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Physical Review A

One-dimensional kinetic Ising model with competing spin-flip and spin-exchange dynamics: Ordering in the case of long-range exchanges

Michel Droz, Zoltàn Ràcz, P. Tartaglia

Published Jun 1, 1990• 35 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.41.6621

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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.
A d=2 dimensional kinetic Ising model that evolves by a combination of spin flips and spin exchanges is investigated. The spin flips satisfy detailed balance for the equilibrium state of the Ising model at temperature T while the spin exchanges are random L\'evy flights of dimension \ensuremath{\sigma}=1.5. Our Monte Carlo (MC) simulations show that the steady state of this system displays a second-order phase transition as T is lowered. Comparing the critical fluctuations of the magnetization to those of an Ising model in which the interaction decays with distance as ${\mathit{r}}^{\mathrm{\ensuremath{-}}3.5}$, we find that, within the resolution of the MC data, the critical exponents and the scaling functions of the two systems coincide. We argue that this coincidence indicates that a recent conjecture about the random L\'evy flights generating long-range interaction of the form Veff{\mathit{V}}_{\mathrm{eff}}Veff​(r)\ensuremath{\sim}${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ is valid not only in the spherical limit and in d=1 but also in d=2.
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.
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.
Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and Kawasaki-type spin-exchange kinetics at infinite temperature T are investigated here in one dimension from the point of view of phase transition and critical behaviour. Branching annihilating random walks with an even number of offspring (on the part of the ferromagnetic domain boundaries), is a decisive process in forming the steady state of the system for a range of parameters, in the family of models considered. A wide variety of quantities characterize the critical behaviour of the system. Results of computer simulations and of a generalized mean field theory are presented and discussed.
Using a quantum formulation of the master equation we study a kinetic Ising model with competing stochastic processes: the Glauber dynamics with probability p and the Kawasaki dynamics with probability 1-p. Introducing explicitly the coupling to a heat bath and the mutual static interaction of the spins the model can be traced back exactly to a Ginzburg–Landau functional when the interaction is of long range order. The dependence of the correlation length on the temperature and on the probability p is calculated. In case that the spins are subject to flip processes the correlation length disappears for each finite temperature. In the exchange dominated case the system is strongly correlated for each temperature.
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.''