Ising model correlation time

An alternative way to describe the phenomenon is to consider that the ground state of a chain is already divided into domains at any temperatures. In order for the system to follow a small variation of the magnetic field some domains have to reverse their spin orientation. This occurs through a random walk of the DWs, that is, equal probability for the DW to move backward or forward, which implies that the DW needs a time proportional to d2 to reach the other end of a domain of length d. Given that d scales as the two spins correlation length, ., which, for the Ising model, is proportional to exp(2///rB7 ), for unitary spins, the same exponential relaxation is found... 

Another model which includes interaction and for which partial results are available on the decay of initial correlations is that of the one dimensional time-dependent Ising model. This model was first suggested by Glauber,18 and analyzed by him for one-dimensional Ising lattices. Let us consider a one-dimensional lattice, each of whose sites contain a spin. The spin on site,/ will be denoted by s/t) where Sj(t) can take on values + 1, and transitions are made randomly between the two states due to interactions with an external heat reservoir. The state of the system is specified by the spin vector s(t) = (..., s- f), s0(t), Ji(0>---)- A- full description of the system is provided by the probability P(s t), but of more immediate interest are the reduced probabilities... 

In Figure 4 we have plotted on logarithmic scales the correlation time of the Wolff algorithm for the 2D Ising model at the critical temperature, over a range of different system sizes. The slope of the line gives us an estimate of the dynamic exponent. Our best fit, given the errors on the data points is... 

Figure 4. The correlation time for the 2D Ising model simulated using the Wolff algorithm. The measurements deviate from a straight line for small system sizes L, but a fit to the larger sizes, indicated by the dashed line, gives a reasonable figure of z = 0.25 0.02 for the dynamic exponent of the algorithm.

In fact, in studies of the Wolff algorithm for the 2D Ising model, one does not usually bother to make use of Eq. (2.5) to calculate r. If we measure time in Monte Carlo steps (i.e., simple cluster flips), we can define the corresponding dynamic exponent zsteps in terms of the correlation time rsteps of Eq. (2.5) thus ... 

Figure 6. The correlation time T,tepl of the 20 Ising model simulated using the Wolff algorithm, measured in units of Monte Carlo steps (i.e., cluster flips). The fit gives us a value of zstep, = 0.50 0.01 for the corresponding dynamic exponent.

R. Biswas and B. Bagchi, A kinetic Ising model study of dynamical correlations in confined fluids emergence of bofli fast and slow time scales. J. Chem. Phys., 133 (2010), 084509-1-7. 

Here we will first compare the implication of the DRIS formalism with the predictions of the kinetic Ising model of Glauber [39] and discuss the results in relation to the theory of Shore and Zwanzig [40]. It should be noted that the model of Glauber is of fundamental importance, inasmuch as this is the first work in which the Markov character of a chain is rigorously considered and an analytical expression is provided for the time decay of correlation functions associated with a linear array of pairwise interdependent units. In the second part of this section, some empirical expressions previously proposed for describing OACFs, will be considered. 

The time dependence of the dynamic correlation function q t) was investigated numerically on the Ising EA model by Ogielski [131], An empirical formula for the decay of q t) was proposed as a combination of a power law at short times and a stretched exponential at long times... 

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