturbation series, both above and below the critical temperature Tc, is examined, and it is shown that there is a shift of Tc from its mean-field value proportional to ${q}^{\ensuremath{-}2}\mathrm{ln}q$, as well as the well-known shift proportional to ${q}^{\ensuremath{-}1}$; here q is the number of mutually interacting particles. It is shown, using renormalization theory, that there is a perturbation series in ${q}^{\ensuremath{-}1}{|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}\frac{1}{2}}$ for which all terms are finite in the limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$, if the shift of Tc is put in correctly. For the two-dimensional model, the shift is shown to be proportional to ${q}^{\ensuremath{-}1}\mathrm{ln}q$. Conditions are derived for a finite system to display critical behavior characteristic of three, two, one, or zero dimensions. It is shown how similar results can be obtained for a model similar to the Heisenberg model and for the standard Ising and Heisenberg models with interactions extending over many neighbors. A comparison is made between previously calculated numerical results for Tc and the asymptotic forms derived here.
Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent Ising models in two and three dimensions. This constitutes the first-order approximation which improves the mean-field theory (the zeroth-order approximation) by incorporating specific short-range correlations into the distribution function. We first calculate the single-spin expectation value. Although the dynamic equation may need to be solved by using a numerical method, its approximate solution in the pretransitional region can be studied in detail. We then calculate the two-spin correlation function. This requires yet another extension of Bethe's method. Again approximate solutions can be obtained in the pretransitional region. Our results for the static pair correlation functions are equivalent to that of Elliott and Marshall, but our derivation is simpler. We also calculate the time-dependent correlation function for the magnetization, and reproduce the fluctuation-dissipation relation.
Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.
We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blöte who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .
We have studied the critical phenomena of the Ising model with long-range interaction and the Ising model with spatially correlated random field, both of which are expected to change the effective dimension of the system continuously.The investigation was conducted from the numerical viewpoint by using the Markov chain Monte Carlo method.Second order phase transitions can be classified into the universality classes and the dimensionality of the Ising model changes the universality class: the one-dimensional Ising model has no long-range order at finite temperatures (lower critical dimension), four or higher dimensional model has the mean-field universality (upper critical dimension), two-dimensional Ising model belongs to different universality from either of them, and so on.Although both of the Ising models with long-range interaction and the Ising model with spatially correlated random field are believed to change their critical behavior from its real spatial dimension into different effective dimensions, their detailed behavior has not been known well because of the notorious difficulties in analysis.In this work, we have obtained the detailed behavior of these models by introducing and developing the various methods.To investigate the Ising model with long-range interaction, we introduced the Fukui-Todo method, which drastically reduces the computational cost from O(N 2 ) to O(N ), where N is the number of spins, and developed the generalized Ewald summation, the improved estimator, and the combined Binder ratio technique, which improves the convergence of the estimation.These techniques lead to the successful estimation of the boundary of the universality classes.We also investigated the Ising model with correlated random field by introducing the method for correlated random number generation and the various techniques for the precise analysis.Although the present numerical investigations suggested that the correlated random field causes a shift of effective dimensionality, they also revealed the difficulty for stronger random-field correlations because of the huge finite-size effects.i Prof. Synge