Metropolis algorithm has been extensively employed for simulating a canonical ensemble and estimating macroscopic properties of a closed system at any desired temperature. A mechanical property, like energy can be calculated by averaging over a large number of micro states of the stationary Markov chain generated by the Metropolis algorithm. However thermal properties like entropy, and free energies are not easily accessible. A method called umbrella sampling was proposed some forty years ago for this purpose. Ever since, umbrella sampling has undergone several metamorphoses and we have now multi canonical Monte Carlo, entropic sampling, flat histogram methods, Wang-Landau algorithm etc. In these talks I shall tell you of Metropolis algorithm for estimating mechanical properties and of Wang-Landau algorithm for estimating both mechanical and thermal properties of an equilibrium system. I shall make these lectures as pedagogical and self-contained as possible.
Recently, the present authors proposed the nonequilibrium-to-equilibrium scaling (NE-ES) scheme for critical Monte Carlo relaxation process, which scales relaxation data in the whole simulation-time regions regardless of functional forms, namely both for the stretched-exponential critical relaxation in cluster algorithms and for the power-law critical relaxation in local-update algorithms. In the present study, we generalize this scheme to off-critical relaxation process, and scale relaxation data for various temperatures in the whole simulation-time regions. This is the first proposal of the …
We investigate the mixed-spin Blume-Capel model with spin-1/2 and spin-$S$ ($S=1$, $2$, and $3$) on the simple cubic and body-centered cubic lattices with single-ion-splitting crystal-field ($Δ$) by using the Metropolis and the Wang-Landau Monte Carlo methods. By numerical simulations, we prove that the tricritical point is spin-independent for both lattices. The positions of the tricritical point in the phase diagram are determined as ($Δ_t/J=2.978(1)$; $k_B T_t/J=0.439(1)$) and ($Δ_t/J=3.949(1)$; $k_B T_t/J=0.854(1)$) for the simple cubic and the body-centered cubic lattices, respectively. A…
We show that a histogram maintained throughout the Wang-Landau (WL) sampling for the energy entries visited during the simulation could be used to make the simulated density of states (DOS) converge. The method is easy to be implemented to the WL sampling with no extra computational cost and bears the advantages of both the WL method and the multicanonical method.
Monte Carlo simulation has been performed in one-dimensional Lebwohl-Lasher model and two dimensional XY-model using the Wang-Landau and the Wang-Landau-Transition-Matrix Monte Carlo methods. Random walk has been performed in the two-dimensional space comprising of energy-order parameter and energy-correlation function and the joint density of states (JDOS) were obtained. From the JDOS the order parameter, susceptibility and correlation function are calculated. Agreement between the results obtained from the two algorithms is very good.
Critical properties of lattice gases with nearest-neighbor exclusion are investigated via the adaptive-window Wang-Landau algorithm on the square and simple cubic lattices, for which the model is known to exhibit an Ising-like phase transition. We study the particle density, order parameter, compressibility, Binder cumulant and susceptibility. Our results show that it is possible to estimate critical exponents using Wang-Landau sampling with adaptive windows. Finite-size-scaling analysis leads to results in fair agreement with exact values (in two dimensions) and numerical estimates (in three …
We present a study on the performance of Wang-Landau algorithm in a lattice model of liquid crystals which is a continuous lattice spin model. We propose a novel method of the spin update scheme in a continuous lattice spin model. The proposed scheme reduces the autocorrelation time of the simulation and results in faster convergence.
We review recent advances in the analysis of the Wang--Landau algorithm, which is designed for the direct Monte Carlo estimation of the density of states (DOS). In the case of a discrete energy spectrum, we present an approach based on introducing the transition matrix in the energy space (TMES). The TMES fully describes a random walk in the energy space biased with the Wang-Landau probability. Properties of the TMES can explain some features of the Wang-Landau algorithm, for example, the flatness of the histogram. We show that the Wang--Landau probability with the true DOS generates a Markov …
We present preliminary results of the investigation of the properties of the Markov random walk in the energy space generated by the Wang-Landau probability. We build transition matrix in the energy space (TMES) using the exact density of states for one-dimensional and two-dimensional Ising models. The spectral gap of TMES is inversely proportional to the mixing time of the Markov chain. We estimate numerically the dependence of the mixing time on the lattice size, and extract the mixing exponent.
We study the mechanism behind dynamical trappings experienced during Wang-Landau sampling of continuous systems reported by several authors. Trapping is caused by the random walker coming close to a local energy extremum, although the mechanism is different from that of critical slowing down encountered in conventional molecular dynamics or Monte Carlo simulations. When trapped, the random walker misses entire or even several stages of Wang-Landau modification factor reduction, leading to inadequate sampling of configuration space and a rough density-of-states even though the modification fact…
We report on numerical simulations of the two-dimensional Blume-Capel ferromagnet embedded in the triangular lattice. The model is studied in both its first- and second-order phase transition regime for several values of the crystal field via a sophisticated two-stage numerical strategy using the Wang-Landau algorithm. Using classical finite-size scaling techniques we estimate with high accuracy phase-transition temperatures, thermal, and magnetic critical exponents and we give an approximation of the phase diagram of the model.
In this work we investigate the behavior of the microcanonical and canonical averages of the two-dimensional Ising model during the Wang-Landau simulation. The simulations were carried out using conventional Wang-Landau sampling and the $1/t$ scheme. Our findings reveal that the microcanonical average should not be accumulated during the initial modification factors \textit{f} and outline a criterion to define this limit. We show that updating the density of states only after every $L^2$ spin-flip trials leads to a much better precision. We present a mechanism to determine for the given model …
Hwang in [Phys. Rev. E 80, 042103 (2009)] suggested that the two transitions of the six-state clock model on the square lattice are not of the Kosterlitz-Thouless type. Here we show from simulations thatat the upper transition, the helicity modulus does make a discontinuous jump to zero. This gives strong evidence for a Kosterlitz-Thouless transition.
We propose a simple method to estimate the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice model from Monte Carlo simulations. The main idea is to use the Wang-Landau flat-histogram algorithm, which allows us to obtain the free energy of a lattice model on a torus as a function of torus radii. The central charge is calculated with a good precision from a free energy scaling at the critical point. We apply the method to the Ising, tricritical Ising (Blume-Capel), Potts and site-diluted Ising models, and also discuss estimation of confo…
It is shown in this work how the Wang-Landau algorithm can be parallelized through the concept of the micromagnetic ensemble, when the Hamiltonian contains both spin interaction and the external field terms, and thus energy-magnetization plane is used for characterizing the density of states. Within this framework random walk is performed on mutually independent micromagnetic lines, and can thus be paralellized on a computer grid, without need for shared memory among individual processes. This approach pushes forward significantly the size of the systems that may be addressed on current comput…
Sampling along reaction coordinates with the Wang-Landau method
最近由Wang和Landau提出的多范围随机游走算法[Phys. Rev. Lett. 86，2050（2001）]适用于沿反应坐标计算分子系统的自由能分布。更一般地说，我们展示了如何提取部分平均值在各种统计合奏，而不调用模拟约束，偏置电位或未知参数。该方法示出了一个模型10维势能面，获得的分析结果。然后将其应用于与气相和四氯化碳溶剂中丁烷分子二面角相关的平均力势。最后，异构化在一个小的岩盐团簇，Na_4F_4，在微正则系综进行了研究，并将结果与平行回火蒙特卡罗。
By combining two generalized-ensemble algorithms, Replica-Exchange Wang-Landau (REWL) method and Multicanonical Replica-Exchange Method (MUCAREM), we propose an effective simulation protocol to determine the density of states with high accuracy. The new protocol is referred to as REWL-MUCAREM, and REWL is first performed and then MUCAREM is performed next. In order to verify the effectiveness of our protocol, we performed simulations of a square-lattice Ising model by the three methods, namely, REWL, MUCAREM, and REWL-MUCAREM. The results showed that the density of states obtained by the REWL-…
We investigate the tricritical scaling behavior of the two-dimensional spin-$1$ Blume-Capel model using the Wang-Landau method measuring the joint density of states for lattice sizes up to $48\times 48$ sites. The first-order transition curve is systematically determined employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. Deep in the first-order area of the phase diagram, we also find that the specific heat exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. At…
In this work we propose a criterion to finish the simulations of the Wang-Landau sampling. Instead of determining a final modification factor for all simulations and every sample sizes, we investigate the behavior of the temperature of the peak of the specific heat during the simulations and finish them when this value varies bellow a given limit. As a result, different runs stop at different final modification factors. We show that in place of the temperature of the peak of the specific heat one can adopt alternatively the integrated heat transfer as a reference quantity. We apply this techni…
Coupled, dynamical spin-lattice models provide a unique test ground for simulations investigating the finite-temperature magnetic properties of materials under the direct influence of the lattice vibrations. These models are constructed by combining a coordinate-dependent interatomic potential with a Heisenberg-like spin Hamiltonian, facilitating the treatment of both the atomic coordinates and spins as explicit phase variables. Using a model parameterized for bcc iron, we study the magnetic phase transition in these complex systems via the recently introduced, massively parallel replica-excha…
Phase transitions are ubiquitous phenomena, exemplified by the melting of ice and spontaneous magnetization of magnetic material. In general, a phase transition is associated with a symmetry breaking of a system; occurs due to the competition between coupling interaction and external fields such as thermal energy. If the phase transition occurs with no latent heat, the system experiences continuous transition, also known as second order phase transition. The ferromagnetic $q$-state Potts model with $r$ extra invisible states, introduced by Tamura, Tanaka, and Kawashima [Prog. Theor. Phys. 124,…
In this work, we present a comparative study of the accuracy provided by the Wang-Landau sampling and the Broad Histogram method to estimate de density of states of the two dimensional Ising ferromagnet. The microcanonical averages used to describe the thermodynamic behaviour and to use the Broad Histogram method were obtained using the single spin-flip Wang-Landau sampling, attempting to convergence issues and accuracy improvements. We compare the results provided by both techniques with the exact ones for thermodynamic properties and critical exponents. Our results, within the Wang-Landau sa…
In this work we develop an implementation of the Wang--Landau algorithm [Phys. Rev. Lett. \textbf{86}, 2050-2053 (2001)]. This algorithm allows us to find the density of states (DOS), a function that, for a given system, describes the proportion of states that have a certain energy. The implementation uses the Python language for the algorithm itself, and it can take advantage of any library, such as the powerful LAMMPS library, for the computation of energy. Therefore, the resulting implementation is simple and flexible without sacrificing efficiency. This implementation also considers recent…
We implement a two-stage approach of the Wang-Landau algorithm to investigate the critical properties of the 3D Ising model with quenched bond randomness. In particular, we consider the case where disorder couples to the nearest-neighbor ferromagnetic interaction, in terms of a bimodal distribution of strong versus weak bonds. Our simulations are carried out for large ensembles of disorder realizations and lattices with linear sizes $L$ in the range $L=8-64$. We apply well-established finite-size scaling techniques and concepts from the scaling theory of disordered systems to describe the natu…
The density of states of continuous models is known to span many orders of magnitudes at different energies due to the small volume of phase space near the ground state. Consequently, the traditional Wang-Landau sampling which uses the same trial move for all energies faces difficulties sampling the low entropic states. We developed an adaptive variant of the Wang-Landau algorithm that very effectively samples the density of states of continuous models across the entire energy range. By extending the acceptance ratio method of Bouzida, Kumar, and Swendsen such that the step size of the trial m…
We investigate the behavior of the deviation of the estimator for the density of states (DOS) with respect to the exact solution in the course of Wang-Landau and Stochastic Approximation Monte Carlo (SAMC) simulations of the two-dimensional Ising model. We find that the deviation saturates in the Wang-Landau case. This can be cured by adjusting the refinement scheme. To this end, the 1/t-modification of the Wang-Landau algorithm has been suggested. A similar choice of refinement scheme is employed in the SAMC algorithm. The convergence behavior of all three algorithms is examined. It turns out…
We propose a method based on the Wang-Landau algorithm to numerically generate the spectral densities of random matrix ensembles. The method employs Dyson's log-gas formalism for random matrix eigenvalues and also enables one to simultaneously investigate the thermodynamic properties. This approach is a powerful alternative to the conventionally used Monte Carlo simulations based on the Boltzmann sampling, and is ideally suited for investigating beta-ensembles.
A simple solid-on-solid model of adsorbate-induced faceting is studied by using a modified Wang-Landau method. The phase diagram for this system is constructed by computing the density of states in a special two-dimensional energy space. A finite--size scaling analysis of transition temperature and specific heat shows that faceting transition is the first order phase transition. Logarithmic dependence of the mean-square width of the surface on system size indicates that surface is rough above the transition temperature.
Wang-Landau Monte Carlo simulation of isotropic-nematic transition in liquid crystals
Wang和Landau最近提出了一种简单而灵活的非Boltzmann Monte Carlo方法来估计态密度，由此可以计算封闭系统的宏观性质。他们通过共连续能谱演示了他们的算法。本文对离散能谱的考虑系统进行了改进。我们发现，当系统具有连续的能量谱时，Wang-Landau算法不能很好地执行。在本文中，我们提出了修改的算法，并证明其性能的液晶系统的晶格模型（Lebwohl-Lasher相互作用具有连续变化的能量），表现出从高温各向同性到低温相变。
Paying attention to the difference of density of states, Δln g(E) = ln g(E+ΔE) - ln g(E), we study the convergence of the Wang-Landau method. We show that this quantity is a good estimator to discuss the errors of convergence, and refer to the $1/t$ algorithm. We also examine the behavior of the 1st-order transition with this difference of density of states in connection with Maxwell's equal area rule. A general procedure to judge the order of transition is given.
When performing a Monte Carlo calculation, the running time should in principle be much longer than the autocorrelation time in order to get reliable results. Among different lattice fermion models, the Holstein model is notorious for its particularly long autocorrelation time. In this work, we employ the Wang-Landau algorithm in the determinant quantum Monte Carlo to achieve the flat-histogram sampling in the "configuration weight space", which can greatly reduce the autocorrelation time by sacrificing some sampling efficiency. The proposal is checked in the Holstein model on both square and …
In this work we apply Wang-Landau simulations to a simple model which has exact solutions both in the microcanonical and canonical formalisms. The simulations were carried out by using an updated version of the Wang-Landau sampling. We consider a homopolymer chain consisting of $N$ monomers units which may assume any configuration on the two-dimensional lattice. By imposing constraints to the moves of the polymers we obtain three different models. Our results show that updating the density of states only after every $N$ monomers moves leads to a better precision. We obtain the specific heat an…
This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and $1/t$ algorithms. The classical algorithms are modified by the use of $m$ independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number $π$ by multiple dimensional integration is used in the continuum approximation. In each case, the error i…
The Wang-Landau (WL) algorithm has been widely used for simulations in many areas of physics. Our analysis of the WL algorithm explains its properties and shows that the difference of the largest eigenvalue of the transition matrix in the energy space from unity can be used to control the accuracy of estimating the density of states. Analytic expressions for the matrix elements are given in the case of the one-dimensional Ising model. The proposed method is further confirmed by numerical results for the one-dimensional and two-dimensional Ising models and also the two-dimensional Potts model.
