The time evolution of two ferromagnetic Ising cubic systems, which differ by the orientation of a single spin at t=0, has been studied in a magnetic field B by computer simulation. The critical field B c (T), above which the previous initial damage does not spread, has been determined. The magnetisation for B=B c is very well approximated by M(B c )=1-exp(-T c /T) in the range of T values investigated. The concentration of minority spins at B c (T) differs strongly from the corresponding concentration for percolation at T.
The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at $T=\ensuremath{\infty}$ and magnetization
M=0, an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization
M0 in one of the configurations upon quenching the system at
TC, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent ${\ensuremath{\theta}}_{D}=1.915(3)$, which is much larger than the exponent $\ensuremath{\theta}=0.197$ characteristic of the initial increase of the magnetization
M(t). Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic
(⟨R2(t)⟩) grows with an exponent ${z}^{\ensuremath{\ast}}\ensuremath{\approx}\ensuremath{\eta}\ensuremath{\approx}1.9$, which is the same, within error bars, as the exponent ${\ensuremath{\theta}}_{D}$. However, the survival probability of the epidemics reaches a plateau so that $\ensuremath{\delta}=0$. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at ${T}_{D}\ensuremath{\simeq}0.51{T}_{C}$, where all the measured observables exhibit power laws with exponents ${\ensuremath{\theta}}_{D}=1.026(3)$, $\ensuremath{\delta}=0.133(1)$, and ${z}^{\ensuremath{\ast}}=1.74(3)$.
We investigate the spreading of damage in the three-dimensional Ising model by means of large-scale Monte Carlo simulations. Within the Glauber dynamics we use different rules for the order in which the sites are updated. We find that the stationary damage values and the spreading temperature are different for different update order. In particular, random update order leads to larger damage and a lower spreading temperature than regular order. Consequently, damage spreading in the Ising model is nonuniversal not only with respect to different update algorithms (e.g., Glauber vs heat-bath dynamics) as already known, but even with respect to the order of sites.
The dynamics of a mixed spin Ising model with a crystal field on the square lattice are investigated by using the damage spreading technique and Metropolis sampling method.The simulation results reveal that the damage(Hamming distance)shows a very similar temperature dependence to that of the magnetization.The Hamming distance decreases with temperature and there is a critical temperature,beyond which the phase space trajectories of two systems with different initial configurations consist with each other.The critical temperature increases when the strength of the crystal field decreases,and the mixed spin Ising model will degenerate to a standard Ising model under a considerable weak crystal field.The critical temperature decreases to zero continuously as the strength of the crystal filed increases.There is not any first order phase transition observed in the above process and thus no tricritical point exists in this system.
By studying the time evolution of two configurations submitted to the same thermal noise, we investigate phase space properties of the three-dimensional ± J Ising spin glass. Our results for the distance between states are consistent with the picture of phase space having a multivalley structure below T c . By fixing the damage at one site, we also study the Hamming distance between the two configurations at its critical point, where the distribution of probability of getting damaged n times is found to be multifractal.
The time evolution of the Hamming distance (damage spreading) for the and Ising models on the square lattice is performed with a special metropolis dynamics algorithm. Two distinct regimes are observed according to the temperature range for both models: a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size; a high-temperature one where the distance vanishes in the long-time limit. Using the finite size scaling method, the dynamical phase transition (damage spreading transition) temperature is obtained as for the Ising model.