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Journal of Physics C Solid State Physics

Dynamics of diluted Ising chains-exact results for a wide class of single-spin-flip dynamics

C K Harris

Published Dec 20, 1983• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/16/35/015

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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.
The author presents an investigation of the dynamic critical behaviour near the percolation bicritical point of the 1D bond-diluted Glauber Ising model. The expected bicritical dynamic scaling form is rigorously established for the decay functions describing the decay of the equilibrium wavevector-dependent correlation function and the decay of the magnetisation from a (non-equilibrium) wavevector-dependent initial state. The corresponding characteristic decay times are evaluated exactly and exhibit a cross over from 'pure' to percolation dominated behaviour on increasing the dilution. The results in the percolation regime are in agreement with recent calculations in which domain boundary diffusion methods are applied to a wider class of kinetic Ising models.
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.
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.
The Glauber dynamics of a bond-diluted Ising model on a Bethe lattice (a random graph with fixed connectivity) is investigated by an approximate theory which provides exact results for equilibrium properties. The time-dependent solutions of the dynamical system derived by this method are in good agreement with the results obtained by Monte Carlo simulations in almost all situations. Furthermore, the derived dynamical system exhibits a remarkable phenomenon that the magnetization shows multi-step relaxations at intermediate time scales in a low-temperature part of the Griffiths phase without bond percolation clusters.