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Journal de physique

Damage spreading in Ising model with increased range of interaction

S.S. Manna

Published Jan 1, 1990• 16 citations
Publisher

Abstract

We have studied the spreading of damage in Ising models with increased range of interaction. Simulating a square lattice of linear size 1000 we find that the spreading temperature and the Curie temperature of magnetic transition are the same for both nearest and next nearest neighbour models whereas for third nearest neighbour model they may not be the same.

DOI
10.1051/jphys:0199000510120126100

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Europhysics Letters (EPL)•Dec 1, 1990•15 citations
We have studied the spreading of damage in Ising models with increased range of interaction. Simulating a square lattice of linear size 1000 we find that the spreading temperature and the Curie temperature of magnetic transition are the same for both nearest and next nearest neighbour models whereas for third nearest neighbour model they may not be the same.
The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.
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.
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{\mathit{T}}_{\mathit{d}}Td​=1.020(2)TC{\mathit{T}}_{\mathit{C}}TC​ (TC{\mathit{T}}_{\mathit{C}}TC​ is the Curie temperature): For T\ensuremath{\ge}Td{\mathit{T}}_{\mathit{d}}Td​ the damage D(t) is insensitive to its initial value D(0), while for TTd{\mathit{T}}_{\mathit{d}}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{\mathit{T}}_{\mathit{d}}Td​. \textcopyright{} 1996 The American Physical Society.
The time evolution of the Hamming distance (damage spreading)for the S=1/2 and S=1 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 Tc ≈1.675±0.025 for the S=1 lsing 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.