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Journal of Physics A Mathematical and General

Dynamical properties of the two- and three-dimensional Ising models by 'damage spreading'

Peter H. Poole, Naeem Jan

Published May 7, 1990• 36 citations
Publisher

Abstract

The method of 'damage spreading' is used to measure the dynamic exponent z for both the two- and three-dimensional Ising models with heat bath dynamics. The authors also measure a new exponent alpha describing the rate at which a damage cloud grows. They report values of z=2.24+or-0.04 and alpha =0.77+or-0.11 in two dimensions, and z=2.202+or-0.03 and alpha =0.92+or-0.10 in three dimensions.

DOI
10.1088/0305-4470/23/9/009

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Journal of Physics A Mathematical and General•Mar 21, 1990•12 citations
The method of 'damage spreading' is used to measure the dynamic exponent z for both the two- and three-dimensional Ising models with heat bath dynamics. The authors also measure a new exponent alpha describing the rate at which a damage cloud grows. They report values of z=2.24+or-0.04 and alpha =0.77+or-0.11 in two dimensions, and z=2.202+or-0.03 and alpha =0.92+or-0.10 in three dimensions.
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 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.
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 authors show that in the three-dimensional Ising model at the critical temperature the fractal dimension of the touching damage, determined by the box counting method, asymptotically converges to the expected value d f =d- beta /v. In contrast, the finite-size scaling analysis indicates for the touching damage an effective fractal dimension whose value is 20% smaller. The expected d f is, however, recovered again when instead of the system size L the quantity L/ln L is used as a scaling variable. This behaviour could be explained by the recently discovered dynamical multiscaling.