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 ${z}^{\ensuremath{\ast}}=1.74(3)$.
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.
Abstract. We present preliminary results of our numerical study of the critical dynamics of percolation observables for the two-dimensional Ising model. We consider the (Monte-Carlo) short-time evolution of the system obtained with a local heat-bath method and with the global Swendsen-Wang algorithm. In both cases, we find qualitatively different dynamic behaviors for the magnetization and Ω, the order parameter of the percolation transition. This may have implications for the recent attempts to describe the dynamics of the QCD phase transition using cluster observables.
We study damage spreading in the ferromagnetic Ising model, on the square lattice, by performing numerical simulations in the microcanonical ensemble. Although the damage spreads for all temperature values, a dynamical critical phenomenon is observed at
Td=1.020(2)
TC (
TC is the Curie temperature): For T\ensuremath{\ge}
Td the damage D(t) is insensitive to its initial value D(0), while for T
Td a weak dependence on D(0) is present in such a way that for each T there are two values of D(t) corresponding to the initial conditions D(0)1/2 and D(0)>1/2. We have also identified a convenient order parameter to characterize the transition occurring at
Td. \textcopyright{} 1996 The American Physical Society.
By studying damage spreading in spin models in a temperature gradient, it is possible to determine not only the critical temperature, and the correlation length exponent v, but also a second critical exponent beta with high precision. This is done by studying the dynamics of damage spreading, in particular the overhangs which develop in the damage front. The authors demonstrate the method for the two-dimensional Ising model using heat bath dynamics, and discuss differences with Metropolis dynamics.
Damage spreading phenomena are defined artificially in the dynamics of the Ising spin systems in computer simulation. Random numbers used in computer simulations play a role to evolve the system to thermal equilibrium states. When the dynamics is taken not only as an artificial tool which realizes an equilibrium state but also as some physical process, damage spreading phenomena can be interpreted as physical objects. It is a main aim to give a concrete physical image to damage spreading phenomena which is artificially defined in original. Key property is a response for thermal noise by the interaction with a heat bath. Damage spreading as a phenomenon in time series is related with a spatial property in the percolation image of the Ising model by interpreting the Fortuin-Kasteleyn representation as a percolation image of the Ising models. This time-space relation reveals the essence of damage spreading as critical phenomena in physical systems.