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 a novel variational approach to solve the dynamics of Ising-like discrete spin systems. The equations are derived under mean-field approximations based on the cluster variational method. Comparison with the actual Glauber dynamics of models defined on Erdos-R\\'enyi random graphs show that some of these approximations are extremely accurate, both at equilibrium and in the off-equilibrium regime, providing the same result of the Monte Carlo simulation in a much shorter time. The models studied are the ferromagnetic kinetic Ising model (both with symmetric and partially asymmetric interactions), the random field Ising model and the Viana-Bray model. Only for the latter model we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states.
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.
We study the time-dependent properties of an Ising chain with random bonds Gaussianly distributed. We obtain some exact as well as Monte Carlo (MC) results. Remanence, as well as a seemingly logarithmic long-time decay of the magnetization and of the energy towards their equilibrium values is observed at low temperatures by means of MC simulation. The remanent values of the magnetization (
31, starting with all spins up) and of the energy are derived. A simple and explicit physical picture of the mechanism behind remanence and the logarithmic relaxation emerges. The equilibrium value of $q(t)={〈〈{\ensuremath{\sigma}}_{i}(0){\ensuremath{\sigma}}_{i}(t)〉〉}_{J}$ is obtained via the MC technique; it also seems to relax logarithmically for low temperatures. In contrast with the two-and three-dimensional cases, it is shown how any MC calculation can start immediately from an "equilibrium state" in this model, a very convenient feature for very-low-temperature MC computations. We show that if
H is switched on at
t=0, then ${[\frac{\ensuremath{\partial}m(H,t)}{\ensuremath{\partial}H}]}_{H=0}=(\frac{1}{\mathrm{kT}})[1\ensuremath{-}q(t)]$ holds exactly in any number of dimensions, where
m is the magnetization per spin. A time-dependent susceptibility $\ensuremath{\chi}(H,t)$ is defined and shown to vanish (as $H\ensuremath{\rightarrow}0$) for low enough temperatures and finite
t. The MC results for $\ensuremath{\chi}$ are in accord with this result and, if graphed versus
T, show a hump at a time-dependent temperature. Finally, we compare the exact equilibrium specific heat,
C, for this model with the results obtained by MC simulation of calorimetric measurements. Thus, a simple explicit case is exhibited of the difficulties which may arise in measurements of
C in spin-glasses due to long-time effects.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.