Rational Construction of Stochastic Numerical Methods for Molecular Sampling

In this article, we focus on the sampling of the configurational Gibbs-Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular $N$-body system modelled at constant temperature. We show how a formal series expansion of the invariant measure of a Langevin dynamics numerical method can be obtained in a straightforward way using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property (4th order accuracy where only 2nd order would be expected) of one method in the high friction limit; this method, moreover, can be reduced to a simple modification of the Euler-Maruyama method for Brownian dynamics involving a non-Markovian (coloured noise) random process. In the Brownian dynamics case, 2nd order accuracy of the invariant density is achieved. All methods considered are efficient for molecular applications (requiring one force evaluation per timestep) and of a simple form. In fully resolved (long run) molecular dynamics simulations, for our favoured method, we observe up to two orders of magnitude improvement in configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared to common alternative methods

Least-biased correction of extended dynamical systems using observational data

We consider dynamical systems evolving near an equilibrium statistical state where the interest is in modelling long term behavior that is consistent with thermodynamic constraints. We adjust the distribution using an entropy-optimizing formulation that can be computed on-the- fly, making possible partial corrections using incomplete information, for example measured data or data computed from a different model (or the same model at a different scale). We employ a thermostatting technique to sample the target distribution with the aim of capturing relavant statistical features while introducing mild dynamical perturbation (thermostats). The method is tested for a point vortex fluid model on the sphere, and we demonstrate both convergence of equilibrium quantities and the ability of the formulation to balance stationary and transient- regime errors.Comment: 27 page

Generating Generalized Distributions from Dynamical Simulation

We present a general molecular-dynamics simulation scheme, based on the Nose' thermostat, for sampling according to arbitrary phase space distributions. We formulate numerical methods based on both Nose'-Hoover and Nose'-Poincare' thermostats for two specific classes of distributions; namely, those that are functions of the system Hamiltonian and those for which position and momentum are statistically independent. As an example, we propose a generalized variable temperature distribution that designed to accelerate sampling in molecular systems.Comment: 10 pages, 3 figure

A molecular-dynamics algorithm for mixed hard-core/continuous potentials

We present a new molecular-dynamics algorithm for integrating the equations of motion for a system of particles interacting with mixed continuous/impulsive forces. This method, which we call Impulsive Verlet, is constructed using operator splitting techniques similar to those that have been used successfully to generate a variety molecular-dynamics integrators. In numerical experiments, the Impulsive Verlet method is shown to be superior to previous methods with respect to stability and energy conservation in long simulations.Comment: 18 pages, 6 postscript figures, uses rotate.st