Wang-Landau Algorithm Research Articles

Компьютерное моделирование двумерной модели Поттса с конкурирующими обменными взаимодействиями

The phase transitions and thermodynamic properties of the two-dimensional Potts model with the number of spin states $q = 4$ on a hexagonal lattice with competing exchange interactions have been investigated on the basis of the Wang-Landau algorithm by the Monte Carlo method. It is shown that taking into account the antiferromagnetic interactions of the next-nearest neighbors leads to a violation of the magnetic ordering and to the appearance of frustration.

Influence of Initial Guess on the Convergence Rate and the Accuracy of Wang–Landau Algorithm

Influence of Initial Guess on the Convergence Rate and the Accuracy of Wang–Landau Algorithm

Reducing autocorrelation time in determinant quantum Monte Carlo using the Wang-Landau algorithm: Application to the Holstein model.

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 paper, 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 honeycomb lattices. Based on such a Wang-Landau assisted determinant quantum Monte Carlo method, some models with long autocorrelation times can now be simulated possibly.

Thermodynamics of hydrophobic-polar model proteins on the face-centered cubic lattice.

The HP model, a coarse-grained protein representation with only hydrophobic (H) and polar (P) amino acids, has already been extensively studied on the simple cubic (SC) lattice. However, this geometry severely restricts possible bond angles, and a simple improvement is to instead use the face-centered cubic (fcc) lattice. In this paper, the density of states and ground state energies are calculated for several benchmark HP sequences on the fcc lattice using the replica-exchange Wang-Landau algorithm and a powerful set of Monte Carlo trial moves. Results from the fcc lattice proteins are directly compared with those obtained from a previous lattice protein folding study with a similar methodology on the SC lattice. A thermodynamic analysis shows comparable folding behavior between the two lattice geometries, but with a greater rate of hydrophobic-core formation persisting into lower temperatures on the fcc lattice.

Numerical estimates of square lattice star vertex exponents.

We implement parallel versions of the generalized atmospheric Rosenbluth methods and Wang-Landau algorithms for stars and for acyclic uniform branched networks in the square lattice. These are models of monodispersed branched polymers, and we estimate the star vertex exponents σ_{f} for f stars, and the entropic exponent γ_{G} for networks with comb and brush connectivity in two dimensions. Our results verify the predicted (but not rigorously proven) exact values of the vertex exponents and we test the scaling relation [B. Duplantier, J. Stat. Phys. 54, 581 (1989)JSTPBS0022-471510.1007/BF01019770]γ_{G}-1=[under ∑]f≥1m_{f}σ_{f}for several acyclic branched networks in two dimensions.

Phase diagram of the Potts model with the number of spin states q = 4 on a kagome lattice

The magnetic structures of the ground state, phase transitions, and the thermodynamic properties of a two-dimensional ferromagnetic Potts model with the number of spin states q = 4 on a kagome lattice are studied using the Wang-Landau algorithm of the Monte Carlo method, taking into account the interactions of the nearest and the next-nearest neighbors. The studies were carried out for the value of the interaction of the next-nearest neighbors in the range 0 ≤ r ≤ 1.0. It is shown that taking into account the antiferromagnetic interactions of the next-nearest neighbor leads to a violation of the magnetic ordering. A phase diagram of the dependence of the critical temperature on the value of the interaction of the next-nearest neighbor is constructed. The analysis of the character of phase transitions is carried out. It was found that in the ranges 0 ≤ r ≤ 0.5 and 0.5 ≤ r ≤ 1.0, a first-order phase transition is observed, and for r = 0.5, frustrations are observed in the system.

Calculation of the residual entropy of Ice Ih by Monte Carlo simulation with the combination of the replica-exchange Wang-Landau algorithm and multicanonical replica-exchange method.

We estimated the residual entropy of Ice Ih by the recently developed simulation protocol, namely, the combination of the replica-exchange Wang-Landau algorithm and multicanonical replica-exchange method. We employed a model with the nearest neighbor interactions on the three-dimensional hexagonal lattice, which satisfied the ice rules in the ground state. The results showed that our estimate of the residual entropy is in accordance with various previous results. In this article, we not only give our latest estimate of the residual entropy of Ice Ih but also discuss the importance of the uniformity of a random number generator in Monte Carlo simulations.

Wang–Landau sampling for estimation of the reliability of physical networks

Modern physical networks, for example in communication and transportation, can be interpreted as directed graphs. Network models are used to identify the probability that given nodes are connected, and therefore the effect of a failure at a given link. This is essential for network design, optimization, and reliability. In this study, we investigated three alternative ensembles for estimating network reliability using the Wang–Landau algorithm. The first performed random walks on a structure function having two possible states: connected and disconnected. The second used random walks on a reliability polynomial. The third combined random walks with the average of connecting probabilities. The accuracy and limitations of the three ensembles were compared by estimating the reliability of three network models: a bridge network, a ladder-type network, and a dodecahedron network. The simulation results showed that the use of a random walk on a structure function failed to produce estimates when applied to highly reliable networks in any of the three network types. The other two approaches performed efficiently for bridge or ladder-type networks at any level of network reliability. The random walk on a probability space using the 1∕t algorithm was the only ensemble that was able to yield accurate estimates for a dodecahedron network, though even this failed at the highest level of network reliability. The other two methods failed to converge within 108 Monte Carlo trials. The use of the average of connecting probabilities required a shorter computation time when applied to a large network. Methods that can reduce variance for large, highly reliable networks require further investigation.

Ising universality in the two-dimensional Blume-Capel model with quenched random crystal field.

Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.

Warp Bridge Sampling: The Next Generation

Bridge sampling is an effective Monte Carlo (MC) method for estimating the ratio of normalizing constants of two probability densities, a routine computational problem in statistics, physics, chemistry, and other fields. The MC error of the bridge sampling estimator is determined by the amount of overlap between the two densities. In the case of unimodal densities, Warp-I, II, and III transformations are effective for increasing the initial overlap, but they are less so for multimodal densities. This article introduces Warp-U transformations that aim to transform multimodal densities into unimodal ones (hence “U”) without altering their normalizing constants. The construction of a Warp-U transformation starts with a normal (or other convenient) mixture distribution that has reasonable overlap with the target density p, whose normalizing constant is unknown. The stochastic transformation that maps back to its generating distribution is then applied to p yielding its Warp-U version, which we denote . Typically, is unimodal and has substantially increased overlap with . Furthermore, we prove that the overlap between and is guaranteed to be no less than the overlap between p and , in terms of any f-divergence. We propose a computationally efficient method to find an appropriate , and a simple but effective approach to remove the bias which results from estimating the normalizing constant and fitting with the same data. We illustrate our findings using 10 and 50 dimensional highly irregular multimodal densities, and demonstrate how Warp-U sampling can be used to improve the final estimation step of the Generalized Wang–Landau algorithm, a powerful sampling and estimation approach. Supplementary materials for this article are available online.

Calculation of Chemical Potential of a Molecule on the Basis of Radial Distribution Functions

The work is devoted to the application of the Hill method for calculating the chemical potential of a molecule in the one-component homogeneous molecular system within computer simulations. This method is based on the double integration of molecular radial distribution functions, which depend on the additional parameter that controls the strength of interactions between the molecules. The results of calculating the chemical potential of the argon molecule by the Hill method are compared with our data obtained by two other methods: the Widom test-particle method and the extended ensemble method within the Wang–Landau algorithm.

Computational statistical mechanics of a confined, three-dimensional Coulomb gas.

The thermodynamic properties of systems with long-range interactions present an ongoing challenge, from the point of view of both theory as well as computer simulation. In this paper we study a model system, a Coulomb gas confined inside a sphere, by using the Wang-Landau algorithm. We have computed the configurational density of states, the thermodynamic entropy, and the caloric curve, and compared with microcanonical Metropolis simulations, while showing how concepts such as the configurational inverse temperature can be used to understand some aspects of thermodynamic behavior. A dynamical multistability behavior is seen at low energies in microcanonical Monte Carlo simulations, suggesting that flat-histogram methods can in fact be useful and complementary alternatives to traditional Metropolis simulation in complex systems.

Monte Carlo Approximation of Bayes Factors via Mixing With Surrogate Distributions

By mixing the target posterior distribution with a surrogate distribution, of which the normalizing constant is tractable, we propose a method for estimating the marginal likelihood using the Wang–Landau algorithm. We show that a faster convergence of the proposed method can be achieved via the momentum acceleration. Two implementation strategies are detailed: (i) facilitating global jumps between the posterior and surrogate distributions via the multiple-try Metropolis (MTM); (ii) constructing the surrogate via the variational approximation. When a surrogate is difficult to come by, we describe a new jumping mechanism for general reversible jump Markov chain Monte Carlo algorithms, which combines the MTM and a directional sampling algorithm. We illustrate the proposed methods on several statistical models, including the log-Gaussian Cox process, the Bayesian Lasso, the logistic regression, and the g-prior Bayesian variable selection. Supplementary materials for this article are available online.

Tricritical point in the mixed-spin Blume-Capel model on three-dimensional lattices: Metropolis and Wang-Landau sampling approaches.

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. We show that the two methods are complementary: The Wang-Landau algorithm is efficient to construct phase diagrams and the Metropolis algorithm allows access to large-sized lattices. By numerical simulations, we prove that the tricritical point is independent of S 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 very strong supercritical slowing down and hysteresis were observed in the Metropolis update close to first-order transitions for Δ>Δ_{t} in the body-centered cubic lattice. In addition, for both lattices we found a line of compensation points, where the two sublattice magnetizations have the same magnitude. We show that the compensation lines are also S independent.

Density of states approach to the hexagonal Hubbard model at finite density

We apply the Linear Logarithmic Relaxation (LLR) method, which generalizes the Wang-Landau algorithm to quantum systems with continuous degrees of freedom, to the fermionic Hubbard model with repulsive interactions on the honeycomb lattice. We compute the generalized density of states of the average Hubbard field and divise two reconstruction schemes to extract physical observables from this result. By computing the particle density as a function of chemical potential we assess the utility of LLR in dealing with the sign problem of this model, which arises away from half filling. We show that the relative advantage over brute-force reweighting grows as the interaction strength is increased and discuss possible future improvements.

Phase Transitions and the Thermodynamic Properties of the Potts Model with the Spin State Number q = 4 at a Kagome Lattice

Phase transitions and the thermodynamic properties of the two-dimensional ferromagnetic Izing model with spin state number q = 4 on a kagome lattice are studied by the Monte Carlo method on the base of the Wang–Landau algorithm. The phase transition character is analyzed using the fourth-order Binder cumulant method and the histogram analysis of the data. It is found that this model undergoes a first-order phase transition.

Phase transitions in the Ising model on a layered triangular lattice in a magnetic field

The influence of the magnetic field on phase transitions and thermodynamic properties of the Ising model on a triangular lattice was studied using the Wang–Landau ​ algorithm and the replica algorithm of the Monte Carlo method. It is shown that in the model under study, depending on the magnitude of the magnetic field h, disordered, partially ordered, and completely ordered phases are observed. The nature of phase transitions was analyzed based on the histogram method of data analysis. It was found that in the range 0≤h≤6 observed second-order phase transition. It was found that for values of the magnetic field h>6 there is no degeneracy of the ground state and the phase transition is destroyed. A plateau was found depending on the magnetization on the magnetic field, equal to 1/3 of the saturation magnetization.

Wang-Landau algorithm: An adapted random walk to boost convergence

The Wang-Landau (WL) algorithm is a recently developed stochastic algorithm computing densities of states of a physical system, and also performing numerical integration in high dimensional spaces. Since its inception, it has been used on a variety of (bio-)physical systems, and in selected cases, its convergence has been proved. The convergence speed of the algorithm is tightly tied to the connectivity properties of the underlying random walk.As such, we propose an efficient random walk that uses geometrical information to circumvent the following inherent difficulties: avoiding overstepping strata, toning down concentration phenomena in high-dimensional spaces, and accommodating multidimensional distributions. These improvements are especially well suited to improve calculations on a per basin basis – included anharmonic ones.Experiments on various models stress the importance of these improvements to make WL effective in challenging cases. Altogether, these improvements make it possible to compute density of states for regions of the phase space of small biomolecules.

Wang-Landau algorithm as stochastic optimization and its acceleration.

We show that the Wang-Landau algorithm can be formulated as a stochastic gradient descent algorithm minimizing a smooth and convex objective function, of which the gradient is estimated using Markov chain Monte Carlo iterations. The optimization formulation provides us another way to establish the convergence rate of the Wang-Landau algorithm, by exploiting the fact that almost surely the density estimates (on the logarithmic scale) remain in a compact set, upon which the objective function is strongly convex. The optimization viewpoint motivates us to improve the efficiency of the Wang-Landau algorithm using popular tools including the momentum method and the adaptive learning rate method. We demonstrate the accelerated Wang-Landau algorithm on a two-dimensional Ising model and a two-dimensional ten-state Potts model.

Phase Transformations and Thermodynamic Properties of the Potts Model with q = 4 on a Hexagonal Lattice with Interactions of Next-Nearest Neighbors

The magnetic structures of the ground state, phase transitions, and the thermodynamic properties of a two-dimensional ferromagnetic Potts model with the number of spin states q = 4 on a hexagonal lattice are studied using the Wang–Landau algorithm of the Monte Carlo method, taking into account the interactions of the nearest and the next-nearest neighbors. It is shown that the inclusion of the antiferromagnetic interactions of the next-nearest neighbors leads to the appearance of frustration and disturbance of magnetic ordering. The phase transition characters are analyzed using the method of fourth-order Binder cumulants and the histogram method. It is found that a first-order phase transition takes place in this model.