Alternative ensemble averages in molecular dynamics simulation of hard spheres

Gauge Invariance of Equilibrium Statistical Mechanics

We identify a recently proposed shifting operation on classical phase space as a gauge transformation for statistical mechanical microstates. The infinitesimal generators of the continuous gauge group form a noncommutative Lie algebra, which induces exact sum rules when thermally averaged. Gauge invariance with respect to finite shifting is demonstrated via Monte Carlo simulation in the transformed phase space which generates identical equilibrium averages. Our results point toward a deeper basis of statistical mechanics than previously known, and they offer avenues for systematic construction of exact identities and of sampling algorithms.

Reduced-variance orientational distribution functions from torque sampling

We introduce a method to sample the orientational distribution function in computer simulations. The method is based on the exact torque balance equation for classical many-body systems of interacting anisotropic particles in equilibrium. Instead of the traditional counting of events, we reconstruct the orientational distribution function via an orientational integral of the torque acting on the particles. We test the torque sampling method in two- and three-dimensions, using both Langevin dynamics and overdamped Brownian dynamics, and with two interparticle interaction potentials. In all cases the torque sampling method produces profiles of the orientational distribution function with better accuracy than those obtained with the traditional counting method. The accuracy of the torque sampling method is independent of the bin size, and hence it is possible to resolve the orientational distribution function with arbitrarily small angular resolutions.

Reduced variance analysis of molecular dynamics simulations by linear combination of estimators

Building upon recent developments of force-based estimators with a reduced variance for the computation of densities, radial distribution functions, or local transport properties from molecular simulations, we show that the variance can be further reduced by considering optimal linear combinations of such estimators. This control variates approach, well known in statistics and already used in other branches of computational physics, has been comparatively much less exploited in molecular simulations. We illustrate this idea on the radial distribution function and the one-dimensional density of a bulk and confined Lennard-Jones fluid, where the optimal combination of estimators is determined for each distance or position, respectively. In addition to reducing the variance everywhere at virtually no additional cost, this approach cures an artifact of the initial force-based estimators, namely, small but non-zero values of the quantities in regions where they should vanish. Beyond the examples considered here, the present work highlights, more generally, the underexplored potential of control variates to estimate observables from molecular simulations.

Implementation of harmonically mapped averaging in LAMMPS, and effect of potential truncation on anharmonic properties

Implementation of the harmonically mapped averaging (HMA) framework in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is presented for on-the-fly computations of the energy, pressure, and heat capacity of crystalline systems during canonical molecular dynamics simulations. HMA has a low central processing unit and storage requirements and is straightforward to use. As a case study, the properties of the Lennard-Jones and embedded-atom model (parameterized for nickel) crystals are computed. The results demonstrate the higher efficiency of the new class compared to the inbuilt LAMMPS classes for calculating these properties. However, HMA loses its effectiveness in systems where diffusion occurs in the crystal, and an example is presented to allow this behavior to be recognized. In addition to its improved precision, HMA is less affected by small errors introduced by having a larger time step in molecular dynamics simulations. We also present an analysis of the effect of potential truncation on anharmonic properties, and show that artifacts of truncation on the HMA averages can be eliminated simply by shifting the potential energy to zero at the truncation radius. Full properties can be obtained by adding easily computed values for the lattice and harmonic properties using the untruncated potential.