Wang-Landau Simulations Research Articles

Reduced critical slowing down for statistical physics simulations.

Wang-Landau simulations offer the possibility to integrate explicitly over a collective coordinate and stochastically over the remainder of configuration space. We propose to choose the so-called "slow mode," which is responsible for large autocorrelation times and thus critical slowing down, for collective integration. We study this proposal for the Ising model and the linear-log-relaxation (LLR) method as simulation algorithm. We first demonstrate supercritical slowing down in a phase with spontaneously broken symmetry and for the heat-bath algorithms, for which autocorrelation times grow exponentially with system size. By contrast, using the magnetization as collective coordinate, we present evidence that supercritical slowing down is absent. We still observe a polynomial increase of the autocorrelation time with volume (critical slowing down), which is, however, reduced by orders of magnitude when compared to local update techniques.

Confinement free energy for a polymer chain: Corrections to scaling.

Spatial confinement of a polymer chain results in a reduction of conformational entropy. For confinement of a flexible N-mer chain in a planar slit or cylindrical pore (confining dimension D), a blob model analysis predicts the asymptotic scaling behavior ΔF/N ∼ D-γ with γ ≈ 1.70, where ΔF is the free energy increase due to confinement. Here, we extend this scaling analysis to include the variation of local monomer density upon confinement giving ΔF/N ∼ D-γ(1 - h(N, D)), where the correction-to-scaling term has the form h ∼ Dy/NΔ with exponents y = 3 - γ ≈ 1.30 and Δ = 3/γ - 1 ≈ 0.76. To test these scaling predictions, we carry out Wang-Landau simulations of confined and unconfined tangent-hard-sphere chains (bead diameter σ) in the presence of a square-well trapping potential. The fully trapped chain provides a common reference state, allowing for an absolute determination of the confinement free energy. Our simulation results for 32 ≤ N ≤ 1024 and 3 ≤ D/σ ≤ 14 are well-described by the extended scaling relation giving exponents of γ = 1.69(1), y = 1.25(2), and Δ = 0.75(6).

All-or-none folding of a flexible polymer chain in cylindrical nanoconfinement.

Geometric confinement of a polymer chain results in a loss of conformational entropy. For a chain that can fold into a compact native state via a first-order-like transition, as is the case for many small proteins, confinement typically provides an entropic stabilization of the folded state, thereby shifting the location of the transition. This allows for the possibility of confinement (entropy) driven folding. Here, we investigate such confinement effects for a flexible square-well-sphere N-mer chain (monomer diameter σ) confined within a long cylindrical pore (diameter D) or a closed cylindrical box (height H = D). We carry out Wang-Landau simulations to construct the density of states, which provides access to the complete thermodynamics of the system. For a wide pore, an entropic stabilization of the folded state is observed. However, as the pore diameter approaches the size of the folded chain (D ∼ N1/3σ), we find a destabilization effect. For pore diameters smaller than the native ground-state, the chain folds into a different, higher energy, ground state ensemble and the T vs D phase diagram displays non-monotonic behavior as the system is forced into different ground states for different ranges of D. In this regime, isothermal reduction of the confinement dimension can induce folding, unfolding, or crystallite restructuring. For the cylindrical box, we find a monotonic stabilization effect with decreasing D. Scaling laws for the confinement free energy in the athermal limit are also investigated.

A First Look at Lattice Effects in Coarse-Grained Protein Models via Wang-Landau Simulations

In order to study the effects of lattice constraints on coarse-grained protein models, we apply Wang-Landau sampling to the continuum analogue of the hydrophobic-polar (HP) lattice protein model. The continuum version is inspired by the AB polymer model but incorporates potentials chosen specifically to mimic those of the lattice case. Because of their relative simplicity, both the lattice and continuum models offer significant computational advantage over all-atom simulations, but the impact of the additional lattice constraint on generic folding behavior is unknown. In this preliminary study, we compare and contrast thermodynamics during the folding process of the continuum model to the original HP lattice protein model for sequences mapped from Crambin, a 46 amino acid plant protein. We find that the folding process for both of these coarse-grained models is quite similar, with major structural transitions occurring at almost exactly the same temperatures.

Prediction of the boiling and critical points of polycyclic aromatic hydrocarbons via Wang-Landau simulations and machine learning

We use molecular simulations and machine learning to determine the vapor-liquid equilibria (VLE) properties of polycyclic aromatic hydrocarbons (PAHs). First, we perform Hybrid Monte Carlo Wang-Landau (HMC-WL) simulations on reference PAHs and evaluate VLE properties including the boiling and critical points, as well as the parameters capturing the dependence of vapor pressure upon temperature. Second, we use machine learning (ML) to extend the simulation predictions to a series of PAHs, with varying shape and number of fused rings. More specifically, HMC-WL results are used to train an artificial neural network using as descriptors molecular data, i.e. molecular weight, moments of inertia and number of aromatic rings. The combined molecular simulation-machine learning approach proposed here provides a self-consistent method to determine the VLE properties of organic compounds.

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.

Selectivity and Desorption Free Energies for Methane–Ethane Mixtures in Covalent Organic Frameworks

We show how expanded Wang–Landau simulations can be used to study the adsorption of methane–ethane mixtures in COF-102, COF-105, and COF-108. This approach has several advantages. First, a single simulation run is performed to determine key thermodynamic properties such as the adsorption isotherms and selectivity. Second, the combination of the expanded method with the Wang–Landau sampling in the grand-canonical ensemble provides direct access to the grand potential Ω = −kBT ln[Θ(μ1,μ2,V,T)] via a numerical evaluation of the grand-canonical partition function. From there, we calculate several thermodynamic quantities of adsorption, including the Gibbs free energy, enthalpy, and entropy, which give important insights into the mechanism of adsorption for the methane–ethane mixtures in covalent organic frameworks (COFs). In particular, using a solution thermodynamics approach, we identify a direct correlation between the separation efficiency (selectivity) of a given COF and its energetic efficiency (through...

Benchmark Free Energies and Entropies for Saturated and Compressed Water

We use molecular simulation to compute the thermodynamic properties of 7 rigid models for water (SPC/E, TIP3P, TIP4P, TIP4P/2005, TIP4P/Ew, TIP5P, OPC) over a wide range of temperature and pressure. Carrying out Expanded Wang-Landau simulations, we obtain a high accuracy estimate for the grand-canonical partition function which, in turn, provides access to all properties, including the free energy and entropy, both at the vapor-liquid coexistence and for compressed water. The results at coexistence highlight the close connection between the behavior of the statistical functions and the thermodynamic properties. They show that the subgroup (SPC/E,TIP4P/2005,TIP4P/Ew) provides the most accurate account of the vapor-liquid equilibrium properties. For compressed water, the comparison of the simulation results to the experimental data establishes that the TIP4P/Ew model performs best among the 7 models considered here, and captures the experimental trends for the dependence of entropy and molar Gibbs free energy on pressure.

Tricriticality in crossed Ising chains.

We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in Nb_{12}O_{29} and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.

Macroscopically constrained Wang-Landau method for systems with multiple order parameters and its application to drawing complex phase diagrams.

A generalized approach to Wang-Landau simulations, macroscopically constrained Wang-Landau, is proposed to simulate the density of states of a system with multiple macroscopic order parameters. The method breaks a multidimensional random-walk process in phase space into many separate, one-dimensional random-walk processes in well-defined subspaces. Each of these random walks is constrained to a different set of values of the macroscopic order parameters. When the multivariable density of states is obtained for one set of values of fieldlike model parameters, the density of states for any other values of these parameters can be obtained by a simple transformation of the total system energy. All thermodynamic quantities of the system can then be rapidly calculated at any point in the phase diagram. We demonstrate how to use the multivariable density of states to draw the phase diagram, as well as order-parameter probability distributions at specific phase points, for a model spin-crossover material: an antiferromagnetic Ising model with ferromagnetic long-range interactions. The fieldlike parameters in this model are an effective magnetic field and the strength of the long-range interaction.

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. V. Impact of an electric field on the thermodynamic properties and ideality contours of water.

Using molecular simulation, we assess the impact of an electric field on the properties of water, modeled with the SPC/E potential, over a wide range of states and conditions. Electric fields of the order of 0.1 V/Å and beyond are found to have a significant impact on the grand-canonical partition function of water, resulting in shifts in the chemical potential at the vapor-liquid coexistence of up to 20%. This, in turn, leads to an increase in the critical temperatures by close to 7% for a field of 0.2 V/Å, to lower vapor pressures, and to much larger entropies of vaporization (by up to 35%). We interpret these results in terms of the greater density change at the transition and of the increased structural order resulting from the applied field. The thermodynamics of compressed liquids and of supercritical water are also analyzed over a wide range of pressures, leading to the determination of the Zeno line and of the curve of ideal enthalpy that span the supercritical region of the phase diagram. Rescaling the phase diagrams obtained for the different field strengths by their respective critical properties allows us to draw a correspondence between these systems for fields of up to 0.2 V/Å.

Impact of Friedel oscillations on vapor-liquid equilibria and supercritical properties in two and three dimensions.

We determine the impact of the Friedel oscillations on the phase behavior, critical properties, and thermodynamic contours in films [two dimensions (2D)] and bulk phases [three dimensions (3D)]. Using expanded Wang-Landau simulations, we calculate the grand-canonical partition function and, in turn, the thermodynamic properties of systems modeled with a linear combination of the Lennard-Jones and Dzugutov potentials, weighted by a parameter X (0<X<1). Varying X allows us to control the height of the first Friedel oscillation and to provide a complete characterization of the effect of the metal-like character in the potential on the thermodynamic properties over a wide range of conditions. For 3D systems, we are able to show that the critical parameters exhibit a linear dependence on X and that the loci for the thermodynamic state points, for which the system shows the same compressibility factor or enthalpy as an ideal gas, are two straight lines spanning the subcritical and supercritical regions of the phase diagram for all X values. Reducing the dimensionality to 2D results in a loss of impact of the Friedel oscillation on the critical properties, as evidenced by the virtually constant critical density across the range of X values. Furthermore, our results establish that the straightness of the two ideality lines is retained in 2D and is independent from the height of the first Friedel oscillation in the potential.

Depletion-driven crystallization of cubic colloids sedimented on a surface.

Cubic colloids, sedimented on a surface and immersed in a solution of depletant molecules, were modeled with a family of shapes which smoothly varies from squares to circles. Using Wang-Landau simulations with expanded ensembles, we observe the formation of rhombic lattices, square lattices, hexagonal lattices, and a fluid phase. This systematic investigation includes locating transitions between all combinations of the three lattice structures upon changing the shape and transitions between the fluid and crystal upon changing the depletant concentration. The rhombic lattice deforms smoothly between square-like and hexagonal-like angles, depending on both the shape and the depletant concentration. Our results on the effect of the depletant concentration, depletant size, and colloid shape to influence the stability of the fluid and the lattice structures may help guide experimental studies with recently synthesized cubic colloids.

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. IV. Performance of many-body force fields and tight-binding schemes for the fluid phases of silicon.

We extend Expanded Wang-Landau (EWL) simulations beyond classical systems and develop the EWL method for systems modeled with a tight-binding Hamiltonian. We then apply the method to determine the partition function and thus all thermodynamic properties, including the Gibbs free energy and entropy, of the fluid phases of Si. We compare the results from quantum many-body (QMB) tight binding models, which explicitly calculate the overlap between the atomic orbitals of neighboring atoms, to those obtained with classical many-body (CMB) force fields, which allow to recover the tetrahedral organization in condensed phases of Si through, e.g., a repulsive 3-body term that favors the ideal tetrahedral angle. Along the vapor-liquid coexistence, between 3000 K and 6000 K, the densities for the two coexisting phases are found to vary significantly (by 5 orders of magnitude for the vapor and by up to 25% for the liquid) and to provide a stringent test of the models. Transitions from vapor to liquid are predicted to occur for chemical potentials that are 10%-15% higher for CMB models than for QMB models, and a ranking of the force fields is provided by comparing the predictions for the vapor pressure to the experimental data. QMB models also reveal the formation of a gap in the electronic density of states of the coexisting liquid at high temperatures. Subjecting Si to a nanoscopic confinement has a dramatic effect on the phase diagram with, e.g. at 6000 K, a decrease in liquid densities by about 50% for both CMB and QMB models and an increase in vapor densities between 90% (CMB) and 170% (QMB). The results presented here provide a full picture of the impact of the strategy (CMB or QMB) chosen to model many-body effects on the thermodynamic properties of the fluid phases of Si.

Thermodynamics and structure of macromolecules from flat-histogram Monte Carlo simulations.

Over the last decade flat-histogram Monte Carlo simulations, especially multi-canonical and Wang-Landau simulations, have emerged as a strong tool to study the statistical mechanics of polymer chains. These investigations have focused on coarse-grained models of polymers on the lattice and in the continuum. Phase diagrams of chains in bulk as well as chains attached to surfaces were studied, for homopolymers as well as for protein-like models. Also, aggregation behavior in solution of these models has been investigated. We will present here the theoretical background for these simulations, explain the algorithms used and discuss their performance and give an overview over the systems studied with these methods in the literature, where we will limit ourselves to studies of coarse-grained model systems. Implementations of these algorithms on parallel computers will be also briefly described. In parallel to the development of these simulation methods, the power of a micro-canonical analysis of such simulations has been recognized, and we present the current state of the art in applying the micro-canonical analysis to phase transitions in nanoscopic polymer systems.

Many-Body Effects on the Thermodynamics of Fluids, Mixtures, and Nanoconfined Fluids

Using expanded Wang-Landau simulations, we show that taking into account the many-body interactions results in sharp changes in the grand-canonical partition functions of single-component systems, binary mixtures, and nanoconfined fluids. The many-body contribution, modeled with a 3-body Axilrod-Teller-Muto term, results in shifts toward higher chemical potentials of the phase transitions from low-density phases to high-density phases and accounts for deviations of more than, e.g., 20% of the value of the partition function for a single-component liquid. Using the statistical mechanics formalism, we analyze how this contribution has a strong impact on some properties (e.g., pressure, coexisting densities, and enthalpy) and a moderate impact on others (e.g., Gibbs or Helmholtz free energies). We also characterize the effect of the 3-body terms on adsorption isotherms and adsorption thermodynamic properties, thereby providing a full picture of the effect of the 3-body contribution on the thermodynamics of nanoconfined fluids.

Sequence Determines Degree of Knottedness in a Coarse-Grained Protein Model

Knots are abundant in globular homopolymers but rare in globular proteins. To shed new light on this long-standing conundrum, we study the influence of sequence on the formation of knots in proteins under native conditions within the framework of the hydrophobic-polar lattice protein model. By employing large-scale Wang-Landau simulations combined with suitable MonteCarlo trial moves we show that even though knots are still abundant on average, sequence introduces large variability in the degree of self-entanglements. Moreover, we are able to design sequences which are either almost always or almost never knotted. Our findings serve as proof of concept that the introduction of just one additional degree of freedom per monomer (in our case sequence) facilitates evolution towards a protein universe in which knots are rare.

Transitions of tethered chain molecules under tension.

An applied tension force changes the equilibrium conformations of a polymer chain tethered to a planar substrate and thus affects the adsorption transition as well as the coil-globule and crystallization transitions. Conversely, solvent quality and surface attraction are reflected in equilibrium force-extension curves that can be measured in experiments. To investigate these effects theoretically, we study tethered chains under tension with Wang-Landau simulations of a bond-fluctuation lattice model. Applying our model to pulling experiments on biological molecules we obtain a good description of experimental data in the intermediate force range, where universal features dominate and finite size effects are small. For tethered chains in poor solvent, we observe the predicted two-phase coexistence at transitions from the globule to stretched conformations and also discover direct transitions from crystalline to stretched conformations. A phase portrait for finite chains constructed by evaluating the density of states for a broad range of solvent conditions and tensions shows how increasing tension leads to a disappearance of the globular phase. For chains in good solvents tethered to hard and attractive surfaces we find the predicted scaling with the chain length in the low-force regime and show that our results are well described by an analytical, independent-bond approximation for the bond-fluctuation model for the highest tensions. Finally, for a hard or slightly attractive surface the stretching of a tethered chain is a conformational change that does not correspond to a phase transition. However, when the surface attraction is sufficient to adsorb a chain it will undergo a desorption transition at a critical value of the applied force. Our results for force-induced desorption show the transition to be discontinuous with partially desorbed conformations in the coexistence region.

Thermodynamics of Phase Coexistence and Metal–Nonmetal Transition in Mercury: Assessment of Effective Potentials via Expanded Wang–Landau Simulations

We present molecular simulation results on the thermodynamics of phase transitions (specifically, the vapor-liquid and metal-nonmetal transitions) in mercury, as predicted by effective potential models. We use a recently developed method, known as Expanded Wang-Landau simulations, to determine the grand-canonical partition function of systems of mercury atoms. Using the statistical mechanics formalism, we are then able to determine all thermodynamic properties of the system, including the Gibbs free energy and entropy. Prior experimental and theoretical work has emphasized the strong interplay between the vapor-liquid coexistence and the metal-nonmetal transition. We therefore start by assessing the accuracy of the effective potentials considered in this work through a comparison to the available experimental data. We then analyze the thermodynamics of the nonmetal liquid-metal liquid transition, characterized by sharp variations in the rate of change of Gibbs free energy and enthalpy as a function of density. We also identify a crossover density (10.5 g/cm(3)) consistent with the results of recent ab initio calculations and with the experiment.

Adsorption of hydrogen in covalent organic frameworks using expanded Wang–Landau simulations

Using expanded Wang–Landau simulations, we study the adsorption of hydrogen in a series of covalent organic frameworks (COF-102, COF-105 and COF-108). In the present implementation, this new method, based on the Wang–Landau sampling, allows for a uniform sampling of the number of molecules adsorbed in the COFs. We show that this method leads to a highly accurate estimate of the grand canonical partition function for the system and gives excellent results for the adsorption isotherms of , both at and , when compared to prior simulation work and experiments. Moreover, various thermodynamic properties of adsorption, including entropy, Gibbs free energy and Helmohltz free energy, as well as the desorption properties are calculated, providing insight into the adsorption phenomenon. We finally propose a ranking for the performance of the three COFs for adsorption, based on the isothermal work required to regenerate the adsorbent , and find the following order for applications at ambient temperature.