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Theoretical and Mathematical Physics

Equation of state for the 3-D Ising model from microscopic level calculations

V. V. Dukhovii, Mykhailo P. Kozlovskii, Ihor V. Pylyuk

Published May 1, 1996• 3 citations
Publisher
DOI
10.1007/bf02071378

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Abstract A method for calculating the phase transition temperature Tc based on the theoretical description of the critical behaviour of the model system on the microscopic level is proposed. The main idea of such a calculation is on based a systematic consideration of the influence of the short‐ and longwavelength fluctuations of the spin density on the value of the phase transition temperature. The equation for Tc is obtained as a result of the analysis of recursion relations for the coefficients of Landau‐Ginzburg‐Wilson‐type “effective forms” and consideration of their singular properties in the critical region. As the interaction potential we use an exponentially decreasing function with effective range b. The equation for the critical temperature is obtained and the value of Tc is calculated. The dependence of Tc on the ratio b/c (c is the lattice constant) is investigated. We obtain the value βcJ = 0.211 in the case when the Ising model is approximated by the quartic measure density (ϱ4 model) and for nearest‐neighbours interaction with the interaction constant J.

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In the companion Letter we have highlighted the dynamical universality class of the Nagle-Kardar model, where a mean-field interaction is added to the one-dimensional nearest-neighbor Ising model. Starting from the microscopic Glauber dynamics, this paper provides a complete derivation of the Fokker-Planck equation that describes the time evolution of the macroscopic variables (magnetization and defect density) appearing in the model's Hamiltonian. The study of the Langevin equation, associated with the Fokker-Planck equation, allowed us to prove that the model belongs to the universal class of systems with diffusive dynamics and non-conserved order parameter (model A). To this goal, we used several features of the model at equilibrium, including the phase diagram and the fluctuations of macroscopic variables, which are here discussed for completeness. The derivation of the Fokker-Planck equation requires the solution of some combinatorial problems that appear in the counting of configurations at an intermediate level between the microscopic Glauber dynamics and the macroscopic Fokker-Planck one. Finally, we apply both the Glauber and the Langevin dynamics to the study of the average first passage time between local equilibrium states. We confirm that this time obeys an exponential Arrhenius law in terms of system's size, offering a direct link between microscopic energy landscapes and macroscopic relaxation mechanisms.

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Abstract A method for calculating the phase transition temperature Tc based on the theoretical description of the critical behaviour of the model system on the microscopic level is proposed. The main idea of such a calculation is on based a systematic consideration of the influence of the short‐ and longwavelength fluctuations of the spin density on the value of the phase transition temperature. The equation for Tc is obtained as a result of the analysis of recursion relations for the coefficients of Landau‐Ginzburg‐Wilson‐type “effective forms” and consideration of their singular properties in the critical region. As the interaction potential we use an exponentially decreasing function with effective range b. The equation for the critical temperature is obtained and the value of Tc is calculated. The dependence of Tc on the ratio b/c (c is the lattice constant) is investigated. We obtain the value βcJ = 0.211 in the case when the Ising model is approximated by the quartic measure density (ϱ4 model) and for nearest‐neighbours interaction with the interaction constant J.
Equation of State and Thermodynamic Functions of the Ising-like Magnet at T>TcT>T_cT>Tc​
The 3D Ising-like system in the external field is described using the non-perturbative collective variables method. The universal as well as nonuniversal system characteristics are obtained within the framework of this approach. The calculations are carried out on the microscopic level starting from the Hamiltonian. They are valid in the whole h−Th-Th−T plane of the critical region. It is established, that the contributions related with wave vector values $\vk\to0$ exhibit the properties of the total system near the critical point. The behaviour of the susceptibility as function of the temperature in the presence of the field is investigated. The locations of the maximums susceptibility on the temperature scale for different values of the field are established.
In the companion Letter we have highlighted the dynamical universality class of the Nagle-Kardar model, where a mean-field interaction is added to the one-dimensional nearest-neighbor Ising model. Starting from the microscopic Glauber dynamics, this paper provides a complete derivation of the Fokker-Planck equation that describes the time evolution of the macroscopic variables (magnetization and defect density) appearing in the model's Hamiltonian. The study of the Langevin equation, associated with the Fokker-Planck equation, allowed us to prove that the model belongs to the universal class of systems with diffusive dynamics and non-conserved order parameter (model A). To this goal, we used several features of the model at equilibrium, including the phase diagram and the fluctuations of macroscopic variables, which are here discussed for completeness. The derivation of the Fokker-Planck equation requires the solution of some combinatorial problems that appear in the counting of configurations at an intermediate level between the microscopic Glauber dynamics and the macroscopic Fokker-Planck one. Finally, we apply both the Glauber and the Langevin dynamics to the study of the average first passage time between local equilibrium states. We confirm that this time obeys an exponential Arrhenius law in terms of system's size, offering a direct link between microscopic energy landscapes and macroscopic relaxation mechanisms.