Wolff algorithm

A convenient side effect, which has certainly contributed to the popularity of the Wolff algorithm, is that it is exceedingly simple to implement. The prescription is as follows ... 

Comparison to the lattice cluster algorithms of Sect. 3 shows that the SW and Wolff algorithms operate in the grand-canonical ensemble, in which the cluster moves do not conserve the magnetization (or the number of particles, in the lattice-gas interpretation), whereas the geometric cluster algorithm... 

MC simulations are performed with N = 10 molecules, each with four n.n. molecules on a 2d square lattice, at constant P and T, and with the same model parameters as for the MF analysis. To each molecules we associate a cell on a square lattice. The Wolffs algorithm is based on the definition of a cluster of variables chosen in such a way to be thermodynamically correlated." To define the Wolffs cluster, a bond index (arm) of a molecule is randomly selected this is the initial element of a stack. The cluster is grown by first checking the remaining arms of the same initial molecule if they are in the same Potts state, then they are added to the stack with probability Psame = ttiin... 

In this section we look at how the Wolff algorithm actually performs in practice. As we will see, it performs extremely well when we are near the critical temperature, but is actually a little slower than the Metropolis algorithm at very high or low temperatures. 

Figure 2. Consecutive states of a 100 x 100 Ising model in equilibrium simulated using the Wolff algorithm. The top row of four states are at a temperature kT = 2.8 J, which is well above the critical temperature the middle row is close to the critical temperature at kT = 2.3 J-, and the bottom row is well below the critical temperature at kT = 1.8 J.

Figure 3. Mean cluster size in the Wolff algorithm as a fraction of the size of the lattice measured as function of temperature. The error bars on the measurements are not shown, because they are smaller than the points. The lines are just a guide to the eye.

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 ... 

The first step in demonstrating Eq. (2.7) is to prove another useful result, about the magnetic susceptibility %. It turns out that, for temperatures T > Tc, the susceptibility is related to the mean size of the clusters flipped by the Wolff algorithm thus ... 

In many simulations using the Wolff algorithm, the susceptibility is measured using the mean cluster size in this way. 

The demonstration of Eq. (2.8) goes like this. Instead of implementing the Wolff algorithm in the way described in this chapter, imagine instead... 

Now consider the average of the size of the clusters which get flipped in the Wolff algorithm. This is not quite the same thing as the average size of the clusters over the entire lattice, because when we choose our seed spin for the Wolff algorithm, it is chosen at random from the entire lattice, which means that the probability pt of it falling in a particular cluster i is proportional to the size of that cluster ... 

The average cluster size in the Wolff algorithm is then given by the average over the probability of the cluster being chosen times the size of that... 

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.

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