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Extra resources for Computer Simulations in Condensed Matter: Systems: From Materials to Chemical Biology. Volume 1
Whereas the distribution is a monotonously decreasing function of cluster size at high temperatures, it becomes bimodal at temperatures around 25% above the critical temperature. The bimodal form is indicative of the formation of large clusters. It turns out to be possible to inﬂuence the cluster-size distribution by placing the pivot in a biased manner. Rather than ﬁrst choosing the pivot location, a particle is selected that will become the ﬁrst member of the cluster. Subsequently, the pivot is placed at random within a cubic box of linear size δ, centered around the position of this particle.
A ddimensional “hypercube” of volume Ld , the correlation length is bounded by the linear system size L. Thus, if the temperature approaches Tc , ξ grows according to (15) until it reaches a maximum value ξmax ∝ L, and for temperatures suﬃciently close to the critical temperature, (14) is replaced by τ ∝ Lz . (16) We thus encounter a phenomenon called critical slowing down. If a system becomes larger, the correlation time grows very rapidly and it becomes increasingly diﬃcult to generate statistically independent conﬁgurations.
Thus, a probability halfway these two extremes is typically chosen in order to maximize the rate at which the system evolves. 2 Indeed, the assignment of bonds involves speciﬁc probabilities, but once the clusters have been formed each of them can be ﬂipped independently without imposing an acceptance criterion that involves the energy change induced by such a collective spin-reversal operation. We note that the absence of an acceptance criterion does not imply that a cluster ﬂip does not entail an energy diﬀerence!