Geometric cluster algorithm

Liu, J., Luijten, E. Rejection-free geometric cluster algorithm for complex fluids. Phys. Rev. Lett. 2004, 92, 035504. 

Geometric Cluster Algorithm for Hard-Sphere Mixtures. 23... 

Generalized Geometric Cluster Algorithm for Interacting Particles. 25... 

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... 

This approach is very general. For example, it is not restricted to monodis-perse systems, and Krauth and co-workers have applied it successfully to binary [17] and polydisperse [18] mixtures. Indeed, conventional simulations of size-asymmetric mixtures typically suffer from jamming problems, in which a very large fraction of all trial moves is rejected because of particle overlaps. In the geometric cluster algorithm particles are moved in a nonlocal fashion, yet overlaps are avoided. 

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]. 

The geometric cluster algorithm described in the previous section is formulated for particles that interact via hard-core repulsions only. Clearly, in order to make this approach widely applicable, a generalization to other t3rpes of pair potentials must be found. Thus, Dress and Krauth [14] suggested to impose a Metropolis-type acceptance criterion, based upon the energy difference induced by the cluster move. Indeed, if a pair potential consists of a hardcore contribution supplemented by an attractive or repulsive tail, such as a... 

Illustration 1 Efficiency of the Generalized Geometric Cluster Algorithm... 

Fig. 3. Efficiency comparison between a conventional local update algorithm (open symbols) and the generalized geometric cluster algorithm (closed symbols), for a binary mixture (see text) with size ratio a. Whereas the autocorrelation time per particle (expressed in us of CPU time per particle move) rapidly increases with size ratio, the GCA features only a weak dependence on a. Reprinted figure with permission from [19], Copyright 2004 by the American Physical Society...

Graphical methods in connection with pattern recognition algorithms, i.e. geometrical or statistical methods, e.g. minimum spanning tree or cluster analysis, are more powerful methods for explorative data analysis than graphical methods alone. 

The objective function values of K-means algorithm with 50 iterative cycles are listed in Table 2, the lowest value is 168.7852. One notices that the behavior of K-means algorithm is influenced by the choice of initial cluster centers, the order in which the samples were taken, and, of course, the geometrical properties of the data. The tendency of sinking into local optima is obvious. Clustering by SA can provide more stable computational results. 

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