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The layered Ising model — mean-field and interfacial approximations

Adam Lipowski, Masuo Suzuki

Physica A Statistical Mechanics and its ApplicationsSep 1, 199330 citations

Random Ising models with transverse field : mean field theory and pair approximation

光史横田

Jan 1, 19910 citations

Ising Models and Mean-Field Theories

WORLD SCIENTIFIC eBooksNov 1, 20210 citations

Pecularities of Bethe-like approximations and long-range-interaction Ising models

James L. Monroe

The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 12, 20032 citations

Introduction to phase transitions

Antoni Planes, Avadh Saxena

The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.

Cambridge University Press eBooksApr 30, 20250 citations

Mean-field solution of a spin-1 Ising-like model

Carla Buzano

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoDec 1, 19812 citations

The Ising Model and Its Basic Characteristics in the Mean Field Approximation

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