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Journal of Physics Conference Series

The parallel Wang-Landau algorithm for joint density of states calculation

V. I. Egorov, Olga G. Maksimova, A R Baidganov

Published Dec 1, 2018• 2 citations
Publisher

Abstract

The Wang-Landau algorithm is used for estimating density of states. The joint density of states (JDOS) with two parameters gives more information than conventional one but it requires a huge computational time. Recently, the developing of fast algorithms for the JDOS calculation is an actual problem. In this work, we employ the modified parallel WL algorithm for JDOS, which combines ideas of global updates and two level methods. This algorithm is checked the three-dimensional Ising and Potts models. The method of obtaining phase diagrams from JDOS is demonstrated for Ising model.

DOI
10.1088/1742-6596/1141/1/012068

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The Wang-Landau algorithm is used for estimating density of states. The joint density of states (JDOS) with two parameters gives more information than conventional one but it requires a huge computational time. Recently, the developing of fast algorithms for the JDOS calculation is an actual problem. In this work, we employ the modified parallel WL algorithm for JDOS, which combines ideas of global updates and two level methods. This algorithm is checked the three-dimensional Ising and Potts models. The method of obtaining phase diagrams from JDOS is demonstrated for Ising model.
We use a two-dimensional Wang-Landau sampling algorithm to calculate the density of states for two discrete spin models and then extract their phase diagrams. The first system is an asymmetric Ising model on a triangular lattice with two- and three-body interactions in an external field. An accurate density of states allows us to locate the critical endpoint accurately in a two-dimensional parameter space. We observe a divergence of the spectator phase boundary and of the temperature derivative of the magnetization coexistence diameter at the critical endpoint in quantitative agreement with theoretical predictions. The second model is a Q-state Potts model in an external field H. We map the phase diagram of this model for Q > 8 and observe a first-order phase transition line that starts at the H = 0 phase transition point and ends at a critical point (Tc,Hc), which must be located in a two-dimensional parameter space. The critical field Hc(Q) is positive and increases with Q, in qualitative agreement with previous theoretical predictions.
The concept of the partition function zeros is one of the most efficient methods for investigating the phase transitions and the critical phenomena in various physical systems. Estimating the partition function zeros requires information on the density of states Ω( E ) as a function of the energy E . Currently, the Wang-Landau Monte Carlo algorithm is one of the best methods for calculating Ω( E ). The partition function zeros in the complex temperature plane of the Ising model on an L × L square lattice ( L = 10 ~ 80) with a periodic boundary condition have been estimated by using the Wang-Landau Monte Carlo algorithm. The efficiency of the Wang-Landau Monte Carlo algorithm and the accuracies of the partition function zeros have been evaluated for three different, 5%, 10%, and 20%, flatness criteria for the histogram H ( E ).
Phase transitions and critical phenomena are the most universal phenomena in nature.To understand the phase transitions and critical phenomena of a given system as a continuous function of temperature and to obtain the partition function zeros in the complex temperature plane indicating most effectively phase transitions and critical phenomena, we need to calculate the density of states.Currently, Wang-Landau Monte Carlo algorithm is one of the most efficient Monte Carlo methods to calculate the approximate density of states.Using Wang-Landau Monte Carlo algorithm, the density of states for the Ising model on L x L square lattices (L = 4 ~ 32) with periodic boundary conditions is obtained, and the partition function zeros of the Ising model are evaluated in the complex temperature plane.By examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), phase transitions and critical phenomena can be much more accurately analyzed.The approximate first zeros of the Ising ferromagnet, obtained from Wang-Landau algorithm, are quite close to the exact ones, indicating that it is a reliable method for calculating the density of states and the first partition function zeros.
Using the Wang-Landau algorithm combined with the analytic method, the density of states of a two-dimensional XY model on square lattices of sizes 16x16, 24x24, and 32x32 is accurately calculated. Thermodynamic quantities, such as internal energy, free energy, entropy, and specific heat are obtained from the resulted density of states by numerical integration. It is found that both the density of states and the extensive thermodynamic quantities obey the scaling law. From the entropy curve, symptoms of phase transition are observed. A general method of calculation of the density of states of continuous models by simulation combined with analytical method is proposed.