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arXiv:cond-mat/0402191 · International Journal of Modern Physics C

EQUILIBRIUM TIMES FOR THE MULTICANONICAL METHOD

M. L. GUERRA, Jose Daniel Muñoz

Published Mar 1, 2004• 3 citations
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Abstract

This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, t eq , of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of t eq =0.27(15) L2.80(13) and with the number of energy levels to explore, k E , of [Formula: see text], in perfect agreement with the above result. In addition, a kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on Markov processes.

DOI
10.1142/s0129183104005851

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This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, t eq , of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of t eq =0.27(15) L2.80(13) and with the number of energy levels to explore, k E , of [Formula: see text], in perfect agreement with the above result. In addition, a kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on Markov processes.
This paper extends to two and three dimensions the recently proposed renormalized multicanonical sampling procedure [D. Yevick, Int. J. Mod. Phys. C 27, 1650033 (2016)IJMPEO0129-183110.1142/S0129183116500339], which generates the density of states of a system from that of a smaller system. The accuracy and efficiency of the method is additionally enhanced through a transition matrix Monte Carlo method that incorporates multicanonical sampling. The performance of the resulting procedure is investigated through calculations of the specific heat of the standard Ising model for square lattices with up to 128^{2} spins and for cubic lattices with up to 24^{3} spins. A comparison of the scaling properties to those of other algorithms based on the transition matrix Monte Carlo method demonstrates that the relative advantage of the renormalized procedure increases rapidly with system size.
I simulated the two-dimensional Ising model at its exact critical temperature with two Markov chain Monte Carlo methods, the single-spin Metropolis algorithm and the Wolff cluster algorithm, on periodic L×L lattices from L = 8 to L = 64. For each method I measured the integrated autocorrelation time of the energy and of the absolute magnetization, using four independent runs per lattice size. Both methods reproduce the exact finite-lattice energy within statistical error. For Metropolis, the autocorrelation time of |M| grows as L^z with z = 2.11(3) for L ≥ 16 and 2.16(5) for L ≥ 24, in line with the accepted value z ≈ 2.17 for local dynamics. For Wolff, measured in sweep-equivalent units, I get z = 0.29(1) from the energy and 0.15(1) from |M| for L ≥ 24, within the range of published values. In CPU time per statistically independent sample of |M|, Wolff was 5 times faster at L = 8 and 275 times faster at L = 64, and the gap grows roughly as L^2. All code and raw results are available, and the full study runs in under an hour on two CPU cores.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point.By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena.As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
We present a measure of the dynamical critical behavior: the ``multiscale dynamical exponent (MDE),'' zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd. Using dynamical renormalization-group concepts we study the relaxation times, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Lambda}}}$, of a family of space scales defined by \ensuremath{\Lambda}=1/k. Assuming dynamical universality we argue, and verify numerically, that zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd has the same value as the usually defined z. We measure zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd in the two-dimensional Ising model with the Metropolis and cluster dynamics and find zmetmd{\mathit{z}}_{\mathrm{met}}^{\mathit{m}\mathit{d}}zmetmd​=2.1\ifmmode\pm\else\textpm\fi{}0.1 and zwolffmd{\mathit{z}}_{\mathrm{wolff}}^{\mathit{m}\mathit{d}}zwolffmd​=0\ifmmode\pm\else\textpm\fi{}0.15, respectively. We note that in our approach zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a method which helps to overcome critical slowing down in the dynamical measurements themselves.
Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.