Wolff cluster algorithm

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... 

In order to emphasize the analogy with the lattice cluster algorithms, we can also formulate a single-cluster (Wolff) variant of the geometric cluster algorithm [15,19]. 

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... 

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 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 ... 

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