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Progress of Theoretical Physics Supplement

Non-Equilibrium Dynamics of the Disordered Ising Model

S. Jain

Published Jan 1, 2000• 2 citations
Publisher

Abstract

The non-equilibrium dynamics of the bond-diluted Ising model in two-dimensions (2d) is investigated numerically for a range of bond concentration 0≤p≤1. The fraction of spins which never flip, P(∞), is found to increase monotonically from zero for p = 0 with increasing bond concentration to a maximum value of about 0.46 for p = 0.5 and then decreases to zero for p = 1. For strong bond-dilution (0≤p≤0.5) we find that r(t) = P(t)-P(∞) decays exponentially to zero at large times. for weak dilution (0.975≤p ≪ 1.0), three distinct regimes are identified: a short time regime where the behaviour is purelike; an intermediate regime where the persistence probability decays nonalgebraically with time; and a long time \lqfrozen \rq regime where the domains cease to grow. Our results for strong dilution are consistent with recent work of Newman and Stein, and suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behaviour would also appear to depend crucially on the strength of the dilution present.

DOI
10.1143/ptps.138.561

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The non-equilibrium dynamics of the bond-diluted Ising model in two-dimensions (2d) is investigated numerically for a range of bond concentration 0≤p≤1. The fraction of spins which never flip, P(∞), is found to increase monotonically from zero for p = 0 with increasing bond concentration to a maximum value of about 0.46 for p = 0.5 and then decreases to zero for p = 1. For strong bond-dilution (0≤p≤0.5) we find that r(t) = P(t)-P(∞) decays exponentially to zero at large times. for weak dilution (0.975≤p ≪ 1.0), three distinct regimes are identified: a short time regime where the behaviour is purelike; an intermediate regime where the persistence probability decays nonalgebraically with time; and a long time \lqfrozen \rq regime where the domains cease to grow. Our results for strong dilution are consistent with recent work of Newman and Stein, and suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behaviour would also appear to depend crucially on the strength of the dilution present.
The nonequilibrium dynamics of the strongly diluted random-bond Ising model in two dimensions (2D) is investigated numerically. The persistence probability, P(t), of spins which do not flip by time t is found to decay to a nonzero, dilution-dependent value P(infinity). We find that p(t)=P(t)-P(infinity) decays exponentially to zero at large times. Furthermore, the fraction of spins which never flip is a monotonically increasing function over the range of bond-dilution considered. Our findings, which are consistent with a recent result of Newman and Stein [Phys. Rev. Lett. 82, 3944 (1999)], suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behavior would also appear to depend crucially on the strength of the dilution present.
Investigates the bicritical dynamics of one-dimensional bond-diluted kinetic Ising models in the percolation limit at criticality. A wide class of single-spin-flip dynamics is applied and domain boundary diffusion arguments are used to identify the dominant processes producing flipping of the ordered finite chain segments constituting the diluted chain. Exact expressions for the rate of flip as a function of length of segment are obtained and are used to derive corresponding expressions for the decay of equilibrium correlations, and of the magnetisation (from a non-equilibrium uniformly magnetised initial state) in the chain. The results are in agreement with those of earlier rigorous calculations restricted to the special case of Glauber dynamics.
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
Abstract We have performed a numerical investigation of the influence of disorder on the dynamical non-equilibrium evolution of a 3D site-diluted Ising model from a low-temperature initial state with magnetization m 0 = 1. It is shown that two-time dependences of the autocorrelation and integrated response functions for systems with spin concentrations p = 1.0, 0.95, 0.8, 0.6 and 0.5 demonstrate ageing properties with anomalous slowing-down relaxation and violation of the fluctuation-dissipation ratio. It was revealed that during non-equilibrium critical dynamics in the long-time regime t − t w ≫ t w ≫ 1 the autocorrelation functions for diluted systems are extremely slow due to the pinning of domain walls on impurity sites. We have found that the fluctuation-dissipation ratio X ∞ = 0 for diluted systems with spin concentration p < 1 while the pure system is characterized by X ∞ = 0.784 ( 7 ) . The autocorrelation function power-law delay becomes the same as for the time dependence of the magnetization in the critical point and is characterized by exponent − β / z ν . Also, for diluted systems we reveal memory effects for critical evolution in the ageing regime with realization of cyclic temperature change and quenching at T < T c .