Search

Combustion simulations are the key aspect for full-scale 3D simulations of modern and perspective engines for aerospace propulsion. The present work studies an opportunity to solve chemical kinetics problems using artificial neural networks. Using classical numerical methods, sets of training data were produced. Choosing among different architectures of multi-layer neural networks and tuning their parameters, a rather simple model was worked out capable to solve this problem. The neural network obtained works in a recursive mode, and it can predict the behavior of chemical multi-species dynamical system by many steps.

We analyze a finite one-dimensional Ising system with periodic boundary conditions taking into account an arbitrary long-range interaction. We examine a discrete spectrum of eigenvalues of the spin connection matrix and a spectrum density of a continuous distribution obtained in the limit $$L \to \infty $$ (L is the linear size of the system). We apply our results to particular cases of long-range interactions decreasing with distance.

The connection matrix of the Ising model on a d -dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.

In the present paper we analyzed a change in the density of states of a two-dimensional Ising model when a next-next-neighbor interaction is introduced. In other words, we examined two-dimensional lattices with diagonal connections. The same as in a three-dimensional model in this case each spin has 6 connections. Since the model is planar, we can calculate the free energy and other characteristics of the system using a polynomial algorithm. We performed computer simulations using the Kasteleyn–Fisher algorithm, which allowed us to study changes of critical values and the density of states when a long-range interaction is taken into account. From the obtained results it follows that the interactions of the type analyzed here result in quantitative changes of the system’s characteristics, but they did not change them qualitatively. In particular, we again obtained a logarithmic divergence of the heat capacity in the critical point.

The smallness parameter determining the possibility of application of the n-vicinity method for calculating the free energy of a spin system has been found. The types of spin systems for which approximation of the n-vicinity method is valid have been determined. It is shown that this method is adequate for studying the properties of spin systems, in which the effective number of nearest neighbors is more than $$\frac{{16}}{3}$$ .

A diagonally connected two-dimensional lattice is the objective of the research. The model draws interest because each of its spins has 6 connects as in a 3D lattice. On the other hand, the planarity of the model allows us to use a strict polynomial algorithm to find its partition function and other characteristics. The Kasteleyn-Fisher algorithm is employed to carry out a computer simulation which enables us to see how the heat capacity behaves with the increasing lattice dimensionality. Given a finite lattice dimensionality, it is impossible to draw a definite conclusion, yet there is every reason to believe that the heat capacity diverges logarithmically at the critical point.

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of it eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions., 10 pages, 1 fugure

For a 1D Ising model, we obtained an exact expression for the spectral density in an n-vicinity of the ground state and explained why our n-vicinity method with the Gaussian approximation of the spectral density did not applicable in this case. We also found an analytical expression for the distribution of magnetization at an arbitrary temperature. When the temperature tends to zero the distribution of magnetization gradually flattens.

The paper gives the results of a numerical simulation of a two-dimensional Ising model built on finite lattices of dimension L = 50, 100, …, 500. Approximate analytical formulae for the spectral energy density are offered. Derived from Onsager’s solution with consideration of the finite size of the system, the formulae agree well with the simulation results.

The relationship between the spectral density and the free energy of a spin system is analyzed. Analytical expressions for calculating the spectral density for exactly solvable models are derived. The approach is tested as applied to the one-dimensional Ising model. Additionally, the approach is used to analyze the spectral density of the two-dimensional Ising model, the Bethe-lattice model, and the mean-field model. It is shown that even a small change in the spectral density is able to radically change the parameters of the system.

Numerical methods are used to examine the thermodynamic characteristics of the twodimensional Ising model as a function of the number of spins N. Onsager’s solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N. Due to the finite extent of the system the critical temperature can only be determined to some accuracy.

The previously developed n-vicinity method allows us to calculate accurately critical values of inverse temperatures for Ising models with short-range interaction. We generalize the method to the case of long-range interactions in spin systems and obtain theoretical formulas for the inverse temperatures in terms of the spin interaction constants. The comparison of our theoretical estimates with computer simulations for the two- and three-dimensional Ising models shows that the larger the dimension of the problem the better their agreement., Comment: 7 pages, 3 tables, and 3 figures

We examine connection matrices of Ising systems with long-rang interaction on d-dimensional hypercube lattices of linear dimensions L. We express the eigenvectors of these matrices as the Kronecker products of the eigenvectors for the one-dimensional Ising system. The eigenvalues of the connection matrices are polynomials of the d-th degree of the eigenvalues for the one-dimensional system. We show that including of the long-range interaction does not remove the degeneracy of the eigenvalues of the connection matrix. We analyze the eigenvalue spectral density in the limit L go to \infty. In the case of the continuous spectrum, for d < 3 we obtain analytical formulas that describe the influence of the long-range interaction on the spectral density and the crucial changes of the spectrum., 10 pages, submitted in J. of Physics A

For the 2D Ising model, we analyzed dependences of thermodynamic characteristics on number of spins by means of computer simulations. We compared experimental data obtained using the Fisher-Kasteleyn algorithm on a square lattice with $N=l{\times}l$ spins and the asymptotic Onsager solution ($N\to\infty$). We derived empirical expressions for critical parameters as functions of $N$ and generalized the Onsager solution on the case of a finite-size lattice. Our analytical expressions for the free energy and its derivatives (the internal energy, the energy dispersion and the heat capacity) describe accurately the results of computer simulations. We showed that when $N$ increased the heat capacity in the critical point increased as $lnN$. We specified restrictions on the accuracy of the critical temperature due to finite size of our system. Also in the finite-dimensional case, we obtained expressions describing temperature dependences of the magnetization and the correlation length. They are in a good qualitative agreement with the results of computer simulations by means of the dynamic Metropolis Monte Carlo method., Comment: 19 pages, 7 figures, 1 table

For a multidimensional Ising model, we expressed eigenvalues of the connection matrix in terms of the spin-spin interaction constants and trigonometric polynomials. In such systems, the eigenvectors are the Kronecker products of the well-known eigenvectors for the one-dimensional case. When boundary conditions are periodic, it is possible to obtain rigorous expressions for the eigenvalues when there is an arbitrary long-range interaction in the system.

We examine multidimensional Ising systems on hypercube lattices and calculate analytically the eigenvalues of their connection matrices. We express the eigenvalues in terms of spin–spin​ interaction constants and the eigenvalues of the one-dimensional Ising connection matrix (the latter are well known). To do this we present the eigenvectors as Kronecker products of the eigenvectors of the one-dimensional Ising connection matrix. For periodic boundary conditions, it is possible to obtain exact results for interactions with an arbitrary large number of neighboring spins. We present exact expressions for the eigenvalues for two- and three-dimensional Ising connection matrices accounting for the first five coordination spheres (that is interactions up to next-next-next-next nearest neighbors). In the case of free-boundary systems, we show that in the two and three dimensions the exact expressions could be obtained only if we account for interactions with spins of not more than first three coordination spheres.

We have investigated the density of states of the Edwards-Anderson model and derived an approximation formula which agrees well with the results of numerical experiments. It is important that the formula can well approximate not only the density of states, but also its first and second derivatives, which are most valuable for obtaining the critical parameters of the system. The evaluations can be further used for examining the behavior of 2D Ising models at different temperatures, particularly for tackling Bayesian inference problems and learning algorithms.