Ising-model Monte Carlo simulations: Density of states and mass gap

Abstract

We have performed Monte Carlo simulations for the three-dimensional Ising model. Using histogram techniques, we calculate the density of states on L block lattices up to size L =14. Statistical jackknife methods are employed to perform a thorough error analysis. We obtain high-precision estimates for the leading zeros of the partition function, which, using finite-size scaling, translate into v=0. 6285+0.0019. Along a different line of approach following recent work in lattice-gauge theories, we accurately determine the mass gap m = I/g (g correlation length) for cylindrical L'L, lattices (with L, =256 and L up to 12). The finite-size-scaling analysis of the mass-gap data leads to v =0.6321%0.0019.

Phase transitions and connectivity in three-dimensional vortex equilibria

“…A flow in lR 3 can be expressed up to the gradient of a scalar function in terms of the vorticity. field {( x) [ 91]: …”

“…Histogram type algorithms have seen occasional use for many years [3], [89], [12], [18] but have recently been revived and improved by Ferrenberg and Swendsen [57], [56].…”

“…The histogram method also estimates a joint probability density P [3,v(Eint,N) of the interaction energy and the number of vortex links at temperature (3 and vortex self-energy ll. As discussed in section 4.3, the probability density can be related to the density of states:…”

Alignment methods for biased multicanonical sampling

“…Methods similar to ours have been previously applied to the Monte Carlo method and to the canonical distribution [16] generated by the Metropolis algorithm. These include successive umbrella sampling [17][18][19], bridge sampling [20], importance sampling, and the weighted histogram analysis method of [21][22][23].…”

Markov Chain Monte Carlo Methods for Simulations of Biomolecules

“…[31] stimulated a renaissance of this approach. Modern work [33,34,35] developed efficient techniques to combine the overlapping distributions into one estimate of the spectral density and to control the statistical errors of the estimate. However, the patching of histograms from canonical simulations faces a number of limitations:…”