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Journal of Physics A Mathematical and General

Static critical exponents from the dynamics of damage spreading and overhangs in the Ising model with temperature gradient

G. G. Batrouni, Alex Hansen

Published Sep 7, 1992• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/25/17/008

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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.
The computer simulations of the critical dynamics of the structurally disordered three-dimensional Ising model are performed by the damage spreading method. For the systems with spin densities p = 0.6 and 0.8, we calculate the critical temperature and dynamic critical exponent z characterizing the behavior of relaxation properties near the critical point. The analysis of the results demonstrates the nonuniversality of the critical dynamics in the disordered Ising model. To interpret such dynamics, we introduce the concept of two universal types of critical behavior corresponding to weak and strong disorder, respectively.
The authors use the method of 'damage spreading' to measure the time taken for the equilibrium damage (the magnetization) to disappear in finite systems (10<or=L<or=103) at the critical temperature of the infinite system. This time is found to scale as L z with an exponent z of 2.16+or-0.02. This value is compatible with the expected value of the dynamic exponent for the 2D Ising model, but they argue that this method is unable to measure fluctuations properly which they show are responsible for the higher values recently reported in the literature. They suggest that the slowest mode of relaxation has an exponent of z>or=2.27.
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 ).