Wang-Landau Algorithm Research Articles

Comparison of the microcanonical population annealing algorithm with the Wang-Landau algorithm

Comparison of the microcanonical population annealing algorithm with the Wang-Landau algorithm

Accuracy and Performance Analysis of the 1/t Wang-Landau Algorithm in the Joint Density of States Estimation

Accuracy and Performance Analysis of the 1/t Wang-Landau Algorithm in the Joint Density of States Estimation

Thermally activated particle motion in biased correlated Gaussian disorder potentials.

Thermally activated particle motion in disorder potentials is controlled by the large-ΔV tail of the distribution of height ΔV of the potential barriers created by the disorder. We employ the optimal fluctuation method to evaluate this tail for correlated quenched Gaussian potentials in one dimension in the presence of a small bias of the potential. We focus on the mean escape time (MET) of overdamped particles averaged over the disorder. We show that the bias leads to a strong (exponential) reduction of the MET in the direction along the bias. The reduction depends both on the bias and on detailed properties of the covariance of the disorder, such as its derivatives and asymptotic behavior at large distances. We verify our theoretical predictions for the large-ΔV tail of the barrier height distribution, as well as earlier predictions of this tail for zero bias, by performing large-deviation simulations of the potential disorder. The simulations employ correlated random potential sampling based on the circulant embedding method and the Wang-Landau algorithm, which enable us to probe probability densities smaller than 10^{-1200}.

Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices

We evaluate, in the large-N limit, the complete probability distribution P(A,m) of the values A of the sum ∑i=1N|λi|m , where λ i ( i=1,2,…,N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N→∞ , the rate function of P(A,m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue ‘evaporates’ from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p=1+1/|m−1| , where 0<m<2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m⩽1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner’s semicircular distribution.

Phase transitions of smectic-isotropic phase in liquid crystals

Smectic phases formed by rod-like molecules with long axes that are parallel and also arranged in planes. The smectic-isotropic phase transition is a phase change from the liquid crystal to the liquid phase. In this work, we use a mobile 6-state Potts model to study the nature of the smectic-isotropic phase transition. Microscopic interactions between neighbouring molecules in this model are supplemented with the Lennard-Jones potential. This study applies Monte Carlo simulation with the Wang-Landau algorithm to determine the characteristics of smectic-isotropic phase transitions. It is shown clearly that the smectic phase goes to the isotropic phase and undergoes a first-order transition. The results also point out that when the temperature increases, molecules on the surface are orientationally disordered, then the molecules gradually lose their positional order. These results are in agreement with experiments that revealed the coexistence of the smectic and isotropic phasesduring the phase transition process in accordance with experimental studies.

Predicting Polymer Brush Behavior in Solvents Using the Steepest-Entropy-Ascent Quantum Thermodynamic Framework.

The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework is utilized to study the effects of temperature on polymer brushes. The brushes are represented by a discrete energy spectrum, and energy degeneracies obtained through the replica-exchange Wang-Landau algorithm. The SEAQT equation of motion is applied to the density of states to establish a unique kinetic path from an initial thermodynamic state to a stable equilibrium state. The kinetic path describes the brush's evolution in state space, as it interacts with a thermal reservoir. The predicted occupation probabilities along the kinetic path are used to determine the expected thermodynamic and structural properties. The polymer density profile of a polystyrene brush in cyclohexane solvent is predicted using the equation of motion, and it agrees qualitatively with the experimental density profiles. The Flory-Huggins parameter chosen to describe brush-solvent interactions affects the solvent distribution in the brush but has a minimal impact on the polymer density profile. Three types of nonequilibrium kinetic paths with differing amounts of entropy production are considered: a heating path, a cooling path, and a heating-cooling path. Properties such as tortuosity, radius of gyration, brush density, solvent density, and brush chain conformations are calculated for each path.

Large-deviation properties of SIR model incorporating protective measures

We simulate spreads of diseases for the susceptible–infected–recovered (SIR) model on contact networks with a particular focus on incorporated protective measures such as face masks. We consider the small-world network model. By using a large-deviation approach, in particular the Wang–Landau algorithm, we obtained the full probability density function (pdf) of the cumulative number C of infected people over the full range of its support. In this way we are able to reach probabilities as small as 10−50. We obtain distinct characteristics in the pdf such as non-analyticities induced by the onset of the protective measures. Still, the results indicate that the mathematical large-deviation principle also holds for this extended SIR model, meaning that the size-dependence enters in P(C) in a simple fashion and the distribution is determined by the so-called rate function. We observe different phases in the pdf, which we investigate by analyzing the corresponding infection courses, i.e. time series, and and their correlations to the observed values of C.

Large-deviations of disease spreading dynamics with vaccination.

We numerically simulated the spread of disease for a Susceptible-Infected-Recovered (SIR) model on contact networks drawn from a small-world ensemble. We investigated the impact of two types of vaccination strategies, namely random vaccination and high-degree heuristics, on the probability density function (pdf) of the cumulative number C of infected people over a large range of its support. To obtain the pdf even in the range of probabilities as small as 10-80, we applied a large-deviation approach, in particular the 1/t Wang-Landau algorithm. To study the size-dependence of the pdfs within the framework of large-deviation theory, we analyzed the empirical rate function. To find out how typical as well as extreme mild or extreme severe infection courses arise, we investigated the structures of the time series conditioned to the observed values of C.

Laplacian roughening model on kagome and honeycomb lattices

We investigate the Laplacian roughening model on kagome and honeycomb lattices. By using the Wang–Landau algorithm, we obtain the density of states, and calculate both the specific heat and the partition function zeros. Critical exponents and transition temperature are measured by means of the finite-size scaling analysis. The results indicate strong evidences that the system undergoes a single first-order phase transition on both lattices. On triangular and square lattices, a single second-order transition was found in our previous studies. It seems that the type of transition in the Laplacian roughening model depends on the substrate lattice structures.

Phase transitions in the four-state Potts model with competing exchange interactions: Application of the Wang-Landau algorithm

Phase transitions in the four-state Potts model with competing exchange interactions: Application of the Wang-Landau algorithm

Predicting defect stability and annealing kinetics in two-dimensional PtSe2 using steepest entropy ascent quantum thermodynamics

The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework was used to calculate the stability of a collection of point defects in 2D PtSe2 and predict the kinetics with which defects rearrange during thermal annealing. The framework provides a non-equilibrium, ensemble-based framework with a self-consistent link between mechanics (both quantum and classical) and thermodynamics. It employs an equation of motion derived from the principle of steepest entropy ascent (maximum entropy production) to predict the time evolution of a set of occupation probabilities that define the states of a system undergoing a non-equilibrium process. The system is described by a degenerate energy landscape of eigenvalues, and the entropy is found from the occupation probabilities and the eigenlevel degeneracies. Scanning tunneling microscopy was used to identify the structure and distribution of point defects observed experimentally in a 2D PtSe2 film. A catalog of observed defects includes six unique point defects (vacancies and anti-site defects on Pt and Se sublattices) and twenty combinations of multiple point defects in close proximity. The defect energies were estimated with density functional theory, while the degeneracies, or density of states, for the 2D film with all possible combinations or arrangements of cataloged defects was constructed using a non-Markovian Monte-Carlo approach (i.e. the Replica–Exchange–Wang–Landau algorithm (Vogel et al 2013 Phys. Rev. Lett. 110 210603)) with a q-state Potts model. The energy landscape and associated degeneracies were determined for a 2D PtSe2 film two molecules thick and unit cells in area (total of 5400 atoms). The SEAQT equation of motion was applied to the energy landscape to determine how an arbitrary density and arrangement of the six defect types evolve during annealing. Two annealing processes were modeled: heating from 77 K (−196 C) to 523 K (250 ∘C) and isothermal annealing at 523 K. The SEAQT framework predicted defect configurations, which were consistent with experimental STM images.

Single-walled Ising nanotube with opposite sign of interactions using Wang–Landau algorithm

The effect of opposite sign of interactions in a single-walled Ising nanotube is investigated using the Wang–Landau algorithm. The thermodynamic observables are calculated from the estimated density of states (DOS) with and without the presence of an external magnetic field. Irrespective of the applied magnetic field, a symmetric trend of DOS is observed for opposite sign of interactions which is in contrast to the asymmetric trend for same sign of interactions. Further, two types of antiferromagnetic (AFM) orderings, namely A-type and C-type antiferromagnetic order, are observed for opposite sign of interactions. These AFM spin orientations are switched to ferromagnetic (FM) phase by increasing the applied magnetic field (B). However, the spin ordering changes from the ordered AFM/FM phase to a disordered paramagnetic phase by increasing the temperature. Phase diagram shows that these three phases coexist around \(B=2.0\). This study indicates that, by properly tuning the magnetic properties, the single-walled nanotube can be used for fabrication of new types of magnetic storage nanomaterials.

Entropy profiles of Schelling’s segregation model from the Wang–Landau algorithm

We evaluate Schelling's segregation outcomes from the square lattice, regular random networks, and clustered regular random networks by situating them in the probability distribution of the entire outcome space of satisfaction and segregation. To do so, we employ the Wang-Landau algorithm and calculate the entropy and the number of states as a function of satisfaction and segregation. According to the results, satisfaction tends to increase with segregation, irrespective of the network structure. Moreover, segregation occurs almost surely when satisfaction is maximized, which we also algebraically derive and confirm on infinite-size networks. The average ratios of the neighbors of the same tag are about 67% for the square lattice and regular random networks and about 73% for clustered regular random networks with a clustering coefficient of 0.37(1). Thus, clustering increases segregation on regular random networks. Further, we find that Schelling's path-dependent process generates sharper segregation than expected by random configurations, an outcome symptomatic of over-optimization from the social welfare perspective. Adopting an alternative rule restricting individuals' choice set may reduce segregation without compromising on satisfaction.

Entropy-driven microstructure evolution predicted with the steepest-entropy-ascent quantum thermodynamic framework

A Potts model and the Replica Exchange Wang–Landau algorithm is used to construct an energy landscape for a crystalline solid containing surfaces and grain boundaries. The energy landscape is used by an equation of motion from the steepest-entropy-ascent quantum thermodynamic (SEAQT) framework to explore the kinetics of three distinct kinds of microstructural evolution: polycrystalline sintering, precipitate coarsening, and grain growth. The steepest-entropy-ascent postulate predicts unique kinetic paths for these non-equilibrium processes without needing any detailed information about the underlying physical mechanisms of the processes. A method is also proposed for associating the kinetic path in state space to a set of smoothly evolving microstructural descriptors. The SEAQT-predicted kinetics agree well with available experimental kinetics for ZrO2 sintering, Al3Li precipitate coarsening, and grain growth in nanocrystalline Pd. The computational cost associated with calculating the energy landscape needed by the approach is comparable to a Monte Carlo simulation. However, the subsequent kinetic calculations from the SEAQT equation of motion are quite modest and save considerable computational resources by obviating the need for averaging multiple kinetic Monte Carlo runs.

Энергетический анализ магнитных структур основного состояния модели Поттса

Based on the Wang-Landau algorithm, the Monte Carlo method was used to perform an energy analysis of the magnetic structures of the ground state of the two-dimensional Potts model with the number of spin states $q=4$ on a triangular lattice, taking into account the exchange interactions of the first $J_1$ and second $J_2$ nearest neighbors. The studies were carried out for the value of the interaction of the second nearest neighbors in the interval $-2.0 \leq J_2 \leq 0.0$. The magnetic structures of the ground state are determined for different values of the interaction of the second nearest neighbors. It is found that a change in the magnitude of the interaction of the second nearest neighbors in this model leads to a change in the magnetic ordering.

Фазовая диаграмма и структура основного состояния трехвершинной модели Поттса на решетке Кагоме

The high-efficient Wang-Landau algorithm of the Monte-Carlo entropic method has used for studies the three-state Potts model on the Kagome lattice, taking into account the ferromagnetic exchange interaction between the nearest neighbors and the competing antiferromagnetic interaction between the next nearest neighbors. The density of states are calculated, the magnetic structures of the ground state are determined and the temperature dependences of various thermodynamic parameters are calculated. It is shown that depending on the ratio of interactions between the nearest and next nearest neighbors, the ground state of the system can be ferromagnetic, highly degenerated frustrated or have a special type of triplet antiferromagnetic ordering. It was established that the phase transition from the ferromagnetic phase to the paramagnetic is the phase transition of the second order, while the transition from the triplet antiferromagnetic phase is the first order. At the frustration point, which divides both regions, the phase transitions does not occur (the system does not go to ordered phase for any temperature close to zero). The critical temperature of the phase transitions calculated and phase diagram of the system are determined.

Comparison of Benchmark HP Model Proteins on Face-Centered Cubic and Simple Cubic Lattices using Replica-Exchange Wang-Landau Sampling

The HP model represents protein sequences by coarse-graining amino acid types as either hydrophobic (H) or polar (P). This model has been studied extensively, where the simple cubic (sc) lattice is a commonly used for simplicity and efficiency, despite a limited ability to geometrically represent protein structures. One straightforward improvement is to instead use the face-centered cubic (fcc) lattice, which offers a more realistic geometry while retaining simplicity. Using the replica-exchange Wang-Landau algorithm, the ground state energies and full density of states are predicted for a set of biologically-motivated HP sequences on the fcc lattice. A direct thermodynamic comparison is made between the folding on fcc and sc lattices, and structural properties are calculated to further examine the folding on the fcc lattice.

A portable and flexible implementation of the Wang–Landau algorithm in order to determine the density of states

In this work we develop an implementation of the Wang–Landau algorithm Wang and Landau (2001) [4]arXiv e-prints. 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 and C++ languages 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 developments in the parallelization of the code for faster computation. We establish the soundness and effectiveness of our implementation by studying well-known systems such as the Ising model, the Lennard–Jones and EAM solids. We have found that our implementation can find the DOS with very good precision in a reasonable amount of time. Therefore, we are equipped with a very powerful and flexible implementation that can be easily used in order to study more realistic models of matter. Program summaryProgram Title: Republica Wang–Landau (RWL)CPC Library link to program files:https://doi.org/10.17632/gcx9k8wnhh.1Licensing provisions: GPLv3Programming language: Python, C++Nature of problem: An implementation of the WL algorithm that is flexible enough to be used for a large variety of systems.Solution method: This implementation separates the actual Wang–Landau code of the abstract implementation of the system. Therefore, any system can be attached as a walker—a Python class that represents the system.Additional comments including restrictions and unusual features: As examples, basic systems such as the Ising model are included plus a wrapper class for a LAMMPS walker to be used for any system supported by LAMMPS.

Энергетический анализ магнитных структур основного состояния модели Поттса с конкурирующими обменными взаимодействиями

Based on the Wang-Landau algorithm, the Monte Carlo method was used to study the magnetic structures of the ground state and the thermodynamic properties of the two-dimensional Potts model with the number of spin states q=4 on a triangular lattice, taking into account the exchange interactions of the first J1 and second J2 nearest neighbors. The studies were carried out for the value of the interaction of the second nearest neighbors in the range -2.0≤ J2≤0.0. The magnetic structures of the ground state are constructed in the interval considered. An energy analysis of the magnetic structures of the ground state has been carried out. A phase diagram of the dependence of the critical temperature on the value of J2 is constructed. It is shown that taking into account the interactions of the second nearest neighbors leads to the appearance of frustrations and violation of the magnetic ordering.

Lattice star and acyclic branched polymer vertex exponents in 3d

Numerical values of lattice star entropic exponents γ f , and star vertex exponents σ f , are estimated using parallel implementations of the PERM and Wang–Landau algorithms. Our results show that the numerical estimates of the vertex exponents deviate from predictions of the ϵ-expansion and confirms and improves on estimates in the literature. We also estimate the entropic exponents of a few acyclic branched lattice networks with comb and brush connectivities. In particular, we confirm within numerical accuracy the Duplantier scaling relation for a comb and two brushes (where m f is the number of nodes of degree f in the network) using our independent estimates of σ f .