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It will be noticed that this acceptance ratio can become larger than one if enough of the spins are aligned opposite their local fields. This problem can trivially be overcome by the standard procedure of dividing the acceptance ratios for a move to proceed in either direction by the larger of the two ratios.
-
It will be noticed that this acceptance ratio can become larger than one if enough of the spins are aligned opposite their local fields. This problem can trivially be overcome by the standard procedure of dividing the acceptance ratios for a move to proceed in either direction by the larger of the two ratios.
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29
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85035224291
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The distance can be defined in various different ways. In their simulations, Barkema and Marko citebarkema defined it using a ``Manhattan'' formula in which the distance between two points ([Formula Presented],[Formula Presented]) and ([Formula Presented],[Formula Presented]) on a two dimensional square lattice is [Formula Presented]-[Formula Presented]+[Formula Presented]-[Formula Presented] (with the obvious generalization in higher dimensions). However, the exact definition of the distance is not important for the working of the algorithm.
-
The distance can be defined in various different ways. In their simulations, Barkema and Marko citebarkema defined it using a ``Manhattan'' formula in which the distance between two points (x1,y1) and (x2,y2) on a two dimensional square lattice is x1-x2+y1-y2 (with the obvious generalization in higher dimensions). However, the exact definition of the distance is not important for the working of the algorithm.
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30
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85035198594
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In their simulations with the LCF, Barkema and Marko used a length scale that was tuned to the correlation length of the equilibrated system at the temperature they were studying. However, this is not a shrewd idea in the present case, since the rapid decrease of the correlation length below [Formula Presented] means that the algorithm would become equivalent to the Metropolis algorithm again at temperatures only a little below the phase transition.
-
In their simulations with the LCF, Barkema and Marko used a length scale that was tuned to the correlation length of the equilibrated system at the temperature they were studying. However, this is not a shrewd idea in the present case, since the rapid decrease of the correlation length below Tc means that the algorithm would become equivalent to the Metropolis algorithm again at temperatures only a little below the phase transition.
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33
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85035227244
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In fact, the speed with which the iteration converges seems to depend on how well the samples that we have taken cover the important ranges of energy. If we try to get the method to converge for values of [Formula Presented] for which the energies of the most likely states fall far away from any of the energies sampled during our Monte Carlo runs, convergence will be slow, and even after convergence the method will not give a very good estimate of the partition function. For this reason, the method is not suitable for extrapolating far beyond the regime in which the simulations were performed or for interpolating between simulations with very different values of the parameters [Formula Presented].
-
In fact, the speed with which the iteration converges seems to depend on how well the samples that we have taken cover the important ranges of energy. If we try to get the method to converge for values of J(n) for which the energies of the most likely states fall far away from any of the energies sampled during our Monte Carlo runs, convergence will be slow, and even after convergence the method will not give a very good estimate of the partition function. For this reason, the method is not suitable for extrapolating far beyond the regime in which the simulations were performed or for interpolating between simulations with very different values of the parameters J(n).
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34
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85035241093
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To be strictly equivalent, we would have to perform our collapse in a range centered around the maximum of the scaling function for the specific heat. But since the difference between the position of this maximum and the positions [Formula Presented] that we have used in our collapses are much smaller than the smallest values of Δ x, we can ignore this point.
-
To be strictly equivalent, we would have to perform our collapse in a range centered around the maximum of the scaling function for the specific heat. But since the difference between the position of this maximum and the positions xc that we have used in our collapses are much smaller than the smallest values of Δ x, we can ignore this point.
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