D. P. Landau

Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States

We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy, and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape.

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

We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. From the density of states at the end of the random walk, we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but we can also estimate the free energy and entropy, quantities that are not directly accessible by conventional Monte Carlo simulations. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. As with the multicanonical Monte Carlo technique, our method overcomes the tunneling barrier between coexisting phases at first-order phase transitions. In this paper, we apply our algorithm to both first- and second-order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to 200 x 200 and Ising models on lattices up to 256 x 256. Our simulational results are compared to both exact solutions and existing numerical data obtained using other methods. Applying this approach to a three-dimensional +/-J spin-glass model, we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method. This simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space.

Square lattice gases with two- and three-body interactions: A model for the adsorption of hydrogen on Pd(100)

Abstract As a model for the adsorption of hydrogen on Pd(100), we consider a square lattice gas with nearest neighbor repulsive and next nearest neighbor attractive interactions. By Monte Carlo methods we obtain predictions for adsorption isotherms, adsorption energies, LEED intensities associated with the c(2 × 2) structure, and phase diagrams. The effect of additional three-body forces and third nearest neighbor forces is also studied, and we show that additional ordered phases may occur such as (2 × 1), (2 × 2), (4 × 2), (4 × 4) and even incommensurate structures. Such phases might be realized in other adsorbate systems. A detailed comparison of our calculations with the experiments of Behm et al. on the H on Pd( uncertainty about the effective hydrogen-hydrogen interactions.

Multicritical phenomena at surfaces

Abstract Phase transitions at the surface of magnetic systems and binary alloys and in adsorbed surface layers are studied in terms of both phenomenological theories and the statistical mechanics of simple models. In both magnetic materials and alloys new types of multicritical phenomena occur if the atomic interactions in the surface layer differ suitably from those in the bulk such that distinct surface (two dimensional) order becomes critical at the bulk (three-dimensional) critical temperature. Appropriate phase diagrams, multicritical exponents, and crossover scaling behavior are discussed. A similar description is shown to be valid for lattice gas models of adsorbed surface layers which are related to two-dimensional metamagnets (the magnetization corresponds to the adatom “coverage”). Monte Carlo calculations over a broad range of temperature and coverage reveal that if next-nearest-neighbor interactions are present tricritical and triple points may appear. The phase diagram may then include several types of simple ordered phases as well as regions of coexisting ordered-ordered and ordered-disordered states.

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

We present a modified Wang-Landau algorithm for models with continuous degrees of freedom. We demonstrate this algorithm with the calculation of the joint density of states of ferromagnet Heisenberg models and a model polymer chain. The joint density of states contains more information than the density of states of a single variable-energy, but is also much more time consuming to calculate. We present strategies to significantly speed up this calculation for large systems over a large range of energy and order parameter.

Monte Carlo Studies of Wetting, Interface Localization and Capillary Condensation

We present a brief review of Monte Carlo simulations of ferromagnetic Ising lattices in a film geometry with surface magnetic fields. The seminal work of Nakanishi and Fisher [Phys. Rev. Lett.49:1565 (1982)] showed how phase transitions in such models are related to wetting in systems with short range forces; and we will show how theoretical concepts about critical and tricritical wetting, interface localization-delocalization, and capillary condensation can be tested in this and similar models. After reviewing the qualitative, phenomenological description of these phenomena on a mean field level, we will summarize predictions of scaling theories. Comments will be made about the models studied and simulation techniques as well as the specific problems that occur in the relevant finite size scaling analysis. The resulting simulational data have prompted considerable new theoretical efforts, but there are still unsolved problems with respect to critical wetting. We will also present results for interface localization-delocalization transitions in both Ising models and lattice polymer mixtures in a thin film geometry and show that theory can account for many, but not all, aspects of the simulations. In systems with asymmetric boundary fields rather complex phase diagrams can result, and these should be relevant for corresponding experiments. The simulational evidence is fully compatible with the scaling predictions of Fisher and Nakanishi [J. Chem. Phys.75:5875 (1981)] on capillary condensation. To conclude we shall summarize the major unanswered theoretical questions in this rich field of inquiry.

Capillary condensation in the lattice gas model: A Monte Carlo study

A three‐dimensional lattice gas model with nearest‐neighbor attractive interaction confined to a slitlike adsorbing capillary of thickness D is studied by computer simulation, varying the chemical potential μ, temperature, T, as well as the strength of the short‐range interaction between the walls of the capillary and the gas. We show that the chemical potential μc(D) at the condensation transition in the capillary is shifted relative to its bulk value μc(∞) according to the Kelvin equation, μc(D)−μc(∞)∝D−1, for large enough D. Deviations are found for small D, however (of the order of 10 lattice spacings), particularly under conditions where for μ=μc(∞) the surfaces of the capillary for D→∞ are wet. For D=16 lattice spacings we also locate a capillary condensation critical point, and study the distortion of the gas–fluid coexistence curve due to the attractive walls. Profiles of density and local energy across the capillary are studied, and data for the temperature dependence of the local density at the ...

Computer simulation studies of critical phenomena

The recent strides which have been made in the use of computer simulations to study phase transitions and critical phenomena are reviewed. We describe how the use of a combination of recently developed histogram techniques and sophisticated Monte Carlo simulation algorithms have allowed the determination of the static critical behavior of the d=3 Ising and Heisenberg models with unprecedented precision. We also discuss the study of dynamic critical behavior in simple spin models by both Monte Carlo and spin dynamics methods. Recent estimates for dynamic critical exponents are reviewed including the first high precision estimate for true dynamics.

Versatile approach to access the low temperature thermodynamics of lattice polymers and proteins.

We show that Wang-Landau sampling, combined with suitable Monte Carlo trial moves, provides a powerful method for both the ground state search and the determination of the density of states for the hydrophobic-polar (HP) protein model and the interacting self-avoiding walk (ISAW) model for homopolymers. We obtain accurate estimates of thermodynamic quantities for HP sequences with >100 monomers and for ISAWs up to >500 monomers. Our procedure possesses an intrinsic simplicity and overcomes the limitations inherent in more tailored approaches making it interesting for a broad range of protein and polymer models.

Monte Carlo study of phase transitions in ferromagnetic bilayers

We have used Monte Carlo computer simulations to study the behavior of an Ising model consisting of two ferromagnetic layers with different interaction constants coupled weakly together. For the range of lattice sizes studied it appears as though the system undergoes a single transition at the transition temperature of an isolated layer with the stronger coupling, but substantial changes in the thermodynamic properties also occur near the transition temperature of an isolated layer with the weaker of the two couplings.