Wang-Landau Method Research Articles

Predicting non-equilibrium folding behavior of polymer chains using the steepest-entropy-ascent quantum thermodynamic framework.

The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework is used to explore the influence of heating and cooling on polymer chain folding kinetics. The framework predicts how a chain moves from an initial non-equilibrium state to stable equilibrium along a unique thermodynamic path. The thermodynamic state is expressed by occupation probabilities corresponding to the levels of a discrete energy landscape. The landscape is generated using the Replica Exchange Wang-Landau method applied to a polymer chain represented by a sequence of hydrophobic and polar monomers with a simple hydrophobic-polar amino acid model. The chain conformation evolves as energy shifts among the levels of the energy landscape according to the principle of steepest entropy ascent. This principle is implemented via the SEAQT equation of motion. The SEAQT framework has the benefit of providing insight into structural properties under non-equilibrium conditions. Chain conformations during heating and cooling change continuously without sharp transitions in morphology. The changes are more drastic along non-equilibrium paths than along quasi-equilibrium paths. The SEAQT-predicted kinetics are fitted to rates associated with the experimental intensity profiles of cytochrome c protein folding with Rouse dynamics.

Simulation of a Four-Component Potts Model on a Hexagonal Lattice by the Wang–Landau Method with Controlled Accuracy

Simulation of a Four-Component Potts Model on a Hexagonal Lattice by the Wang–Landau Method with Controlled Accuracy

Wang-Landau simulations with non-flat distributions

We show how the well-known Wang-Landau method can be modified to produce non-flat distributions. Through the choice of a suitable profile this can lead to an increase in efficiency for some systems. Examples for such an enhancement are provided.

An entropic simulational study of the spin-1 Baxter–Wu model in a crystal field

We investigate the critical behavior of the two-dimensional spin-1 Baxter–Wu model in a crystal field using entropic sampling simulations, based on the Wang–Landau method, with the joint density of states. We obtain the temperature–crystal field phase diagram, which includes a tetracritical line ending at a pentacritical point. A finite-size scaling analysis of the maximum of the specific heat, while changing the crystal field anisotropy, is used to obtain a precise location of the pentacritical point. Our results give the critical temperature and crystal field as Tpc=0.98030(10) and Dpc=1.68288(62). We also detect that at the first-order region of the phase diagram, the specific heat exhibits a double peak structure as in the Schottky-like anomaly, which is associated with an order–disorder transition.

Phase transition of Laplacian roughening model on a triangular lattice using Wang–Landau Monte Carlo simulation and Fisher zeros

Phase transition of the discrete Laplacian roughening model was investigated on a triangular lattice of various sizes by using the Wang–Landau (WL) Monte Carlo simulation and finite-size scaling analysis of partition-function zeros. The density of states was calculated using the WL method, and the internal energy and specific heat were calculated as a function of temperature. A single transition was observed at T c = 1.855(5) with a new universality class of the correlation exponent ν = 0.70(1) and the specific heat exponent α = 0.60(4). The results were confirmed by the finite size scaling analysis of the first zeros for the partition function. The critical exponents satisfy the scaling relation α = 2 − dν very well.

Physics-Based Computational Protein Design: An Update.

We describe methods for physics-based protein design and some recent applications from our work. We present the physical interpretation of a MC simulation in sequence space and show that sequences and conformations form a well-defined statistical ensemble, explored with Monte Carlo and Boltzmann sampling. The folded state energy combines molecular mechanics for solutes with continuum electrostatics for solvent. We usually assume one or a few fixed protein backbone structures and discrete side chain rotamers. Methods based on molecular dynamics, which introduce additional backbone and side chain flexibility, are under development. The redesign of a PDZ domain and an aminoacyl-tRNA synthetase enzyme were successful. We describe a versatile, adaptive, Wang-Landau MC method that can be used to design for substrate affinity, catalytic rate, catalytic efficiency, or the specificity of these properties. The methods are transferable to all biomolecules, can be systematically improved, and give physical insights.

Statistical Physics Approach to Liquid Crystals: Dynamics of Mobile Potts Model Leading to Smectic Phase, Phase Transition by Wang-Landau Method.

We study the nature of the smectic–isotropic phase transition using a mobile 6-state Potts model. Each Potts state represents a molecular orientation. We show that with the choice of an appropriate microscopic Hamiltonian describing the interaction between individual molecules modeled by a mobile 6-state Potts spins, we observe the smectic phase dynamically formed when we cool the molecules from the isotropic phase to low temperatures (T). In order to elucidate the order of the transition and the low-T properties, we use the high-performance Wang–Landau flat energy-histogram technique. We show that the smectic phase goes to the liquid (isotropic) phase by melting/evaporating layer by layer starting from the film surface with increasing T. At a higher T, the whole remaining layers become orientationally disordered. The melting of each layer is characterized by a peak of the specific heat. Such a succession of partial transitions cannot be seen by the Metropolis algorithm. The successive layer meltings/evaporations at low T are found to have a first-order character by examining the energy histogram. These results are in agreement with experiments performed on some smectic liquid crystals.

Flat-histogram method comparison on the two-dimensional Ising model

We compare the convergence of several flat-histogram methods applied to the two-dimensional Ising model, including the recently introduced stochastic approximation with a dynamic update factor (SAD) method. We compare this method to the Wang-Landau (WL) method, the 1/t variant of the WL method, and standard stochastic approximation Monte Carlo (SAMC). In addition, we consider a procedure WL followed by a "production run" with fixed weights that refines the estimation of the entropy. We find that WL followed by a production run does converge to the true density of states, in contrast to pure WL. Three of the methods converge robustly: SAD, 1/t-WL, and WL followed by a production run. Of these, SAD does not require a priori knowledge of the energy range. This work also shows that WL followed by a production run performs superior to other forms of WL while ensuring both ergodicity and detailed balance.

Investigation of the Models of Magnetic Dendrimers by the Wang–Landau Method

The thermodynamic properties of the models of magnetic dendrimers were studied by the Monte Carlo method. The system state density is calculated; the ground state structures are determined. The temperature dependences of magnetization m, entropy S, internal energy E, and heat capacity C are calculated. It is shown that in the investigated model of dendrimer, the effect of the surface sites on the general behavior of the system does not become weaker with growing sizes of the system.

Global Thermodynamic Properties of Complex Spin Systems Calculated from Density of States and Indirectly by Thermodynamic Integration Method

Evaluation of global thermodynamic properties such as the entropy or the free energy of complex systems featuring a high degree of frustration or disorder is often desirable. Nevertheless, they cannot be measured directly in standard Monte Carlo simulation. Therefore, they are either evaluated indirectly from the directly measured quantities, for example by the thermodynamic integration method (TIM), or by applying more sophisticated simulation methods, such as the Wang-Landau (WL) algorithm, which can directly sample density of states. In the present investigation we compare the performance of the WL and TIM methods for the calculation of the entropy of an Ising antiferromagnetic system on a Kagome lattice – a typical example of a complex spin system with high geometrical frustration resulting in a non-zero residual entropy the value of which is exactly known. It is found that the easier to implement TIM can yield results of comparable accuracy with that of the more involved WL method.

Monte Carlo simulations of Heisenberg magnets by the Wang-Landau method

In this work, we investigate the critical behavior of the three dimensional Heisenberg model by parallel Wang-Landau algorithm developed with using of the OpenMP technology.

Efficient simulation protocol for determining the density of states: Combination of replica-exchange Wang-Landau method and multicanonical replica-exchange method.

By combining two generalized-ensemble algorithms, the replica-exchange Wang-Landau (REWL) method and the 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. In order to verify the effectiveness of our protocol, we performed simulations of a square-lattice Ising model using the three methods, namely REWL, MUCAREM, and REWL-MUCAREM. The results showed that the density of states obtained by REWL-MUCAREM is more accurate than that is estimated by the two methods separately.

Wang–Landau approach to the simulation of water clusters

ABSTRACTThe Wang–Landau Monte Carlo method [Wang FG, Landau DP. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys Rev Lett. 2001;86:2050-2053; Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. Phys Rev E. 2001;64:056101] has been established as an effective computational approach for studying complex systems. In this paper, we review the properties of 6 popular empirical water potentials, including a recently proposed explicit 3-body potential, obtained by applying the Wang–Landau method to the simulation of water clusters. Comparing to numerous classical and first principle calculations, we show that several classical water models can provide a consistent description of the structural and thermodynamical properties of water clusters partially in agreement of ab initio results, although they were designed to reproduce macroscopic properties of water at ambient conditions.

Theory and calculation of colloidal depletion interaction

Colloidal dispersion is composed of particles with size ranging from 1 nm to [Formula: see text]m dispersed in solvents. There are the volume exclusion interaction and other interactions between colloidal particles, of which the former interaction causes the depletion effect. When a big sphere is immersed in the colloidal system of small spheres, there is a depletion layer around the big sphere where the center of small sphere cannot enter. The depletion layers of two big spheres overlap if they are close to each other, increasing the free volume accessed by small spheres and thereby enlarging the entropy of the system. As a result, an effective interaction between the two big spheres is resulted from the change of entropy as a function of their distance, which is referred to as the depletion interaction. This paper first introduces the concept and scenario of the depletion interaction in colloidal systems. Then we briefly introduce various numerical or simulations methods of the depletion interaction of hard sphere systems, such as the acceptance ratio method, Wang–Landau method, and the density functional theory method. Taking the Asakura–Oosawa model as an example, we introduce a useful approximation method, Derjaguin approximation as well as the derivation of some approximate formula for the depletion interaction of different hardcore colloidal systems, such as between a pair of spheres in mono-disperse small spheres, between a hard sphere and a hard wall in a liquid of small spheres, and between a pair of hard spheres in a liquid of thin rods and thin disks.

A practical guide to replica-exchange Wang—Landau simulations

This paper is based on a series of tutorial lectures about the replica-exchange Wang-Landau (REWL) method given at the IX Brazilian Meeting on Simulational Physics (BMSP 2017). It provides a practical guide for the implementation of the method. A complete example code for a model system is available online. In this paper, we discuss the main parallel features of this code after a brief introduction to the REWL algorithm. The tutorial section is mainly directed at users who have written a single-walker Wang–Landau program already but might have just taken their first steps in parallel programming using the Message Passing Interface (MPI). In the last section, we answer “frequently asked questions” from users about the implementation of REWL for different scientific problems.

Partition function zeros of the p-state clock model in the complex temperature plane.

We investigate the partition function zeros of the two-dimensional p-state clock model in the complex temperature plane by using the Wang-Landau method. For p=5, 6, 8, and 10, we propose a modified energy representation to enumerate exact irregular energy levels for the density of states without any binning artifacts. Comparing the leading zeros between different p's, we provide strong evidence that the upper transition at p=6 is indeed of the Berezinskii-Kosterlitz-Thouless (BKT) type in contrast to the claim of the previous Fisher zero study [Phys. Rev. E 80, 042103 (2009)10.1103/PhysRevE.80.042103]. We find that the leading zeros of p=6 at the upper transition collapse onto the zero trajectories of the larger p's including the XY limit while the finite-size behavior of p=5 differs from the converged behavior of p≥6 within the system sizes examined. In addition, we argue that the nondivergent specific heat in the BKT transition is responsible for the small partition function magnitude that decreases exponentially with increasing system size near the leading zero, fundamentally limiting access to large systems in search for zeros with an estimator under finite statistical fluctuations.

Density of states for systems with multiple order parameters: a constrained Wang-Landau method

A macroscopically constrained Wang-Landau Monte Carlo method was recently proposed to calculate the joint density of states (DOS) for systems with multiple order parameters. Here we demonstrate results for a nearest-neighbor Ising antiferromagnet with ferromagnetic long-range interactions (a model spin-crossover material). Its two relevant order parameters are the magnetization M and the staggered magnetization Ms. The joint DOS, g(E, M, Ms) where E is the total system energy, is calculated for zero external field and long-range interaction strength, and then obtained for arbitrary values of these two field-like model parameters by a simple transformation of E. Illustrations are shown for several parameter sets.

Lectin pathway proteins and association with complement activation and disease activity in patients with Systemic Lupus Erythematosus

Isomerization, ionization and fragmentation of molecular compounds in the interstellar medium can be triggered by stellar radiation and cosmic rays. In the present contribution, we examine the propensity for isomerization and the relative stability of aromatic rings in the pyrene and coronene molecules at various degrees of dehydrogenation by means of molecular modeling. Using the AIREBO reactive force field and advanced Monte Carlo techniques such as the Wang–Landau method based on suitable order parameters, entire free-energy profiles describing the isomerization pathways and equilibrium properties were calculated as a function of temperature or total energy. We generally find that hydrogenation significantly stabilizes the fully polycyclic aromatic hydrocarbon (PAH) structure, even though local dehydrogenation next to an aromatic ring favors ring opening. The formation of pentagonal rings, a typical defect motif in the polycyclic carbon skeleton, is predicted to be actually competitive with the loss of a hydrogen atom. Our investigation emphasizes the likely presence of defects in astrophysical PAHs, whose spectral features remain to be better characterized and understood.

Entropy of diluted antiferromagnetic Ising models on frustrated lattices using the Wang-Landau method.

We use a Monte Carlo simulation to study the diluted antiferromagnetic Ising model on frustrated lattices including the pyrochlore lattice to show the dilution effects. Using the Wang-Landau algorithm, which directly calculates the energy density of states, we accurately calculate the entropy of the system. We discuss the nonmonotonic dilution concentration dependence of residual entropy for the antiferromagnetic Ising model on the pyrochlore lattice, and compare it to the generalized Pauling approximation proposed by Ke etal. [Phys. Rev. Lett. 99, 137203 (2007)PRLTAO0031-900710.1103/PhysRevLett.99.137203]. We also investigate other frustrated systems, the antiferromagnetic Ising model on the triangular lattice and the kagome lattice, demonstrating the difference in the dilution effects between the system on the pyrochlore lattice and that on other frustrated lattices.

Dehydrogenation effects on the stability of aromatic units in polycyclic aromatic hydrocarbons in the interstellar medium: A computational study at finite temperature

Isomerization, ionization and fragmentation of molecular compounds in the interstellar medium can be triggered by stellar radiation and cosmic rays. In the present contribution, we examine the propensity for isomerization and the relative stability of aromatic rings in the pyrene and coronene molecules at various degrees of dehydrogenation by means of molecular modeling. Using the AIREBO reactive force field and advanced Monte Carlo techniques such as the Wang–Landau method based on suitable order parameters, entire free-energy profiles describing the isomerization pathways and equilibrium properties were calculated as a function of temperature or total energy. We generally find that hydrogenation significantly stabilizes the fully polycyclic aromatic hydrocarbon (PAH) structure, even though local dehydrogenation next to an aromatic ring favors ring opening. The formation of pentagonal rings, a typical defect motif in the polycyclic carbon skeleton, is predicted to be actually competitive with the loss of a hydrogen atom. Our investigation emphasizes the likely presence of defects in astrophysical PAHs, whose spectral features remain to be better characterized and understood.