Generic, hierarchical framework for massively parallel Wang-Landau sampling

Monte Carlo algorithm for free energy calculation

We propose a Monte Carlo algorithm for the free energy calculation based on configuration space sampling. An upward or downward temperature scan can be used to produce F(T). We implement this algorithm for the Ising model on a square lattice and triangular lattice. Comparison with the exact free energy shows an excellent agreement. We analyze the properties of this algorithm and compare it with the Wang-Landau algorithm, which samples in energy space. This method is applicable to general classical statistical models. The possibility of extending it to quantum systems is discussed.

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.

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.

Examining the phase transition behavior of amphiphilic lipids in solution using statistical temperature molecular dynamics and replica-exchange Wang-Landau methods

Two different techniques - replica-exchange Wang-Landau (REWL) and statistical temperature molecular dynamics (STMD) - were applied to systematically study the phase transition behavior of self-assembling lipids as a function of temperature using an off-lattice lipid model. Both methods allow the direct calculation of the density of states with improved efficiency compared to the original Wang-Landau method. A 3-segment model of amphiphilic lipids solvated in water has been studied with varied particle interaction energies (ε) and lipid concentrations. The phase behavior of the lipid molecules with respect to bilayer formation has been characterized through the calculation of the heat capacity as a function of temperature, in addition to various order parameters and general visual inspection. The simulations conducted by both methods can go to very low temperatures with the whole system exhibiting well-ordered structures. With optimized parameters, several bilayer phases are observed within the temperature range studied, including gel phase bilayers with frozen water, mixed water (i.e., frozen and liquid water), and liquid water, and a more fluid bilayer with liquid water. The results obtained from both methods, STMD and REWL, are consistently in excellent agreement with each other, thereby validating both the methods and the results.