Wang-Landau algorithm for continuous models and joint density of states

Mean Field Initialization of the Annealed Importance Sampling Algorithm for an Efficient Evaluation of the Partition Function Using Restricted Boltzmann Machines

Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset.

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

The development of new algorithms for simulations in physics is as important as the development of new analytical methods. In this paper, we present a comparison of the recently developed microcanonical population annealing (MCPA) algorithm with the rather mature Wang-Landau algorithm. The comparison is performed on two cases of the Potts model that exhibit a first-order phase transition. We compare the simulation results of both methods with exactly known results, including the finite-dimensional dependence of the maximum of the specific heat capacity. We evaluate the Binder cumulant minimum, the ratio of peaks in the energy distribution at the critical temperature, the energies of the ordered and disordered phases, and interface tension. Both methods exhibit similar accuracy at selected sets of modeling parameters.

Aggregation and crystallization of small alkanes

We present a computer simulation study of the aggregation and ordering of short alkane chains using a united atom model description. Our simulation approach allows us to determine the density of states of our systems and, from those, their thermodynamics for all temperatures. All systems show a first order aggregation transition followed by a low-temperature ordering transition. For a few chain aggregates of intermediate lengths (up to N = 40), we show that these ordering transitions resemble the quaternary structure formation in peptides. In an earlier publication, we have already shown that single alkane chains fold into low-temperature structures, best described as secondary and tertiary structure formation, thus completing this analogy here. The aggregation transition in the thermodynamic limit can be extrapolated in pressure to the ambient pressure for which it agrees well with experimentally known boiling points of short alkanes. Similarly, the chain length dependence of the crystallization transition agrees with known experimental results for alkanes. For small aggregates, for which volume and surface effects are not yet well separated, our method allows us to identify the crystallization in the core of the aggregate and at its surface, individually.

Topological phase transitions in two-dimensional bent-core liquid crystal models

Two-dimensional liquid crystal (LC) models of interacting V-shaped bent-core molecules with two rigid rodlike identical segments connected at a fixed angle (θ=112^{∘}) are investigated. The model assigns equal biquadratic tensor coupling among constituents of the interacting neighboring molecules on a square lattice, allowing for reorientations in three dimensions (d=2, n=3). We find evidence of two temperature-driven topological transitions mediated by point disclinations associated with the three ordering directors, condensing the LC medium successively into uniaxial and biaxial phases. With Monte Carlo simulations, temperature dependencies of the system energy, specific heat, orientational order parameters, topological order parameters, and densities of unbound topological defects of the different ordering directors are computed. The high-temperature transition results in topological ordering of disclinations of the primary director, imparting uniaxial symmetry to the phase. The low-temperature transition precipitates simultaneous topological ordering of defects of the remaining directors, resulting in biaxial symmetry. The correlation functions, quantifying spatial variations of the orientational correlations of the molecular axes show exponential decays in the high-temperature (disordered) phase, and power-law decays in the low-temperature (biaxial) phase. Differing temperature dependencies of the topological parameters point to a significant degree of cross coupling among the uniaxial and biaxial tensors of interacting molecules. This simplified Hamiltonian leaves θ as the only free model parameter, and the system traces a θ-dependent trajectory in a plane of the phenomenological parameter space.

The Measurement of Information and Free Energy in Mechanical-Force-Driven Coil-Globule Transitions

We study the role of information (the relative entropy) for polymers undergoing coil-globule transitions driven by a time-dependent force. Pulling experiments at various speeds are performed by Brownian dynamics simulations. We obtain the work distributions for the forward and time-reversed backward processes and information stored at the end of the nonequilibrium pulling processes. We present the systematic method to measure the information from the pulling experiments and extract the information by analyzing slowly relaxing modes. When the information is incorporated, the work distributions modified by the information allow access to the proper free energy via the formulation of the generalized fluctuation theorems even if the initial states of the forward and time-reversed backward processes are out of equilibrium. This demonstrates that the work-information conversion works well for a single-molecule system with many degrees of freedom.

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.

Free energy measurements by the generalized fluctuation theorems: Theory and numerical study of a model filament

We measure the free energy of a model filament, which undergoes deformations and structural transitions, as a function of its extension, in silico. We perform Brownian Dynamics (BD) simulations of pulling experiments at various speeds, following a protocol close to experimental ones. The results from the fluctuation theorems are compared with the estimates from Monte Carlo (MC) simulation, where the rugged free energy landscape is produced by the density of states method. The fluctuation theorems (FT) give accurate estimates of the free energy up to moderate pulling speeds. At higher pulling speeds, the work distributions do not efficiently sample the domain of small work and FT slightly overestimates free energy. In order to comprehend the differences, we analyze the work distributions from the BD simulations in the framework of trajectory thermodynamics and propose the generalized fluctuation theorems that take into account the information (relative entropy) evaluated in the expanded phase space. The measured work - free energy relation is consistent with the results obtained from the generalized fluctuation theorems. We discuss operational methods to improve the estimates at high pulling speed.

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.

Adsorption of a Helical Filament Subject to Thermal Fluctuations

We consider semiflexible chains governed by preferred curvature and twist and their flexural and twist moduli. These filaments possess a helical rather than straight three-dimensional (3D) ground state and we call them helical filaments (H-filament). Depending on the moduli, the helical shape may be smeared by thermal fluctuations. Secondary superhelical structures are expected to form on top of the specific local structure of biofilaments, as is documented for vimentin. We study confinement and adsorption of helical filaments utilizing both a combination of numerical simulations and analytical theory. We investigate overall chain shapes, transverse chain fluctuations, loop and tail distributions, and energy distributions along the chain together with the mean square average height of the monomers 〈 z 2 〉 . The number fraction of adsorbed monomers serves as an order parameter for adsorption. Signatures of adsorbed helical polymers are the occurrence of 3D helical loops/tails and spiral or wavy quasi-flat shapes. None of these arise for the Worm-Like-Chain, whose straight ground state can be embedded in a plane.

Two Phase Transitions in the Two-Dimensional Nematic Three-Vector Model with No Quasi-Long-Range Order: Monte Carlo Simulation of the Density of States

The presence of stable topological defects in a two-dimensional (d=2) liquid crystal model allowing molecular reorientations in three dimensions (n=3) was largely believed to induce a defect-mediated Berzenskii-Kosterlitz-Thouless-type transition to a low temperature phase with quasi-long-range order. However, earlier Monte Carlo (MC) simulations could not establish certain essential signatures of the transition, suggesting further investigations. We study this model by computing its equilibrium properties through MC simulations, based on the determination of the density of states of the system. Our results show that, on cooling, the high temperature disordered phase deviates from its initial progression towards the topological transition, crossing over to a new fixed point, condensing into a nematic phase with exponential correlations of its director fluctuations. The thermally induced topological kinetic processes continue, however, limited to the length scales set by the nematic director fluctuations, and lead to a second topological transition at a lower temperature. It is argued that in the (d=2, n=3) system with an attractive biquadratic Hamiltonian, the presence of additional molecular degrees of freedom and local Z_{2} symmetry associated with lattice sites together promote the onset of an additional relevant scaling field at matching length scales in the high temperature region, leading to a crossover.

Phase transitions, order by disorder, and finite entropy in the Ising antiferromagnetic bilayer honeycomb lattice

We present an analytical and numerical study of the Ising model on a bilayer honeycomb lattice including interlayer frustration and coupling with an external magnetic field. First, we discuss the exact T=0 phase diagram, where we find finite entropy phases for different magnetizations. Then, we study the magnetic properties of the system at finite temperature using complementary analytical techniques (Bethe lattice) and two types of Monte Carlo algorithms (Metropolis and Wang-Landau). We characterize the phase transitions and discuss the phase diagrams. The system presents a rich phenomenology: There are first- and second-order transitions, low-temperature phases with extensive degeneracy, and order-by-disorder state selection.

Complex free-energy landscapes in biaxial nematic liquid crystals and the role of repulsive interactions: A Wang-Landau study

General quadratic Hamiltonian models, describing the interaction between liquid-crystal molecules (typically with D_{2h} symmetry), take into account couplings between their uniaxial and biaxial tensors. While the attractive contributions arising from interactions between similar tensors of the participating molecules provide for eventual condensation of the respective orders at suitably low temperatures, the role of cross coupling between unlike tensors is not fully appreciated. Our recent study with an advanced Monte Carlo technique (entropic sampling) showed clearly the increasing relevance of this cross term in determining the phase diagram (contravening in some regions of model parameter space), the predictions of mean-field theory, and standard Monte Carlo simulation results. In this context, we investigated the phase diagrams and the nature of the phases therein on two trajectories in the parameter space: one is a line in the interior region of biaxial stability believed to be representative of the real systems, and the second is the extensively investigated parabolic path resulting from the London dispersion approximation. In both cases, we find the destabilizing effect of increased cross-coupling interactions, which invariably result in the formation of local biaxial organizations inhomogeneously distributed. This manifests as a small, but unmistakable, contribution of biaxial order in the uniaxial phase. The free-energy profiles computed in the present study as a function of the two dominant order parameters indicate complex landscapes. On the one hand, these profiles account for the unusual thermal behavior of the biaxial order parameter under significant destabilizing influence from the cross terms. On the other, they also allude to the possibility that in real systems, these complexities might indeed be inhibiting the formation of a low-temperature biaxial order itself-perhaps reflecting the difficulties in their ready realization in the laboratory.

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.

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.

Extending the parQ transition matrix method to grand canonical ensembles

Phase coexistence properties as well as other thermodynamic features of fluids can be effectively determined from the grand canonical density of states (DOS). We present an extension of the parQ transition matrix method in combination with the efasTM method as a very fast approach for determining the grand canonical DOS from the transition matrix. The efasTM method minimizes the deviation from detailed balance in the transition matrix using a fast Krylov-based equation solver. The method allows a very effective use of state space transition data obtained by different exploration schemes. An application to a Lennard-Jones system produces phase coexistence properties of the same quality as reference data.

Free-energy landscape of mono- and dinucleosomes: Enhanced rotational flexibility of interconnected nucleosomes

The nucleosome represents the basic unit of eukaryotic genome organization, and its conformational fluctuations play a crucial role in various cellular processes. Here we provide insights into the flipping transition of a nucleosome by computing its free-energy landscape as a function of the linking number and nucleosome orientation using the density-of-states Monte Carlo approach. To investigate how the energy landscape is affected by the presence of neighboring nucleosomes in a chromatin fiber, we also compute the free-energy landscape for a dinucleosome array. We find that the mononucleosome is bistable between conformations with negatively and positively crossed linkers while the conformation with open linkers appears as a transition state. The dinucleosome exhibits a markedly different energy landscape in which the conformation with open linkers populates not only the transition state but also the global minimum. This enhanced stability of the open state is attributed to increased rotational flexibility of nucleosomes arising from their mechanical coupling with neighboring nucleosomes. Our results provide a possible mechanism by which chromatin may enhance the accessibility of its DNA and facilitate the propagation and mitigation of DNA torsional stresses.

Reexamination of the mean-field phase diagram of biaxial nematic liquid crystals: Insights from Monte Carlo studies

Investigations of the phase diagram of biaxial liquid-crystal systems through analyses of general Hamiltonian models within the simplifications of mean-field theory (MFT), as well as by computer simulations based on microscopic models, are directed toward an appreciation of the role of the underlying molecular-level interactions to facilitate its spontaneous condensation into a nematic phase with biaxial symmetry. Continuing experimental challenges in realizing such a system unambiguously, despite encouraging predictions from MFT, for example, are requiring more versatile simulational methodologies capable of providing insights into possible hindering barriers within the system, typically gleaned through its free-energy dependences on relevant observables as the system is driven through the transitions. The recent paper from this group [Kamala Latha et al., Phys. Rev. E 89, 050501(R) (2014)], summarizing the outcome of detailed Monte Carlo simulations carried out employing an entropic sampling technique, suggested a qualitative modification of the MFT phase diagram as the Hamiltonian is asymptotically driven toward the so-called partly repulsive regions. It was argued that the degree of (cross) coupling between the uniaxial and biaxial tensor components of neighboring molecules plays a crucial role in facilitating a ready condensation of the biaxial phase, suggesting that this could be a plausible factor in explaining the experimental difficulties. In this paper, we elaborate this point further, providing additional evidence from curious variations of free-energy profiles with respect to the relevant orientational order parameters, at different temperatures bracketing the phase transitions.

First-order phase transition and tricritical scaling behavior of the Blume-Capel model: A Wang-Landau sampling approach

We investigate the tricritical scaling behavior of the two-dimensional spin-1 Blume-Capel model by using the Wang-Landau method of measuring the joint density of states for lattice sizes up to 48×48 sites. We find that the specific heat deep in the first-order area of the phase diagram exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. The first-order transition curve is systematically determined by employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. At the tricritical point, we characterize the tricritical exponents through finite-size-scaling analysis including the phenomenological finite-size scaling with thermodynamic variables. Our estimation of the tricritical eigenvalue exponents, yt=1.804(5), yg=0.80(1), and yh=1.925(3), provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.

Dynamical traps in Wang-Landau sampling of continuous systems: Mechanism and solution

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 the critical slowing-down encountered in conventional molecular dynamics or Monte Carlo simulations. When trapped, the random walker misses the entire or even several stages of Wang-Landau modification factor reduction, leading to inadequate sampling of the configuration space and a rough density of states, even though the modification factor has been reduced to very small values. Trapping is dependent on specific systems, the choice of energy bins, and the Monte Carlo step size, making it highly unpredictable. A general, simple, and effective solution is proposed where the configurations of multiple parallel Wang-Landau trajectories are interswapped to prevent trapping. We also explain why swapping frees the random walker from such traps. The efficacy of the proposed algorithm is demonstrated.

Basis function sampling: a new paradigm for material property computation

Wang-Landau sampling, and the associated class of flat histogram simulation methods have been remarkably helpful for calculations of the free energy in a wide variety of physical systems. Practically, convergence of these calculations to a target free energy surface is hampered by reliance on parameters which are unknown a priori. Here, we derive and implement a method built upon orthogonal functions which is fast, parameter-free, and (importantly) geometrically robust. The method is shown to be highly effective in achieving convergence. An important feature of this method is its ability to attain arbitrary levels of description for the free energy. It is thus ideally suited to in silico measurement of elastic moduli and other material properties related to free energy perturbations. We demonstrate the utility of such applications by applying our method to calculate the Frank elastic constants of the Lebwohl-Lasher model of liquid crystals.

Scalable replica-exchange framework for Wang-Landau sampling

We investigate a generic, parallel replica-exchange framework for Monte Carlo simulations based on the Wang-Landau method. To demonstrate its advantages and general applicability for massively parallel simulations of complex systems, we apply it to lattice spin models, the self-assembly process in amphiphilic solutions, and the adsorption of molecules on surfaces. While of general current interest, the latter phenomena are challenging to study computationally because of multiple structural transitions occurring over a broad temperature range. We show how the parallel framework facilitates simulations of such processes and, without any loss of accuracy or precision, gives a significant speedup and allows for the study of much larger systems and much wider temperature ranges than possible with single-walker methods.

Detection of an intermediate biaxial phase in the phase diagram of biaxial liquid crystals: entropic sampling study

We investigate the phase sequence of biaxial liquid crystals, based on a general quadratic model Hamiltonian over the relevant parameter space, with a Monte Carlo simulation which constructs equilibrium ensembles of microstates, overcoming possible (free) energy barriers (combining entropic and frontier sampling techniques). The resulting phase diagram qualitatively differs from the universal phase diagram predicted earlier from mean-field theory (MFT), as well as the Monte Carlo simulations with the Metropolis algorithm. The direct isotropic-to-biaxial transition predicted by the MFT is replaced in certain regions of the space by the onset of an additional intermediate biaxial phase of very low order, leading to the sequence N(B)-N(B1)-I. This is due to inherent barriers to fluctuations of the components comprising the total energy, and may explain the difficulties in the experimental realization of these phases.

Molecular Dynamics in the Multicanonical Ensemble: Equivalence of Wang-Landau Sampling, Statistical Temperature Molecular Dynamics, and Metadynamics

We show a direct formal relationship between the Wang-Landau iteration [PRL 86, 2050 (2001)], metadynamics [PNAS 99, 12562 (2002)], and statistical temperature molecular dynamics (STMD) [PRL 97, 050601 (2006)] that are the major work-horses for sampling from generalized ensembles. We demonstrate that STMD, itself derived from the Wang-Landau method, can be made indistinguishable from metadynamics. We also show that Gaussian kernels significantly improve the performance of STMD, highlighting the practical benefits of this improved formal understanding.

Wang-Landau sampling with logarithmic windows for continuous models

We present a modified Wang-Landau sampling (MWLS) for continuous statistical models by partitioning the energy space into a set of windows with logarithmically shrinking width. To demonstrate its necessity and advantages, we apply this sampling to several continuous models, including the two-dimensional square XY spin model, triangular J1-J2 spin model, and Lennard-Jones cluster model. Given a finite number of bins for partitioning the energy space, the conventional Wang-Landau sampling may not generate sufficiently accurate density of states (DOS) around the energy boundaries. However, it is demonstrated that much more accurate DOS can be obtained by this MWLS, and thus a precise evaluation of the thermodynamic behaviors of the continuous models at extreme low temperature (kBT<0.1) becomes accessible. The present algorithm also allows efficient computation besides the highly reliable data sampling.

Dynamically optimized Wang-Landau sampling with adaptive trial moves and modification factors

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 move and acceptance rate are adapted in an energy-dependent fashion, the random walker efficiently adapts its sampling according to the local phase space structure. The Wang-Landau modification factor is also made energy dependent in accordance with the step size, enhancing the accumulation of the density of states. Numerical simulations show that our proposed method performs much better than the traditional Wang-Landau sampling.

Generic, hierarchical framework for massively parallel Wang-Landau sampling

We introduce a parallel Wang-Landau method based on the replica-exchange framework for Monte Carlo simulations. To demonstrate its advantages and general applicability for simulations of complex systems, we apply it to different spin models including spin glasses, the Ising model, and the Potts model, lattice protein adsorption, and the self-assembly process in amphiphilic solutions. Without loss of accuracy, the method gives significant speed-up and potentially scales up to petaflop machines.

Density of states-based molecular simulations

One of the central problems in statistical mechanics is that of finding the density of states of a system. Knowledge of the density of states of a system is equivalent to knowledge of its fundamental equation, from which all thermodynamic quantities can be obtained. Over the past several years molecular simulations have made considerable strides in their ability to determine the density of states of complex fluids and materials. In this review we discuss some of the more promising approaches proposed in the recent literature along with their advantages and limitations.

Difference of energy density of states in the Wang-Landau algorithm

Paying attention to the difference of density of states, Δln g(E)≡ln g(E+ΔE)-lng(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 first-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.

Evaporation-condensation transition of the two-dimensional Potts model in the microcanonical ensemble

The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method. An intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite-size scaling of them to indicate the clear tendency of numerical data to converge to the infinite-size limit predicted by phenomenological theory for the isotherm lattice gas model.

Phase transitions of a single polymer chain: A Wang-Landau simulation study

A single flexible homopolymer chain can assume a variety of conformations which can be broadly classified as expanded coil, collapsed globule, and compact crystallite. Here we study transitions between these conformational states for an interaction-site polymer chain comprised of N=128 square-well-sphere monomers with hard-sphere diameter sigma and square-well diameter lambdasigma. Wang-Landau sampling with bond-rebridging Monte Carlo moves is used to compute the density of states for this chain and both canonical and microcanonical analyses are used to identify and characterize phase transitions in this finite size system. The temperature-interaction range (i.e., T-lambda) phase diagram is constructed for lambda<or=1.30. Chains assume an expanded coil conformation at high temperatures and a crystallite structure at low temperatures. For lambda>1.06 these two states are separated by an intervening collapsed globule phase and thus, with decreasing temperature a chain undergoes a continuous coil-globule (collapse) transition followed by a discontinuous globule-crystal (freezing) transition. For well diameters lambda<1.06 the collapse transition is pre-empted by the freezing transition and thus there is a direct first-order coil-crystal phase transition. These results confirm the recent prediction, based on a lattice polymer model, that a collapsed globule state is unstable with respect to a solid phase for flexible polymers with sufficiently short-range monomer-monomer interactions.

Collapse transitions in a flexible homopolymer chain: application of the Wang-Landau algorithm

The thermodynamic behavior of a continuous homopolymer is described using the Wang-Landau algorithm for chain lengths up to N=561. The coil-globule and liquid-solid transitions are analyzed in detail with traditional thermodynamic and structural quantities. The behavior of the coil-globule transition is well within theoretical and computational predictions for all chain lengths, while the behavior of the liquid-solid transition is much more susceptible to finite-size effects. Certain "magic number" lengths (N=13,55,147,309,561) , whose minimal energy states offer unique icosahedral geometries, are discussed along with chains residing between these special cases. The low temperature behavior near the liquid-solid transition is rich in structural transformations for certain chain lengths, showing many similarities to the behavior of classical clusters with similar interaction potentials. General comments are made on this size dependent behavior and how it affects transition behavior in this model.

Temperature and anharmonic effects on the infrared absorption spectrum from a quantum statistical approach: application to naphthalene

A method is developed to calculate the finite-temperature infrared absorption spectrum of polyatomic molecules with energy levels described by a second-order Dunham expansion. The anharmonic couplings are fully incorporated in the calculation of the quantum density of states, achieved using a Wang-Landau Monte Carlo procedure, as well as in the determination of transition energies. Additional multicanonical simulations provide the microcanonical absorption intensity as a function of both the absorption wavelength and the internal energy of the molecule. The finite-temperature spectrum is finally obtained by Laplace transformation of this microcanonical histogram. The present scheme is applied to the infrared spectrum of naphthalene, for which we quantify the shifting, broadening, and third-order effects as a continuous function of temperature. The influence of anharmonicity and couplings is manifested on the nontrivial variations of these features with increasing temperature.

Thermodynamics of a conformational change using a random walk in energy-reaction coordinate space: Application to methane dimer hydrophobic interactions

A random walk sampling algorithm allows the extraction of the density of states distribution in energy-reaction coordinate space. As a result, the temperature dependences of thermodynamic quantities such as relative energy, entropy, and heat capacity can be calculated using first-principles statistical mechanics. The strategies for optimal convergence of the algorithm and control of its accuracy are proposed. We show that the saturation of the error [Q. Yan and J. J. de Pablo, Phys. Rev. Lett. 90, 035701 (2003); E. Belardinelli and V. D. Pereyra, J. Chem. Phys. 127, 184105 (2007)] is due to the use of histogram flatness as a criterion of convergence. An application of the algorithm to methane dimer hydrophobic interactions is presented. We obtained a quantitatively accurate energy-entropy decomposition of the methane dimer cavity potential. The presented results confirm the previous results, and they provide new information regarding the thermodynamics of hydrophobic interactions. We show that the finite-difference approximation, which is widely used in molecular dynamic simulations for the energy-entropy decomposition of a free energy potential, can lead to a significant error.

Improving Wang-Landau sampling with adaptive windows

Wang-Landau sampling (WLS) of large systems requires dividing the energy range into "windows" and joining the results of simulations in each window. The resulting density of states (and associated thermodynamic functions) is shown to suffer from boundary effects in simulations of lattice polymers and the five-state Potts model. Here, we implement WLS using adaptive windows. Instead of defining fixed energy windows (or windows in the energy-magnetization plane for the Potts model), the boundary positions depend on the set of energy values on which the histogram is flat at a given stage of the simulation. Shifting the windows each time the modification factor f is reduced, we eliminate border effects that arise in simulations using fixed windows. Adaptive windows extend significantly the range of system sizes that may be studied reliably using WLS.

Optimal modification factor and convergence of the Wang-Landau algorithm

We propose a strategy to achieve the fastest convergence in the Wang-Landau algorithm with varying modification factors. With this strategy, the convergence of a simulation is at least as good as the conventional Monte Carlo algorithm, i.e., the statistical error vanishes as 1/sqrt t, where t is a normalized time of the simulation. However, we also prove that the error cannot vanish faster than 1/t . Our findings are consistent with the 1/t Wang-Landau algorithm discovered recently, and we argue that one needs external information in the simulation to beat the conventional Monte Carlo algorithm.

Critical endpoint behavior in an asymmetric Ising model: application of Wang-Landau sampling to calculate the density of states

Using the Wang-Landau sampling method with a two-dimensional random walk we determine the density of states for an asymmetric Ising model with two- and three-body interactions on a triangular lattice, in the presence of an external field. With an accurate density of states we were able to map out the phase diagram accurately and perform quantitative finite-size analyses at, and away from, the critical endpoint. We observe a clear divergence of the curvature of the spectator phase boundary and of the magnetization coexistence diameter derivative at the critical endpoint, and the exponents for both divergences agree well with previous theoretical predictions.