Langevin and Brownian simulations play a prominent role in computational research, and state of the art integration algorithms provide trajectories with different stability ranges and accuracy in reproducing statistical averages. The practical usability of integrators is an important aspect to allow choosing large time steps while ensuring numerical stability and overall computational efficiency. In this work, different use cases and practical features are selected in order to perform a cumulative comparison of integrators with a focus on evaluating the derived velocity and position autocorrelation functions, a comparison that is often disregarded in the literature. A standard industrial open-source software methodology is suggested to compare systematically the different algorithms.
Path reweighting is a principally exact method to estimate dynamic properties from biased simulations - provided that the path probability ratio matches the stochastic integrator used in the simulation. Previously reported path probability ratios match the Euler-Maruyama scheme for overdamped Langevin dynamics. Since MD simulations use Langevin dynamics rather than overdamped Langevin dynamics, this severely impedes the application of path reweighting methods. Here, we derive the path probability ratio $M_L$ for Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. Thi…
We present MIST, the Molecular Integration Simulation Toolkit, a lightweight and efficient software library written in C++ which provides an abstract in- terface to common molecular dynamics codes, enabling rapid and portable development of new integration schemes for molecular dynamics. The initial release provides plug-in interfaces to NAMD-Lite, GROMACS and Amber, and includes several standard integration schemes, a constraint solver, tem- perature control using Langevin Dynamics, and two tempering schemes. We describe the architecture and functionality of the library and the C and For- tra…
Two of the most widely used Langevin integrators for molecular dynamics simulations are the GROMACS Stochastic Dynamics (GSD) integrator and the splitting method BAOAB. In this letter, we show that the GROMACS Stochastic Dynamics integrator is equal to the less frequently used splitting method BAOA. It immediately follows that GSD and BAOAB sample the same configurations and have the same high configurational accuracy. Our numerical results indicate that GSD/BAOA has higher kinetic accuracy than BAOAB.
Energy-correction method is proposed as an addition to mainstream integrators for equations of motion of systems of classical spins. This solves the problem of non-conservation of energy in long computations and makes mainstream integrators competitive with symplectic integrators for spin systems that for different-site interactions conserve the energy explicitly. The proposed method is promising for spin systems with single-site interactions for which symplectic integrators do not conserve energy and thus have no edge against mainstream integrators. From the energy balance in the spin system …
We reformulate the algorithm of Grønbech-Jensen and Farago (GJF) for Langevin dynamics simulations at constant temperature. The GJF algorithm has become increasingly popular in molecular dynamics simulations because it provides robust (i.e., insensitive to variations in the time step) and accurate configurational sampling of the phase space with larger time steps than other Langevin thermostats. In the original derivation [Mol. Phys. {\bf 111}, 983 (2013)], the algorithm was formulated as a velocity-Verlet type integrator with an in-site velocity variable. Here, we reformulate it as a leap fro…
The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present …
There has been considerable interest in designing Markov chain Monte Carlo algorithms by exploiting numerical methods for Langevin dynamics, which includes Hamiltonian dynamics as a deterministic case. A prominent approach is Hamiltonian Monte Carlo (HMC), where a leapfrog discretization of Hamiltonian dynamics is employed. We investigate a recently proposed class of irreversible sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS), which uses an augmented target density similarly as in HMC, but involves a flexible proposal scheme and a carefully formulated acceptance-re…
The problem of mass diffusion in layered systems has relevance to applications in different scientific disciplines, e.g., chemistry, material science, soil science, and biomedical engineering. The mathematical challenge in these type of model systems is to match the solutions of the time-dependent diffusion equation in each layer, such that the boundary conditions at the interfaces between them are satisfied. As the number of layers increases, the solutions may become increasingly complicated. Here, we describe an alternative computational approach to multi-layer diffusion problems, which is b…
Efficient Boltzmann-sampling using first-principles methods is challenging for extended systems due to the steep scaling of electronic structure methods with the system size. Stochastic approaches provide a gentler system-size dependency at the cost of introducing "noisy" forces, which serve to limit the efficiency of the sampling. In the first-order Langevin dynamics (FOLD), efficient sampling is achievable by combining a well-chosen preconditioning matrix S with a time-step-bias-mitigating propagator (Mazzola et al., Phys. Rev. Lett., 118, 015703 (2017)). However, when forces are noisy, S is…
In a recent work of Wu, Wang, Sun and Liu, a second-order explicit symplectic integrator was proposed for the integrable Kerr spacetime geometry. It is still suited for simulating the nonintegrable dynamics of charged particles moving around the Kerr black hole embedded in an external magnetic field. Its successful construction is due to the contribution of a time transformation. The algorithm exhibits a good long-term numerical performance in stable Hamiltonian errors and computational efficiency. As its application, the dynamics of order and chaos of charged particles is surveyed. In some ci…
