Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram

Equilibrium thermodynamics from basin-sampling

We present a "basin-sampling" approach for calculation of the potential energy density of states for classical statistical models. It combines a Wang-Landau-type uniform sampling of local minima and a novel approach for approximating the relative contributions from local minima in terms of the volumes of basins of attraction. We have employed basin-sampling to study phase changes in atomic clusters modeled by the Lennard-Jones potential and for ionic clusters. The approach proves to be efficient for systems involving broken ergodicity and has allowed us to calculate converged heat capacity curves for systems that could previously only be treated using the harmonic superposition approximation. Benchmarks are also provided by comparison with parallel tempering and Wang-Landau simulations, where these proved feasible.

Thermodynamic properties of Ag2O--B2O3 glasses by a modified scale-transformed energy space sampling Monte Carlo method

The density of states of Ag(2)O--B(2)O(3) glasses has been calculated by using a modified scale-transformed energy space sampling algorithm. This algorithm combines the scale-transformed energy space sampling algorithm and the Wang-Landau method. It is shown how the two algorithms can be combined to improve the efficiency of calculation. The thermodynamic properties, in particular the specific heat C(V), of the above-mentioned glass system is studied. At temperatures above 80 K, the value of specific heat C(v) is close to 22 J/mol/K. At low temperatures, the deviations of C(v) from a T(3) behavior are discernible, that is, C(v)/T(3) exhibits a hump at T = 7 K, which is in good agreement with the reported experimental behavior.

Determination of surface tension in binary mixtures using transition-matrix Monte Carlo

We present a methodology based on grand-canonical transition-matrix Monte Carlo and finite-size scaling analysis to calculate surface tensions in binary mixtures. In particular, mixture transition-matrix Monte Carlo is first used to calculate apparent, system-size-dependent free-energy barriers separating coexisting fluid phases. Finite-size scaling is then used to extrapolate these values to the infinitely large system limit to determine the true thermodynamic surface tension. A key distinction of the methodology is that it yields the entire isothermal surface-tension curve for a binary mixture in a relatively small number of simulations. We demonstrate the utility of the method by calculating surface-tension curves for three binary Lennard-Jones mixtures. While we have only examined the surface tension of simple fluids in this work, the method is general and can be extended to molecular fluids as well as to determine interfacial tensions of liquid-liquid interfaces.

Efficient combination of Wang–Landau and transition matrix Monte Carlo methods for protein simulations

An efficient combination of the Wang-Landau and transition matrix Monte Carlo methods for protein and peptide simulations is described. At the initial stage of simulation the algorithm behaves like the Wang-Landau algorithm, allowing to sample the entire interval of energies, and at the later stages, it behaves like transition matrix Monte Carlo method and has significantly lower statistical errors. This combination allows to achieve fast convergence to the correct values of density of states. We propose that the violation of TTT identities may serve as a qualitative criterion to check the convergence of density of states. The simulation process can be parallelized by cutting the entire interval of simulation into subintervals. The violation of ergodicity in this case is discussed. We test the algorithm on a set of peptides of different lengths and observe good statistical convergent properties for the density of states. We believe that the method is of general nature and can be used for simulations of other systems with either discrete or continuous energy spectrum.

Lack of self-averaging of the specific heat in the three-dimensional random-field Ising model

We apply the recently developed critical minimum-energy subspace scheme for the investigation of the random-field Ising model. We point out that this method is well suited for the study of this model. The density of states is obtained via the Wang-Landau and broad histogram methods in a unified implementation by employing the N-fold version of the Wang-Landau scheme. The random fields are obtained from a bimodal distribution (hi = +/-2), and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from L=4 to L=32. Observing the finite-size scaling behavior of the maxima of the specific heats we examine the question of saturation of the specific heat. The lack of self-averaging of this quantity is fully illustrated, and it is shown that this property may be related to the question mentioned above.

Entropic sampling via Wang-Landau random walks in dominant energy subspaces

Dominant energy subspaces of statistical systems are defined with the help of restrictive conditions on various characteristics of the energy distribution, such as the probability density and the fourth order Binder's cumulant. Our analysis generalizes the ideas of the critical minimum energy subspace (CRMES) technique, applied previously to study the specific heat's finite-size scaling. Here, we illustrate alternatives that are useful for the analysis of further finite-size anomalies and the behavior of the corresponding dominant subspaces is presented for the two-dimensional (2D) Baxter-Wu and the 2D and 3D Ising models. In order to show that a CRMES technique is adequate for the study of magnetic anomalies, we study and test simple methods which provide the means for an accurate determination of the energy-order-parameter (E,M) histograms via Wang-Landau random walks. The 2D Ising model is used as a test case and it is shown that high-level Wang-Landau sampling schemes yield excellent estimates for all magnetic properties. Our estimates compare very well with those of the traditional Metropolis method. The relevant dominant energy subspaces and dominant magnetization subspaces scale as expected with exponents alpha/nu and gamma/nu, respectively. Using the Metropolis method we examine the time evolution of the corresponding dominant magnetization subspaces and we uncover the reasons behind the inadequacy of the Metropolis method to produce a reliable estimation scheme for the tail regime of the order-parameter distribution.

Optimized ensemble Monte Carlo simulations of dense Lennard-Jones fluids

We apply the recently developed adaptive ensemble optimization technique to simulate dense Lennard-Jones fluids and a particle-solvent model by broad-histogram Monte Carlo techniques. Equilibration of the simulated fluid is improved by sampling an optimized histogram in radial coordinates that shifts statistical weight towards the entropic barriers between the shells of the liquid. Interstitial states in the vicinity of these barriers are identified with unprecedented accuracy by sharp signatures in the quickly converging histogram and measurements of the local diffusivity. The radial distribution function and potential of mean force are calculated to high precision.

Computational characterization of the sequence landscape in simple protein alphabets

We characterize the "sequence landscapes" in several simple, heteropolymer models of proteins by examining their mutation properties. Using an efficient flat-histogram Monte Carlo search method, our approach involves determining the distribution in energy of all sequences of a given length when threaded through a common backbone. These calculations are performed for a number of Protein Data Bank structures using two variants of the 20-letter contact potential developed by Miyazawa and Jernigan [Miyazawa S, Jernigan WL. Macromolecules 1985;18:534], and the 2-monomer HP model of Lau and Dill [Lau KF, Dill KA. Macromolecules 1989;22:3986]. Our results indicate significant differences among the energy functions in terms of the "smoothness" of their landscapes. In particular, one of the Miyazawa-Jernigan contact potentials reveals unusual cooperative behavior among its species' interactions, resulting in what is essentially a set of phase transitions in sequence space. Our calculations suggest that model-specific features can have a profound effect on protein design algorithms, and our methods offer a number of ways by which sequence landscapes can be quantified.

Fast flat-histogram method for generalized spin models

We present a Monte Carlo method that efficiently computes the density of states for spin models having any number of interaction per spin. By combining a random walk in the energy space with collective updates controlled by the microcanonical temperature, our method yields dynamic exponents close to their ideal random-walk values, reduced equilibrium times, and very low statistical error in the density of states. The method can host any density of states estimation scheme, including the Wang-Landau algorithm and the transition matrix method. Our approach proves remarkably powerful in the numerical study of models governed by long-range interactions, where it is shown to reduce the algorithm complexity to that of a short-range model with the same number of spins. We apply the method to the -state Potts chains with power-law decaying interactions in their first-order regime; we find that conventional local-update algorithms are outperformed already for sizes above a few hundred spins. By considering chains containing up to spins, which we simulated in fairly reasonable time, we obtain estimates of transition temperatures correct to five-figure accuracy. Finally, we propose several efficient schemes aimed at estimating the microcanonical temperature.

Direct evaluation of multicomponent phase equilibria using flat-histogram methods

We present a method for directly locating density-driven phase transitions in multicomponent systems. Phase coexistence conditions are determined through manipulation of a total density probability distribution evaluated over a density range that includes both coexisting phases. Saturation quantities are determined through appropriate averaging of density-dependent mean values of a given property of interest. We discuss how to implement the method in both the grand-canonical and isothermal-isobaric semigrand ensembles. Calculations can be conducted using any of the recently introduced flat-histogram techniques. Here, we combine the general algorithm with a transition-matrix approach to produce an efficient self-adaptive technique for determining multicomponent phase equilibrium properties. To assess the performance of the new method, we generate phase diagrams for a number of binary and ternary Lennard-Jones mixtures.

Overcoming the slowing down of flat-histogram Monte Carlo simulations: cluster updates and optimized broad-histogram ensembles

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased Markovian random walk in energy space. For the d = 1, 2, 3 Ising model, the mean first-passage time tau scales with the number of spins N = L(d) as tau proportional N2L(z). The exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z > 0 for finite d can be overcome by two complementary approaches--cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that tau proportional N2 up to logarithmic corrections for the d = 1, 2 Ising model.

Ground states and thermal states of the random field Ising model

The random field Ising model is studied numerically at both zero and positive temperature. Ground states are mapped out in a region of random field and external field strength. Thermal states and thermodynamic properties are obtained for all temperatures using the Wang-Landau algorithm. The specific heat and susceptibility typically display sharp peaks in the critical region for large systems and strong disorder. These sharp peaks result from large domains flipping. For a given realization of disorder, ground states and thermal states near the critical line are found to be strongly correlated--a concrete manifestation of the zero temperature fixed point scenario.

Wang-Landau Monte Carlo simulation of isotropic-nematic transition in liquid crystals

Wang and Landau proposed recently, a simple and flexible non-Boltzmann Monte Carlo method for estimating the density of states, from which the macroscopic properties of a closed system can be calculated. They demonstrated their algorithm by considering systems with discrete energy spectrum. We find that the Wang-Landau algorithm does not perform well when the system has continuous energy spectrum. We propose in this paper modifications to the algorithm and demonstrate their performance on a lattice model of liquid crystalline system (with Lebwohl-Lasher interaction having continuously varying energy), exhibiting transition from high temperature isotropic to low temperature nematic phase.

Structure and stability of a model three-helix-bundle protein on tailored surfaces

The interaction of protein molecules with surfaces is important in numerous applications. Theoretical work on protein adsorption has been limited. In particular, it is difficult to obtain quantitative predictions about the structure and stability of proteins on surfaces. In this study, density-of-states-based simulations were performed on a Gō-like model of a three-helix-bundle fragment from protein A (PDB ID: 1bdd). Both mechanical and thermal stability were investigated on neutral and attractive surfaces and compared to that in the absence of a surface. It was found that attaching the peptide to any type of surface decreases its melting temperature by as much as 9 K, depending upon orientation. Calorimetric cooperativity, as measured by van't Hoff to calorimetric enthalpy ratios, similarly decreased. It was also found that the mechanical strength of the peptide attached to surfaces is degraded to varying extents, depending upon the surface type and protein orientation. A comparison of mechanical and thermal stability showed that the two are not synonymous, but occur through different pathways, and that system configurations that are more thermally stable are not always so mechanically.

A unified methodological framework for the simulation of nonisothermal ensembles

A general framework is developed for the simulation of nonisothermal statistical-mechanical ensembles. This framework is intended to synthesize the formulation of advanced Monte Carlo simulation methods such as multihistogram reweighting, replica-exchange methods, and expanded ensemble techniques so that they can be applied to different nonisothermal ensembles. Using Lennard-Jones systems as test cases, novel implementations of these methods are demonstrated with different ensembles including the microcanonical, isobaric-isoenthalpic, and isobaric-semigrand ensembles. In particular, it is shown that the use of multiensemble methods allows the efficient simulation of microcanonical density of states, entropies, vapor-liquid and solid-liquid equilibrium for pure component systems, and fluid-phase coexistence for binary mixtures. In these applications, comparisons are also presented that highlight the advantages of the proposed multiensemble implementations over alternative methods used before.

Wang-Landau sampling with self-adaptive range

We report a self-adapting version of the Wang-Landau algorithm that is ideally suited for application to systems with a complicated structure of the density of states. Applications include determination of two-dimensional densities of states and high-precision numerical integration of sharply peaked functions on multidimensional integration domains.

Density of states of classical spin systems with continuous degrees of freedom

In the last years different studies have revealed the usefulness of a microcanonical analysis of finite systems when dealing with phase transitions. In this approach the quantities of interest are exclusively expressed as derivatives of the entropy S=ln Omega where Omega is the density of states. Obviously, the density of states has to be known with very high accuracy for this kind of analysis. Important progress has been achieved recently in the computation of the density of states of classical systems, as new types of algorithms have been developed. Here we extend one of these methods, originally formulated for systems with discrete degrees of freedom, to systems with continuous degrees of freedom. As an application we compute the density of states of the three-dimensional XY model and demonstrate that critical quantities can directly be determined from the density of states of finite systems in cases where the degrees of freedom take continuous values.

Entropy and density of states from isoenergetic nonequilibrium processes

Two identities in statistical mechanics involving entropy differences (or ratios of densities of states) at constant energy are derived. The first provides a nontrivial extension of the Jarzynski equality to the microcanonical ensemble [C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)], which can be seen as a "fast-switching" version of the adiabatic switching method for computing entropies [M. Watanabe and W. P. Reinhardt, Phys. Rev. Lett. 65, 3301 (1990)]. The second is a thermodynamic integration formula analogous to a well-known expression for free energies, and follows after taking the quasistatic limit of the first. Both identities can be conveniently used in conjunction with a scaling relation (herein derived) that allows one to extrapolate measurements taken at a single energy to a wide range of energy values. Practical aspects of these identities in the context of numerical simulations are discussed.

Multicanonical schemes for mapping out free-energy landscapes of single-component and multicomponent systems

Multicanonical (MUCA) sampling is a powerful approach for simulating large domains of thermodynamic macrostate space that relies on mapping out either the density of states or a free energy of the system as a function of a suitable "order parameter." The purpose of this study is to extend and apply to more complex systems the method introduced in a previous paper [M. K. Fenwick and F. A. Escobedo, J. Chem. Phys. 120, 3066 (2004)] that uses Bennett's acceptance ratio method for estimating MUCA free energies. Four types of MUCA schemes are considered according to what order parameter is adopted and how the macrostate space is traversed: a la grand canonical ensemble, a la semigrand canonical ensemble, a la semigrand isothermal-isobaric ensemble, and a la isothermal-isobaric ensemble. Two types of systems are studied, the first is a two-component Lennard-Jones mixture that exhibits a vapor-liquid transition, and the second is a hard-cuboid containing system that exhibits an isotropic-liquid crystalline transition. These systems are simulated with different MUCA schemes and the resulting free-energy profiles are used to determine phase-coexistence conditions. For the Lennard-Jones systems, it is also demonstrated that different types of MUCA simulations can be conveniently performed over different macrostate regions and the results can be subsequently pieced together into a continuous weighting function.

Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid

Using extensive Monte Carlo simulations and a simple approximation in density functional theory, we study the phase behavior of a fluid of nematogenic molecules with centers of mass constrained to lie in a plane but with axes free to rotate in any direction, both with and without an external disorienting field perpendicular to the plane. We find that simulation predicts the existence of an order-disorder phase transition belonging to the Berezinskii-Kosterlitz-Thouless type, along with a low temperature gas-liquid transition. In contrast to the simulation results, density functional theory predicts a first-order orientational phase transition coupled continuously with a first-order gas-liquid transition. The approximate theoretical approach qualitatively reproduces the field dependence of the order-disorder and gas-liquid transitions but is far from quantitative.

First-order interface localization-delocalization transition in thin Ising films using Wang-Landau sampling

Using extensive Monte Carlo simulations, we study the interface localization-delocalization transition of a thin Ising film with antisymmetric competing walls for a set of parameters where the transition is strongly first order. This is achieved by estimating the density of states (DOS) of the model by means of Wang-Landau sampling (WLS) in the space of energy, using both single-spin-flip as well as N-fold way updates. From the DOS we calculate canonical averages related to the configurational energy, like the internal energy and the specific heat, as well as the free energy and the entropy. By sampling micro-canonical averages during simulations we also compute thermodynamic quantities related to magnetization like the reduced fourth-order cumulant of the order parameter. We estimate the triple temperatures of infinitely large systems for three different film thicknesses via finite size scaling of the positions of the maxima of the specific heat, the minima of the cumulant, and the equal weight criterion for the energy probability distribution. The wetting temperature of the semi-infinite system is computed with help of the Young equation. In the limit of large film thicknesses the triple temperatures are seen to converge toward the wetting temperature of the corresponding semi-infinite Ising model in accordance with standard capillary wave theory. We discuss the slowing down of WLS in energy space as observed for the larger film thicknesses and lateral linear dimensions. In the case of WLS in the space of total magnetization we find evidence that the slowing down is reduced and can be attributed to persisting free energy barriers due to shape transitions.

Density guided importance sampling: application to a reduced model of protein folding

MOTIVATION

Monte Carlo methods are the most effective means of exploring the energy landscapes of protein folding. The rugged topography of folding energy landscapes causes sampling inefficiencies however, particularly at low, physiological temperatures.

RESULTS

A hybrid Monte Carlo method, termed density guided importance sampling (DGIS), is presented that overcomes these sampling inefficiencies. The method is shown to be highly accurate and efficient in determining Boltzmann weighted structural metrics of a discrete off-lattice protein model. In comparison to the Metropolis Monte Carlo method, and the hybrid Monte Carlo methods, jump-walking, smart-walking and replica-exchange, the DGIS method is shown to be more efficient, requiring no parameter optimization. The method guides the simulation towards under-sampled regions of the energy spectrum and recognizes when equilibrium has been reached, avoiding arbitrary and excessively long simulation times.

AVAILABILITY

Fortran code available from authors upon request.

CONTACT

m.j.parker@leeds.ac.uk.

Stretching a heteropolymer

We study the elastic properties of single heteropolymers. By means of exact enumeration of conformations, Monte Carlo (MC) simulation, and variational principles, we calculate equilibrium force-extension curves of heterocopolymers for specific arrangements of the monomer types along the sequence. At a given extension z, the time averaged measured force is the weighted sum of restoring forces for various configurations. Using variational principles, we calculate force-extension (f-z) curves of heteropolymers with fixed extensions z. These results are compared with f-z curves obtained from MC simulations and exact enumeration of all conformations. Typical random sequences manifest several piecewise unfoldings of blocks of various size, which are overlapping due to thermal fluctuations. The shape of the elastic response of a heteropolymer reflects the disorder in the primary block structure and the binding energies of these blocks.

Comparison of sequence-based and structure-based energy functions for the reversible folding of a peptide

We used computer simulations to compare the reversible folding of a 20-residue peptide, as described by sequence-based and structure-based energy functions. Sequence-based energy functions are transferable and can be used to describe the behavior of different proteins, since interactions are defined between atomic species. Conversely, structure-based energy functions are not transferable, since the interactions are defined relative to the native conformation, which is assumed to correspond to the global minimum of the energy. Our results indicate that the sequence-based and the structure-based descriptions are in qualitative agreement in characterizing the two-state behavior of the peptide that we studied. We also found, however, that several equilibrium properties, including the free-energy landscape, can be significantly different in the various models. These results suggest that the fact that a model describes the native state of a polypeptide chain does not necessarily imply that the thermodynamic and kinetic properties will also be reproduced correctly.

Determination of fluid-phase behavior using transition-matrix Monte Carlo: Binary Lennard-Jones mixtures

We present a novel computational methodology for determining fluid-phase equilibria in binary mixtures. The method is based on a combination of highly efficient transition-matrix Monte Carlo and histogram reweighting. In particular, a directed grand-canonical transition-matrix Monte Carlo scheme is used to calculate the particle-number probability distribution, after which histogram reweighting is used as a postprocessing procedure to determine the conditions of phase equilibria. To validate the methodology, we have applied it to a number of model binary Lennard-Jones systems known to exhibit nontrivial fluid-phase behavior. Although we have focused on monatomic fluids in this work, the method presented here is general and can be easily extended to more complex molecular fluids. Finally, an important feature of this method is the capability to predict the entire fluid-phase diagram of a binary mixture at fixed temperature in a single simulation.

Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

In complex systems with many degrees of freedom such as spin glass and biomolecular systems, conventional simulations in canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble performs a random walk in potential energy space and overcomes this difficulty. From only one simulation run, one can obtain canonical ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST), and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present five new generalized-ensemble algorithms which are extensions of the above methods.

Estimation of critical behavior from the density of states in classical statistical models

We present a simple and efficient approximation scheme which greatly facilitates the extension of Wang-Landau sampling (or similar techniques) in large systems for the estimation of critical behavior. The method, presented in an algorithmic approach, is based on a very simple idea, familiar in statistical mechanics from the notion of thermodynamic equivalence of ensembles and the central limit theorem. It is illustrated that we can predict with high accuracy the critical part of the energy space and by using this restricted part we can extend our simulations to larger systems and improve the accuracy of critical parameters. It is proposed that the extensions of the finite-size critical part of the energy space, determining the specific heat, satisfy a scaling law involving the thermal critical exponent. The method is applied successfully for the estimation of the scaling behavior of specific heat of both square and simple cubic Ising lattices. The proposed scaling law is verified by estimating the thermal critical exponent from the finite-size behavior of the critical part of the energy space. The density of states of the zero-field Ising model on these lattices is obtained via a multirange Wang-Landau sampling.

Density of states determined from Monte Carlo simulations

We describe a method for calculating the density of states by combining several canonical Monte Carlo runs. We discuss how critical properties reveal themselves in g (epsilon) and demonstrate this by applying the method to several different phase transitions. We also demonstrate how this can used to calculate the conformal charge, where the dominating numerical method has traditionally been the transfer matrix.

Estimation of the density of states by multicanonical molecular dynamics simulation

We formulate a procedure for calculating the density of states (DOS) from a multicanonical molecular dynamics (MMD) simulation. DOS cannot be obtained directly from the result of MMD simulation, because the Gaussian thermostat that is used in MMD simulation restricts the system to a spherical surface in momentum space. We perform MMD simulation for liquid Ar with Lennard-Jones potentials and evaluate DOS. Some physical quantities are estimated as a function of temperature from that DOS. The internal energy, entropy, and Helmholtz free energy are in good agreement with experiment. The quantity related to the fluctuation--the specific heat at constant volume--does not agree with experiment, which is ascribed to insufficient accuracy of DOS.

Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations

We present an adaptive algorithm which optimizes the statistical-mechanical ensemble in a generalized broad-histogram Monte Carlo simulation to maximize the system's rate of round trips in total energy. The scaling of the mean round-trip time from the ground state to the maximum entropy state for this local-update method is found to be O ( [N ln N](2) ) for both the ferromagnetic and the fully frustrated two-dimensional Ising model with N spins. Our algorithm thereby substantially outperforms flat-histogram methods such as the Wang-Landau algorithm.

Free-energy landscape and the critical velocity of superfluid films

We briefly review some important properties of superfluid flow, especially the problem of critical velocity. We then present new numerical simulation results for a mesoscopic model of superfluids shedding light on the free-energy landscape, the critical velocity and the formation of vortices, which destroy the superflow when the velocity is high.

Phase behavior of n-alkanes in supercritical solution: A Monte Carlo study

We present a coarse-grained model for n-alkanes in a supercritical solution, which is exemplified by a mixture of hexadecane and CO2. For pure hexadecane, the Monte Carlo simulations of the coarse-grained model reproduce the experimental phase diagram and the interfacial tension with good accuracy. For the mixture, the phase behavior sensitively depends on the compatibility of the polymer with the solvent. We present a global phase diagram with critical lines, which is in semiquantitative agreement with experiments. In this context we developed two computational schemes: The first adopts Wang-Landau sampling to the off-lattice grand canonical ensemble, the second combines umbrella sampling with an extrapolation scheme to determine the weight function. Additionally, we use Wertheim's theory (TPT1) to obtain the equation of state for our coarse-grained model of supercritical mixtures and discuss the behavior for longer alkanes.

New results for phase transitions from catastrophe theory

Catastrophe theory predicts that in certain limits universal relations should exist between barrier heights, curvatures and the positions of local maxima and minima on a potential or free energy surface. In the present work we investigate these relations for both first- and second-order phase transitions, revealing that the ideal ratios often hold quite well over a wide range of conditions. This elementary catastrophe theory is illustrated for the melting transition of an atomic cluster, the isotropic-to-nematic transition in a liquid crystal, and the ferromagnetic-to-paramagnetic phase transition in the two-dimensional Ising model.

Density-of-states Monte Carlo simulation of a binary glass

A newly proposed Monte Carlo formalism has been used to simulate a glass-forming liquid above and below the glass transition temperature. The heat capacity exhibits a sharp peak at a temperature lower than that reported from extensive molecular dynamics simulations. Its height is larger than that reported earlier. At temperatures below mode coupling, the average inherent-structure energy of the configurations generated in this work is significantly lower than that reported in the literature. The entropy of the supercooled liquid is calculated directly from our simulations and that of a disordered solid is calculated by a normal-mode analysis. We find that at low temperatures these two entropy curves become essentially parallel. They do not intersect each other, raising questions about the existence of a Kauzmann temperature in this glass-forming mixture.

Generalized 1/k-ensemble algorithm

We generalize the 1/k-ensemble algorithm so that it can be used for both discrete and continuous systems, and show that the generalization is correct numerically and mathematically. We also compare the efficiencies of the generalized 1/k-ensemble algorithm and the generalized Wang-Landau algorithm through a neural network example. The numerical results favor to the generalized 1/k-ensemble algorithm.

Adaptive integration method for Monte Carlo simulations

We present an adaptive sampling method for computing free energies, radial distribution functions, and potentials of mean force. The method is characterized by simplicity and accuracy, with the added advantage that the data are obtained in terms of quasicontinuous functions. The method is illustrated and tested with simulations on a high density fluid, including a stringent consistency test involving an unusual thermodynamic cycle that highlights its advantages.

Reconstructing the density of states by history-dependent metadynamics

We present a novel method for the calculation of the energy density of states D(E) for systems described by classical statistical mechanics. The method builds on an extension of a recently proposed strategy that allows the free-energy profile of a canonical system to be recovered within a preassigned accuracy [Proc. Natl. Acad. Sci. U.S.A. 99, 12562 (2002)]]. The method allows a good control over the error on the recovered system entropy. This fact is exploited to obtain D(E) more efficiently by combining measurements at different temperatures. The accuracy and efficiency of the method are tested for the two-dimensional Ising model (up to size 50 x 50) by comparison with both exact results and previous studies. This method is a general one and should be applicable to more realistic model systems.

Mixtures of lattice polymers with structured monomers

The influence of monomer structure on the thermodynamic properties of lattice model polymer blends is investigated through Monte Carlo computations. The model of lattice polymers with monomer structure has been used extensively in the context of the lattice cluster theory (LCT), a thermodynamic theory for polymer mixtures in the liquid state. The Monte Carlo computations provide the first unequivocal test of the accuracy of the LCT predictions for binary mixtures of polymers with structured monomers. Four types of monomer structures are analyzed, corresponding to to the monomers of polyethylene, polypropylene, polyethylethylene, and polyisobutylene (PIB). Most computations use chains with M=12 and 24 beads and the total volume fraction of the beads is phi=0.6. Both structurally symmetric and asymmetric blends are investigated. For the symmetric case, the predictions of the LCT for the energies of mixing and the liquid-liquid coexistence curves are in qualitative agreement with the Monte Carlo computations, except for the PIB/PIB symmetric blend. For structurally asymmetric blends, the LCT does not capture contributions to the energy of mixing arising solely from structural differences between the components. Computational estimates of the nonideal entropy of mixing indicate that the LCT also underestimates the entropic cost of mixing chains with different structures, thus explaining some discrepancies between the theoretical and the Monte Carlo liquid--liquid coexistence curves.