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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Damage spreading in the Ising model with a microcanonical constraint

F. G. Brady Moreira, Adauto J. F. de Souza, Ananias M. Mariz

Published Jan 1, 1996• 11 citations

Abstract

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 ${\mathit{T}}_{\mathit{d}}$=1.020(2)${\mathit{T}}_{\mathit{C}}$ (${\mathit{T}}_{\mathit{C}}$ is the Curie temperature): For T\ensuremath{\ge}${\mathit{T}}_{\mathit{d}}$ the damage D(t) is insensitive to its initial value D(0), while for T${\mathit{T}}_{\mathit{d}}$ 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 ${\mathit{T}}_{\mathit{d}}$. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreve.53.332

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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

{\mathit{T}}_{\mathit{d}}

=1.020(2)

{\mathit{T}}_{\mathit{C}}

(

{\mathit{T}}_{\mathit{C}}

is the Curie temperature): For T\ensuremath{\ge}

{\mathit{T}}_{\mathit{d}}

the damage D(t) is insensitive to its initial value D(0), while for T

{\mathit{T}}_{\mathit{d}}

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

{\mathit{T}}_{\mathit{d}}

. \textcopyright{} 1996 The American Physical Society.

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

{M}_{0}

in one of the configurations upon quenching the system at

{T}_{C}

, 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

(⟨{R}^{2}(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 authors study damage spreading in the Ising model, on a square lattice, by a Monte Carlo approach, using a general 'f-dynamics' (0<or=f<or=1) which reproduces, for f=0 the well known heat-bath dynamics and for f=1 Glauber dynamics. They find, for initially small damage, a critical value f c =0.50+or-0.03 as the threshold for damage spreading. For f c <f<or=1 the damage is non-zero only for T c <T<T 0 (f), where T c is the usual critical temperature and T 0 (f) rises with f like T 0 (f)-T c infinity (f-f c ) alpha ( alpha =0.83+or-0.08). For T c <or=T the damage D(T,f) seems to satisfy the scaling relation D(T,f) approximately mod 1n(f-f c ) mod -1 T((1-T c /T)/(f-f c ) alpha ).

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.

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.