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

Flat histogram Monte Carlo sampling for mechanical variables and conjugate thermodynamic fields with example applications to strongly correlated electronic systems

We present a general and unifying framework for deriving Monte Carlo acceptance rules which facilitate flat histogram sampling. The framework yields uniform sampling rules for thermodynamic states given by the mechanically extensive variables appearing in the Hamiltonian. Likewise, Monte Carlo schemes which uniformly sample the thermodynamic fields that are conjugate to the mechanical variables can be derived within this framework. We apply these different, yet equivalent sampling schemes to the extended Hubbard model in the atomic limit with explicit electron spin. Results for the full density-of-states, the charge-order parameter distribution, and phase diagrams for different ratios of the on-site Hubbard repulsion and the intersite interaction are presented. A tricritical point at half-filling of the lattice is located using finite-size scaling techniques.

Optimized expanded ensembles for simulations involving molecular insertions and deletions. II. Open systems

In the Grand Canonical, osmotic, and Gibbs ensembles, chemical potential equilibrium is attained via transfers of molecules between the system and either a reservoir or another subsystem. In this work, the expanded ensemble (EXE) methods described in part I [F. A. Escobedo and F. J. Martinez-Veracoechea, J. Chem. Phys. 127, 174103 (2007)] of this series are extended to these ensembles to overcome the difficulties associated with implementing such whole-molecule transfers. In EXE, such moves occur via a target molecule that undergoes transitions through a number of intermediate coupling states. To minimize the tunneling time between the fully coupled and fully decoupled states, the intermediate states could be either: (i) sampled with an optimal frequency distribution (the sampling problem) or (ii) selected with an optimal spacing distribution (staging problem). The sampling issue is addressed by determining the biasing weights that would allow generating an optimal ensemble; discretized versions of this algorithm (well suited for small number of coupling stages) are also presented. The staging problem is addressed by selecting the intermediate stages in such a way that a flat histogram is the optimized ensemble. The validity of the advocated methods is demonstrated by their application to two model problems, the solvation of large hard spheres into a fluid of small and large spheres, and the vapor-liquid equilibrium of a chain system.

Recent developments in methodologies for calculating the entropy and free energy of biological systems by computer simulation

The Helmholtz free energy, F, plays an important role in proteins because of their rugged potential energy surface, which is 'decorated' with a tremendous number of local wells (denoted microstates, m). F governs protein folding, whereas differences DeltaF(mn) determine the relative populations of microstates that are visited by a flexible cyclic peptide or a flexible protein segment (e.g. a surface loop). Recently developed methodologies for calculating DeltaF(mn) (and entropy differences, DeltaS(mn)) mainly use thermodynamic integration and calculation of the absolute F; interesting new approaches in these categories are the adaptive integration method and the hypothetical scanning molecular dynamics method, respectively.

Simulation of the density of states in isothermal and adiabatic ensembles

This paper provides a unified treatment of the fundamental methods used to obtain the density of states via molecular simulations with isothermal ensembles (IEs) and adiabatic ensembles (AEs). Our analysis and results show that provides a natural bridge to go back and forth between IE and AE simulation data. They also underline the difference between the density of states of potential energy macrostates and that of total energy macrostates Omega, even though both provide access to the thermodynamic properties of the system. Visited-states approaches and transition matrix methods are described and applied to the Lennard-Jones fluid to target omega and Omega as functions of energy and volume macrostates. It is shown that one can obtain omega via a generalized acceptance-ratio formula that is applicable regardless of the conditions at which the ensemble is simulated. In this way, one can obtain while performing conventional IE or AE simulations, and do it at no extra cost and with a higher accuracy than is achievable with histogram methods.

On the use of transition matrix methods with extended ensembles

Different extended ensemble schemes for non-Boltzmann sampling (NBS) of a selected reaction coordinate lambda were formulated so that they employ (i) "variable" sampling window schemes (that include the "successive umbrella sampling" method) to comprehensibly explore the lambda domain and (ii) transition matrix methods to iteratively obtain the underlying free-energy eta landscape (or "importance" weights) associated with lambda. The connection between "acceptance ratio" and transition matrix methods was first established to form the basis of the approach for estimating eta(lambda). The validity and performance of the different NBS schemes were then assessed using as lambda coordinate the configurational energy of the Lennard-Jones fluid. For the cases studied, it was found that the convergence rate in the estimation of eta is little affected by the use of data from high-order transitions, while it is noticeably improved by the use of a broader window of sampling in the variable window methods. Finally, it is shown how an "elastic" window of sampling can be used to effectively enact (nonuniform) preferential sampling over the lambda domain, and how to stitch the weights from separate one-dimensional NBS runs to produce a eta surface over a two-dimensional domain.

A general framework for non-Boltzmann Monte Carlo sampling

Non-Boltzmann sampling (NBS) methods have been extensively employed in recent years, mainly due to their ability to enhance ergodicity in simulations of complex systems. In addition, they make possible reliable computation of equilibrium properties (ensemble averages, free-energy differences, and potentials of mean force) over continuous ranges of thermodynamic conditions. In this work, we put forward a general and systematic framework for NBS methods that allows a single set of equations and procedures to be applied to diverse systems. Moreover, we show how to exploit simulation data most effectively by obtaining continuous profiles of any mechanical properties, including structural quantities not directly related to the ensemble parameters. Finally, we demonstrate the usefulness of the developed formulation by applying it to spin systems, Lennard-Jones fluids, and a model protein molecule (both in isolation and in the proximity of a flat wall).

Phase Behavior of Colloidal Hard Tetragonal Parallelepipeds (Cuboids): A Monte Carlo Simulation Study

The impact of particle geometry on the phase behavior of hard colloidal tetragonal parallelepipeds (TPs) was studied by using Monte Carlo simulations in continuum space. TPs or "cuboids" of aspect ratios varying from 0.25 to 8 were simulated by approximating their shapes with multisite objects, i.e., via rigid clusters of hard spheres. Using equation of state curves, order parameters, radial distribution functions, particle distribution functions along three directions, and visual analysis of configurations, an approximate phase diagram for the TPs was mapped out as a function of aspect ratio (r) and volume fraction. For r > 3 and intermediate concentrations, the behavior of the TPs was similar to that of spherocylinders, exhibiting similar liquid crystalline mesophases (e.g., nematic and smectic phases). For r = 1, a cubatic phase occurs with orientational order along the three axes but with little translational order. For 1 < r < 4, the TPs exhibit a cubatic-like mesophase with a high degree of order along three axes where the major axes of the particles are not all aligned in the same direction. For r < 1, the TPs exhibit a smectic-like phase where the particles have rotational freedom in each layer but form stacks with tetratic order. The equation of state for perfect hard cubes (r = 1) was also simulated and found to be consistent with that of the rounded-edge r = 1 TPs, except for its lack of discontinuity at the cubatic-solid transition.

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.