Dynamical origin of uniform sampling in multicanonical ensemble

Generalized simulated tempering for exploring strong phase transitions

An extension of the simulation tempering algorithm is proposed. It is shown to be particularly suited to the exploration of first-order phase transition systems characterized by the backbending or S-loop in the statistical temperature or a microcanonical caloric curve. A guided Markov process in an auxiliary parameter space systematically combines a set of parametrized Tsallis-weight ensemble simulations, which are targeted to transform unstable or metastable energy states of canonical ensembles into stable ones and smoothly join ordered and disordered phases across phase transition regions via a succession of unimodal energy distributions. The inverse mapping between the sampling weight and the effective temperature enables an optimal selection of relevant Tsallis-weight parameters. A semianalytic expression for the biasing weight in parameter space is adaptively updated "on the fly" during the simulation to achieve rapid convergence. Accelerated tunneling transitions with a comprehensive sampling for phase-coexistent states are explicitly demonstrated in systems subject to strong hysteresis including Potts and Ising spin models and a 147 atom Lennard-Jones cluster.

Generalized replica exchange method

We present a powerful replica exchange method, particularly suited to first-order phase transitions associated with the backbending in the statistical temperature, by merging an optimally designed generalized ensemble sampling with replica exchanges. The key ingredients of our method are parametrized effective sampling weights, smoothly joining ordered and disordered phases with a succession of unimodal energy distributions by transforming unstable or metastable energy states of canonical ensembles into stable ones. The inverse mapping between the sampling weight and the effective temperature provides a systematic way to design the effective sampling weights and determine a dynamic range of relevant parameters. Illustrative simulations on Potts spins with varying system size and simulation conditions demonstrate a comprehensive sampling for phase-coexistent states with a dramatic acceleration of tunneling transitions. A significant improvement over the power-law slowing down of mean tunneling times with increasing system size is obtained, and the underlying mechanism for accelerated tunneling is discussed.

Optimal replica exchange method combined with Tsallis weight sampling

A unified framework integrating the generalized ensemble sampling associated with the Tsallis weight [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] and the replica exchange method (REM) has been proposed to accelerate the convergence of the conventional temperature REM (t-REM). Using the effective temperature formulation of the Tsallis weight sampling, it is shown that the average acceptance probability for configurational swaps between neighboring replicas in the combination of Tsallis weight sampling and REM (Tsallis-REM) is directly proportional to an overlap integral of the energy distributions of neighboring replicas as in the t-REM. Based on this observation, we suggest a robust method to select optimal Tsallis parameters in the conventional parametrization scheme and present new parametrization schemes for the Tsallis-REM, which significantly improves the acceptance of configurational swaps by systematically modulating energy overlaps between neighboring replicas. The distinguished feature of our method is that all relevant parameters in the Tsallis-REM are automatically determined from the equilibrium phase simulation using the t-REM. The overall performance of our method is explicitly demonstrated for various simulation conditions for the Lennard-Jones 31 atom clusters, exhibiting a double-funneled energy landscape.

Structure optimization and folding mechanisms of off-lattice protein models using statistical temperature molecular dynamics simulation: Statistical temperature annealing

The recently proposed statistical temperature molecular dynamics (STMD) algorithm [Kim, Phys. Rev. Lett. 97, 050601 (2006)] is used as the core of an optimization algorithm, statistical temperature annealing (STA), for finding low-lying energy minima of complex potential energy landscapes. Since STMD realizes a random walk in energy, the idea is simply to initiate repeated minimizations from configurations in the low-energy segments of STMD trajectories. STA is tested in structural optimization of various off-lattice AB and extended AB protein models in two and three dimensions with different chain lengths. New putative ground states were found for the two- and three-dimensional AB 55-mer, and for the three-dimensional extended AB 21-mer and 55-mer. The distinct folding features of the models are analyzed in terms of the statistical temperature and other representations of the structure of the potential energy landscape. It is shown that the characteristic behavior of the statistical temperature undergoes a qualitative change with the inclusion of a torsional potential in the extended AB model, as the more rigid backbone makes the potential energy landscape more funnel-like.

Statistical temperature molecular dynamics: Application to coarse-grained β-barrel-forming protein models

Recently the authors proposed a novel sampling algorithm, "statistical temperature molecular dynamics" (STMD) [J. Kim et al., Phys. Rev. Lett. 97, 050601 (2006)], which combines ingredients of multicanonical molecular dynamics and Wang-Landau sampling. Exploiting the relation between the statistical temperature and the density of states, STMD generates a flat energy distribution and efficient sampling with a dynamic update of the statistical temperature, transforming an initial constant estimate to the true statistical temperature T(U), with U being the potential energy. Here, the performance of STMD is examined in the Lennard-Jones fluid with diverse simulation conditions, and in the coarse-grained, off-lattice BLN 46-mer and 69-mer protein models, exhibiting rugged potential energy landscapes with a high degree of frustration. STMD simulations combined with inherent structure (IS) analysis allow an accurate determination of protein thermodynamics down to very low temperatures, overcoming quasiergodicity, and illuminate the transitions occurring in folding in terms of the energy landscape. It is found that a thermodynamic signature of folding is significantly suppressed by accurate sampling, due to an incoherent contribution from low-lying non-native IS in multifunneled landscapes. It is also shown that preferred accessibility to such IS during the collapse transition is intimately related to misfolding or poor foldability.

Statistical-temperature Monte Carlo and molecular dynamics algorithms

A simulation method is presented that achieves a flat energy distribution by updating the statistical temperature instead of the density of states in Wang-Landau sampling. A novel molecular dynamics algorithm (STMD) applicable to complex systems and a Monte Carlo algorithm are developed from this point of view. Accelerated convergence for large energy bins, essential for large systems, is demonstrated in tests on the Ising model, the Lennard-Jones fluid, and bead models of proteins. STMD shows a superior ability to find local minima in proteins and new global minima are found for the 55 bead AB model in two and three dimensions. Calculations of the occupation probabilities of individual protein inherent structures provide new insights into folding and misfolding.

Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

In complex systems with many degrees of freedom such as spin glass and biomolecular systems, conventional simulations in canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble performs a random walk in potential energy space and overcomes this difficulty. From only one simulation run, one can obtain canonical ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST), and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present five new generalized-ensemble algorithms which are extensions of the above methods.

Multicanonical molecular dynamics algorithm employing an adaptive force-biased iteration scheme

We present one effective multicanonical molecular dynamics (MCMD) algorithm accelerating the convergence of rough energy landscapes simulations via an adaptive force-biased iteration scheme. Our method utilizes several short MCMD simulations with dynamically updated weights and combines them to estimate the density of states via multiple histogram technique. The key step of our algorithm is the adaptive refinement for the derivative of multicanonical weight, which allows the system to enlarge the sampling energy range maintaining the statistical accuracy. The performance of our method has been validated for atomic Lennard-Jones clusters.

Generalized simulated tempering realized on expanded ensembles of non-Boltzmann weights

A generalized version of the simulated tempering operated in the expanded ensembles of non-Boltzmann weights has been proposed to mitigate a quasiergodicity problem occurring in simulations of rough energy landscapes. In contrast to conventional simulated tempering employing the Boltzmann weight, our method utilizes a parametrized, generalized distribution as a workhorse for stochastic exchanges of configurations and subensembles transitions, which allows a considerable enhancement for the rate of convergence of Monte Carlo and molecular dynamics simulations using delocalized weights. A feature of our method is that the exploration of the parameter space encouraging subensembles transitions is greatly accelerated using the dynamic update scheme for the weight via the average guide specific to the energy distribution. The performance and characteristic feature of our method have been validated in the liquid-solid transition of Lennard-Jones clusters and the conformational sampling of alanine dipeptide by taking two types of Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] expanded ensembles associated with different parametrization schemes.

Dynamical origin of enhanced conformational searches of Tsallis statistics sampling

The characteristic sampling dynamics of importance samplings driven by the Tsallis weight [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] has been analyzed in terms of recently developed Langevin stochastic model by considering the effects of the density of states and the potential smoothing of the Tsallis transformation. Our study reveals that the fixed points, which are determined by the crossing points of the statistical temperature and the Tsallis effective temperature, play a critical role in overall dynamics of the Tsallis statistics sampling. The dynamical origin of enhanced conformational searches of the Tsallis weight has been investigated by unveiling the intimate relationship between the sampling dynamics and the stability change of corresponding fixed points. Based on this stochastic analysis, we propose one effective method to realize a broad energy distribution in the Tsallis statistics sampling by determining optimal Tsallis parameters systematically based on preliminary canonical samplings. The effectiveness of our method has been validated in the folding simulation of Met-Enkephalin and liquid-solid transition simulation of Lennard-Jones cluster systems.

Stochastic formulation of sampling dynamics in generalized ensemble methods

The fundamental relations of the sampling process in the multicanonical ensemble and simulated tempering have been studied in the expanded ensembles formalism. The simulated tempering is identified as the multicanonical sampling with the generalized weight determined by a Laplace transform of the temperature weight. The characteristic dynamics of both sampling methods has been verified in the stochastic formulation of the sampling process. Our study gives a necessary and sufficient condition for the weights to realize a uniform sampling in the energy and temperature spaces. Based on the stochastic model, robust force biased iteration schemes have been proposed to allow automatic determinations of uniform sampling weights.

Determination of multicanonical weight based on a stochastic model of sampling dynamics

Based on the stochastic interpretation of the sampling process modeled by a Langevin equation, we present an effective iteration scheme to determine the weight in multicanonical molecular dynamics. Our method enables an automatic determination of the weight producing a uniform energy sampling via an iterative cancellation of the deterministic force in a Langevin equation. The deterministic force has been calculated from the energy trajectory by identifying the moments of the transition probability of a Fokker-Planck equation associated with a Langevin equation. The intimate relationship between the sampling process and the stochastic dynamics has been verified by applying the iteration scheme to a helix-coil transition of the 8-polyalanine system in a gas phase.