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Journal of Physics A Mathematical and General

Nonequilibrium stationary states in the 1-d BEG model: first- and second-order phase transitions

José F. F. Mendes, Eduardo Lage

Published Aug 7, 1993• 0 citations
Publisher

Abstract

Monte Carlo simulations and finite-size scaling were used to study the one-dimensional BEG (Blume-Emergy-Griffiths) model under the influence of two competing dynamics: single spin flip (Glauber dynamics at finite temperature T) and random exchange to two spins (Kawasaki dynamics at infinite temperature). The phase diagram shows first- and second-order phase transitions. The critical exponents are obtained for different values of the rates and the range of the exchanges. The authors find beta =0.325+or-0.020, gamma =0.996+or-0.004 and nu =1.71+or-0.01 for long-range exchanges and equal intrinsic rates. However, for different rates and the nearest-neighbour exchanges they obtain beta =0.07+or-0.03, gamma =1.60+or-0.06 and nu =1.53+or-0.08, thus exhibiting non-universal behaviour.

DOI
10.1088/0305-4470/26/15/033

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Monte Carlo simulations and finite-size scaling were used to study the one-dimensional BEG (Blume-Emergy-Griffiths) model under the influence of two competing dynamics: single spin flip (Glauber dynamics at finite temperature T) and random exchange to two spins (Kawasaki dynamics at infinite temperature). The phase diagram shows first- and second-order phase transitions. The critical exponents are obtained for different values of the rates and the range of the exchanges. The authors find beta =0.325+or-0.020, gamma =0.996+or-0.004 and nu =1.71+or-0.01 for long-range exchanges and equal intrinsic rates. However, for different rates and the nearest-neighbour exchanges they obtain beta =0.07+or-0.03, gamma =1.60+or-0.06 and nu =1.53+or-0.08, thus exhibiting non-universal behaviour.
The authors investigate the one-dimensional Blume-Emery-Griffiths model with Kawasaki dynamics, using domain-wall arguments (DWA) and Monte Carlo simulations (MCS) (conventional and Gillespie algorithm), after a quench from a disordered state to low temperatures. They observe that this model exhibits domain scaling behaviour, controlled by a universal exponent (x=1/3) as in other dimensions for model B. However, they find also that the critical exponent z is not universal and depends on the coupling constants of the Hamiltonian. The results of DWA are consistent with those of MCS.
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.
We analyze the stationary nonequilibrium state of a spin system evolving under combined flips at temperature ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}$ and exchanges at \ensuremath{\beta}=0. Computer simulations indicate that the phase transition changes, as the exchange rate \ensuremath{\Gamma} increases, from second to first order. We interpret this as a changeover from Ising behavior for small \ensuremath{\Gamma} to mean-field behavior for \ensuremath{\Gamma}\ensuremath{\rightarrow}\ensuremath{\infty} where the magnetization is discontinuous in the ``continuum limit.''
A family of nonequilibrium kinetic Ising models, introduced earlier, evolving under the competing effect of spin flips at zero temperature and nearest-neighbour random spin exchanges, is further investigated. By increasing the range of spin exchanges and/or their strength the nature of the phase transition 'Ising-to-active' becomes of (dynamic) mean-field type and a first-order tricritical point is located at the Glauber ( delta =0) limit. Corrections to the mean-field theory are evaluated up to sixth order in a cluster approximation and found to give good results concerning the phase boundary and the critical exponent beta of the order parameter which is obtained as beta approximately=1.0.
Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.
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.