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Neimenggu Shi-da xuebao. Zhexue shehui kexue hanwen ban

Monte Carlo Calculation of the Ising Model on the Square Lattice Under Glauber and Kawasaki Dynamics

Wang Chun-an

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The critical temperature of the paramagnetic-ferromagnetic phase transition on the Ising model of the square lattice is studied by means of the Monte Carlo method.The Glauber and Kawasaki dynamics are considered in the calculation for the stochastic and sequent processes and their effects on the critical temperatures are discussed.It's shown that the critical temperature can be calculated in the stochastic and sequent processes when the Glauber dynamic is in the lead.But only in the stochastic process the critical temperature can be calculated when the Kawasaki dynamic is in the lead.

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The critical temperature of the paramagnetic-ferromagnetic phase transition on the Ising model of the square lattice is studied by means of the Monte Carlo method.The Glauber and Kawasaki dynamics are considered in the calculation for the stochastic and sequent processes and their effects on the critical temperatures are discussed.It's shown that the critical temperature can be calculated in the stochastic and sequent processes when the Glauber dynamic is in the lead.But only in the stochastic process the critical temperature can be calculated when the Kawasaki dynamic is in the lead.
We consider a two-dimensional antiferromagnet Ising system interacting with a heat bath at temperature T. The dynamics of the system is simulated by two competing stochastic processes: the two-spin-exchange Kawasaki kinetics at temperature T>0 and the one-spin-flip Glauber dynamics at T(G)-->0(-), which mimics the increase of the energy of the system. These two processes have probabilities 1-p and p, respectively. Monte Carlo simulations were employed to determine the phase diagram for the stationary states of the model and the corresponding critical exponents. Contrary to the ferromagnetic case, the phase diagram obtained does not exhibit the phenomenon of self-organization: for any nonzero value of the competing parameter p, and for any value of T, the only stationary phase which remains is the ferromagnetic one. At the phase transition between the antiferromagnetic and paramagnetic phases, at p=0, the values found for the critical exponents agree with those of the corresponding equilibrium Ising model.
This paper studies, by means of Monte Carlo simulations, an open ferromagnetic spin-3/2 Ashkin-Teller model defined on a square lattice. This model can be viewed as the superposition of two Ising models which are coupled by a four-spin interaction strength K4while in each Ising model, the nearest-neighbor interaction constant is K2. The model is here submitted to two stochastic dynamics that consist of two processes; the first one is the Glauber dynamics which simulates the contact of the system with a heat bath at temperature T and the second one is the Kawasaki dynamics which simulates the continuous energy flux into the system from an external source. The Glauber single-spin flip process happens with the probability p while the Kawasaki spin-exchange process between neighboring sites occurs with the probability q = 1− p. The temperature-dependence of the system magnetizations for fixed coupling constants and lattice anisotropy has been thoroughly investigated. Several thermodynamic phases and phase transitions have been obtained as well as multicritical points. When the Kawasaki dynamics becomes dominant, self-organized antiferromagnetic phases are generated. Several phase diagrams have been devised to illustrate model thermodynamic properties.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
By use of the Monte Carlo method, a study is made on the square lattices Ising model for different boundary conditions. Combined with the finite size scalar theory, the relevant transition temperatures are obtained and the dynamic processes of the square lattices Ising model in different boundary conditions near the critical point are observed simulatedly. It shows that the phase transition temperatures of the finite Ising systems in different boundary conditions are different, and the speeds getting equilibrium of the systems are different as well, but the difference vanishes at the dynamic limit.