Estimating thermodynamic expectations and free energies in expanded ensemble simulations: Systematic variance reduction through conditioning.

Markov chain Monte Carlo methods are primarily used for sampling from a given probability distribution and estimating multi-dimensional integrals based on the information contained in the generated samples. Whenever it is possible, more accurate estimates are obtained by combining Monte Carlo integration and integration by numerical quadrature along particular coordinates. We show that this variance reduction technique, referred to as conditioning in probability theory, can be advantageously implemented in expanded ensemble simulations.

Using Bayes formula to estimate rates of rare events in transition path sampling simulations.

Transition path sampling is a method for estimating the rates of rare events in molecular systems based on the gradual transformation of a path distribution containing a small fraction of reactive trajectories into a biased distribution in which these rare trajectories have become frequent. Then, a multistate reweighting scheme is implemented to postprocess data collected from the staged simulations. Herein, we show how Bayes formula allows to directly construct a biased sample containing an enhanced fraction of reactive trajectories and to concomitantly estimate the transition rate from this sample.

Path factorization approach to stochastic simulations.

The computational efficiency of stochastic simulation algorithms is notoriously limited by the kinetic trapping of the simulated trajectories within low energy basins. Here we present a new method that overcomes kinetic trapping while still preserving exact statistics of escape paths from the trapping basins. The method is based on path factorization of the evolution operator and requires no prior knowledge of the underlying energy landscape.

Free energy calculations from adaptive molecular dynamics simulations with adiabatic reweighting.

We propose an adiabatic reweighting algorithm for computing the free energy along an external parameter from adaptive molecular dynamics simulations. The adaptive bias is estimated using Bayes identity and information from all the sampled configurations. We apply the algorithm to a structural transition in a cluster and to the migration of a crystalline defect along a reaction coordinate.

Estimating time-correlation functions by sampling and unbiasing dynamically activated events.

Transition path sampling is a rare-event method that estimates state-to-state time-correlation functions in many-body systems from samples of short trajectories. In this framework, it is proposed to bias the importance function using the lowest Jacobian eigenvalue moduli along the dynamical trajectory. A lowest eigenvalue modulus is related to the lowest eigenvalue of the Hessian matrix and is evaluated here using the Lanczos algorithm as in activation-relaxation techniques.

Molecular Monte Carlo Simulations Using Graphics Processing Units: To Waste Recycle or Not?

In the waste recycling Monte Carlo (WRMC) algorithm, (1) multiple trial states may be simultaneously generated and utilized during Monte Carlo moves to improve the statistical accuracy of the simulations, suggesting that such an algorithm may be well posed for implementation in parallel on graphics processing units (GPUs). In this paper, we implement two waste recycling Monte Carlo algorithms in CUDA (Compute Unified Device Architecture) using uniformly distributed random trial states and trial states based on displacement random-walk steps, and we test the methods on a methane-zeolite MFI framework system to evaluate their utility. We discuss the specific implementation details of the waste recycling GPU algorithm and compare the methods to other parallel algorithms optimized for the framework system.

Waste-recycling Monte Carlo with optimal estimates: application to free energy calculations in alloys.

The estimator proposed recently by Delmas and Jourdain for waste-recycling Monte Carlo achieves variance reduction optimally with respect to a control variate that is evaluated directly using the simulation data. Here, the performance of this estimator is assessed numerically for free energy calculations in generic binary alloys and is compared to those of other estimators taken from the literature. A systematic investigation with varying simulation parameters of a simplified system, the anti-ferromagnetic Ising model, is first carried out in the transmutation ensemble using path-sampling.

Simulating structural transitions by direct transition current sampling: the example of LJ38.

Reaction paths and probabilities are inferred, in a usual Monte Carlo or molecular dynamic simulation, directly from the evolution of the positions of the particles. The process becomes time-consuming in many interesting cases in which the transition probabilities are small. A radically different approach consists of setting up a computation scheme where the object whose time evolution is simulated is the transition current itself.

Multiple-replica exchange with information retrieval.

The parallel tempering simulation method was recently extended to allow for possible exchanges between non-adjacent replicas. We introduce a multiple-exchange variant which naturally incorporates the information from all replicas when calculating statistical averages, building on the related virtual-move method of Coluzza and Frenkel (ChemPhysChem 2005, 6, 1779). The method is extensively tested on three model systems, namely, a Lennard-Jones cluster exhibiting a finite size phase transition, the Lennard-Jones fluid, and the 2D ferromagnetic Ising model.

Measurement of nonequilibrium entropy from space-time thermodynamic integration.

The entropy of a system transiently driven out of equilibrium by a time-inhomogeneous stochastic dynamics is first expressed as a transient response function generalizing the nonlinear Kawasaki-Crooks response. This function is then reformulated into three statistical averages defined over ensembles of nonequilibrium trajectories. The first average corresponds to a space-time thermodynamic perturbation relation, while the two following ones correspond to space-time thermodynamic integration relations.