Generalized Bragg-Williams equation for systems with an arbitrary long-range interaction

Kryzhanovsky, Boris; Litinskii, Leonid

doi:10.1134/s1064562414070357

Cited by 26 publications

(41 citation statements)

References 7 publications

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“…In a general case, the distribution of the energies of the states from the n-vicinity is unknown. However, in [12,13] we found the exact expressions for the mean energy n E and the variance 2 n  in terms of the characteristics of the connection matrix. Numerous computer simulations show that the distribution of energies from n  is a single-mode one and the Gaussian density with the calculated parameters n E and 2 n  approximates it rather accurately (see [13]).…”

Section: Introductionmentioning

confidence: 95%

Applicability of n-vicinity method for calculation of free energy of Ising model

Kryzhanovsky

Litinskii

2017

Physica A: Statistical Mechanics and its Applications

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In a previous work, the n-vicinity method for approximate calculation of the partition function of a spin system was proposed. The equation of state was obtained in the most general form. In the present paper, we analyze the applicability of this method for the Ising model on a D-dimensional cubic lattice. The equation of state is solved for an arbitrary dimension D and the behavior of the free energy is analyzed. As expected, for large dimensions, 3 D  , the system demonstrates a phase transition of the second kind. In this case, we obtain an analytical expression for the critical value of the inverse temperature. When 37 D  this expression is in a very good agreement with the results of computer simulations. In the case of small dimensions 3 D  , there is a noticeable discrepancy with the known exact results.

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Section: Introductionmentioning

confidence: 95%

Applicability of n-vicinity method for calculation of free energy of Ising model

Kryzhanovsky

Litinskii

2017

Physica A: Statistical Mechanics and its Applications

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“…(2.1). That is we use the n-vicinity method [8,9] and the same as in [7] we approximated it by polynomials of even degrees.…”

Section: Discussionmentioning

confidence: 99%

“…In the framework of this method, we choose an initial configuration 0  N sR and all other configurations we group in its n-vicinities Here the subscript "nv" denotes the n-vicinity. Justification of the n-vicinity method we published in [8], the boundaries of its applicability for Ising models on hypercubes we analyzed in [9]. In Section 3, for the planar Ising model we discuss the results following from this approach.…”

Section: Introductionmentioning

confidence: 99%

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Spectral density and calculation of free energy

Litinskii

Kryzhanovsky

2018

Physica A: Statistical Mechanics and its Applications

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For planar and cubic Ising models, we examined two ways of approximation of a spectral density that describes a degeneracy of energy levels. We approximated the exponent of the spectral density by polynomials of even degrees and using our n-vicinity method [8,9]. According our analysis, the free energy is almost independent of the chosen method of approximation. However, its derivatives depend on the way of approximation and substantially differ in the neighborhood of a critical temperature. Our calculations showed that when approximating by polynomials the system necessarily finds itself in the ground state at a finite temperature, which is forbidden. The n-vicinity method approximates the derivatives of the free energy correctly for the cubic Ising model and it works poorly in the planar case.Key words: Ising model, free energy, spectral density, n-vicinity method.In the both formulas, J is the interaction constant between neighboring spins.1 The factor N in the exponent means that we calculate the energy () E s per one spin (see Eq. (1.4).

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“…In neuroinformatics, we have to know the global minimum when developing associative memory systems [18][19][20][21] and constructing neural networks and neural network minimization algorithms [22][23][24]. In physics, the knowledge of the global energy minimum is most frequently necessary when studying the behavior of spin glass systems [25][26][27][28][29][30][31][32][33][34][35] and even when describing four-photon mixing in nonlinear media [36,37]. The question of calculation of the global minimum depth has been discussed over the years.…”

Section: Introductionmentioning

confidence: 99%

Global Minimum Depth in Edwards-Anderson Model

Karandashev¹,

Kryzhanovsky²

2019

Engineering Applications of Neural Networks

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In the literature the most frequently cited data are quite contradictory, and there is no consensus on the global minimum value of 2D Edwards-Anderson (2D EA) Ising model. By means of computer simulations, with the help of exact polynomial Schraudolph-Kamenetsky algorithm, we examined the global minimum depth in 2D EA-type models. We found a dependence of the global minimum depth on the dimension of the problem N and obtained its asymptotic value in the limit N→∞. We believe these evaluations can be further used for examining the behavior of 2D Bayesian models often used in machine learning and image processing.

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Generalized Bragg-Williams equation for systems with an arbitrary long-range interaction

Cited by 26 publications

References 7 publications

Applicability of n-vicinity method for calculation of free energy of Ising model

Applicability of n-vicinity method for calculation of free energy of Ising model

Spectral density and calculation of free energy

Global Minimum Depth in Edwards-Anderson Model

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