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

Performance limitations of flat-histogram methods

We determine the optimal scaling of local-update flat-histogram methods with system size by using a perfect flat-histogram scheme based upon the exact density of states of 2D Ising models. The typical tunneling time needed to sample the entire bandwidth does not scale with the number of spins N as the minimal N2 of an unbiased random walk in energy space. While the scaling is power law for the ferromagnetic and fully frustrated Ising model, for the +/-J nearest-neighbor spin glass the distribution of tunneling times is governed by a fat-tailed Fréchet extremal value distribution that obeys exponential scaling. Furthermore, the shape parameters of these distributions indicate that statistical sample means become ill defined already for moderate system sizes within these complex energy landscapes.

A scalable parallel Monte Carlo method for free energy simulations of molecular systems

We present a method of parallelizing flat histogram Monte Carlo simulations, which give the free energy of a molecular system as an output. In the serial version, a constant probability distribution, as a function of any system parameter, is calculated by updating an external potential that is added to the system Hamiltonian. This external potential is related to the free energy. In the parallel implementation, the simulation is distributed on to different processors. With regular intervals the modifying potential is summed over all processors and distributed back to every processor, thus spreading the information of which parts of parameter space have been explored. This implementation is shown to decrease the execution time linearly with added number of processors.

Configurational temperature density of states simulations of proteins

A novel method has been implemented to compute the density of states of proteins. A united atom representation and the CHARMM (Brooks et al., 1983) force-field parameters have been adopted for all the simulations. In this approach, an intrinsic temperature is computed based on configurational information about the protein. A random walk is performed in potential energy space and the configurational temperature is collected as a function of potential energy of the system. The density of states is then calculated by integrating the reciprocal of temperature. Unlike previously available methods, this approach does not involve calculations based on histograms of stochastic visits to distinct energy states. It is found that the proposed method is more efficient than earlier, related schemes for simulation of protein folding. Furthermore, it directly provides thermodynamic information, including free energies. The usefulness of the method is discussed by presenting results of simulations of the 16-residue beta-hairpin taken from the C-terminal fragment (41-56) of protein G.

Determination of the interaction potential from the pair distribution function: an inverse Monte Carlo technique

In this work we present an efficient procedure to evaluate effective pair potentials compatible with "experimental" distribution functions using a Monte Carlo simulation scheme. Using computer simulation results for the pair distribution functions, we have applied the method to a Lennard-Jones fluid and to a model of liquid aluminum. In both cases the procedure was able to recover with high accuracy the actual interaction potential of the systems. Moreover, the procedure can easily incorporate additional information, for instance, thermodynamic properties, in order to improve the reliability of the results.

Avoiding boundary effects in Wang-Landau sampling

A simple modification of the "Wang-Landau sampling" algorithm removes the systematic error that occurs at the boundary of the range of energy over which the random walk takes place in the original algorithm.

Continuous demixing at liquid-vapor coexistence in a symmetrical binary fluid mixture

We report a Monte Carlo finite-size scaling study of the demixing transition of a symmetrical Lennard-Jones binary fluid mixture. For equal concentration of species, and for a choice of the unlike-to-like interaction ratio delta=0.7, this transition is found to be continuous at liquid-vapor coexistence. The associated critical end point exhibits an Ising-like universality. These findings confirm those of earlier smaller scale simulation studies of the same model, but contradict the findings of recent integral equation and hierarchical reference theory investigations.

Flat histogram methods for quantum systems: algorithms to overcome tunneling problems and calculate the free energy

We present a generalization of the classical Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)]] to quantum systems. The algorithm proceeds by stochastically evaluating the coefficients of a high temperature series expansion or a finite temperature perturbation expansion to arbitrary order. Similar to their classical counterpart, the algorithms are efficient at thermal and quantum phase transitions, greatly reducing the tunneling problem at first order phase transitions, and allow the direct calculation of the free energy and entropy.

Fast calculation of the density of states of a fluid by Monte Carlo simulations

Two related methods are proposed to calculate the density of states of a fluid from Monte Carlo simulations. In contrast to previous approaches, which require that histograms be accumulated in a stochastic manner, the methods proposed here rely on evaluation of the instantaneous temperature. In the first method, the temperature is calculated from the gradient of the forces. In the second, it is estimated from the kinetic contribution to the total energy. The validity and usefulness of the new approaches are demonstrated by presenting results from simulations of a Lennard-Jones fluid. It is shown that the new methods are considerably faster than previously available techniques.

Dynamical origin of uniform sampling in multicanonical ensemble

The stochastic model describing the sampling process in multicanonical ensemble has been derived by considering the sampling process as an overdamped Brownian motion on the free energy surface. The essential dynamics of the multicanonical sampling has been characterized by a Langevin equation in a piecewise multivalleyed free energy landscape, modulated by a temperature-dependent curvature. Based on the stochastic model we showed that the multicanonical weight can be determined by interpolating maximum probability energy points of the canonical samplings at different temperatures.

Generalization of the Wang-Landau method for off-lattice simulations

We present a rigorous derivation for off-lattice implementations of the so-called "random-walk" algorithm recently introduced by Wang and Landau [Phys. Rev. Lett. 86, 2050 (2001)]. Originally developed for discrete systems, the algorithm samples configurations according to their inverse density of states using Monte Carlo moves; the estimate for the density of states is refined at each simulation step and is ultimately used to calculate thermodynamic properties. We present an implementation for atomic systems based on a rigorous separation of kinetic and configurational contributions to the density of states. By constructing a "uniform" ensemble for configurational degrees of freedom-in which all potential energies, volumes, and numbers of particles are equally probable-we establish a framework for the correct implementation of simulation acceptance criteria and calculation of thermodynamic averages in the continuum case. To demonstrate the generality of our approach, we perform sample calculations for the Lennard-Jones fluid using two implementation variants and in both cases find good agreement with established literature values for the vapor-liquid coexistence locus.

Broad histogram relation for the bond number and its applications

We discuss Monte Carlo methods based on the cluster (graph) representation for spin models. We derive a rigorous broad histogram relation (BHR) for the bond number; a counterpart for the energy was derived by Oliveira previously. A Monte Carlo dynamics based on the number of potential moves for the bond number is proposed. We show the efficiency of the BHR for the bond number in calculating the density of states and other physical quantities.