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

Stochastic approximation Monte Carlo with a dynamic update factor

We present a Monte Carlo algorithm based on the stochastic approximation Monte Carlo (SAMC) algorithm for directly calculating the density of states. The proposed method is stochastic approximation with a dynamic update factor (SAD), which dynamically adjusts the update factor γ_{t} during the course of the simulation. We test this method on a square-well fluid and a 31-atom Lennard-Jones cluster and compare the convergence behavior of several related Monte Carlo methods. We find that both the SAD and 1/t-Wang-Landau (1/t-WL) methods rapidly converge to the correct density of states without the need for the user to specify an arbitrary tunable parameter t_{0} as in the case of SAMC. SAD requires as input the temperature range of interest, in contrast with 1/t-WL, which requires that the user identify the interesting range of energies. The convergence of the 1/t-WL method is very sensitive to the energy range chosen for the low-temperature heat capacity of the Lennard-Jones cluster. Thus, SAD is more powerful in the common case in which the range of energies is not known in advance.

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.

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.

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.

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.