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Journal of Physics A Mathematical and General

Mixing heat-bath and Glauber dynamics: damage spreading in the Ising model

Ananias M. Mariz, Hans Jürgen Herrmann

Published Nov 21, 1989• 17 citations
Publisher

Abstract

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 ).

DOI
10.1088/0305-4470/22/22/006

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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 ).
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.
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 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 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.