Complete set of stochastic Verlet-type thermostats for correct Langevin simulations

Abstract

We present the complete set of stochastic Verlet-type algorithms that can provide correct statistical measures for both configurational and kinetic sampling in discretetime Langevin systems. The approach is a brute-force general representation of the Verlet-algorithm with free parameter coefficients that are determined by requiring correct Boltzmann sampling for linear systems, regardless of time step. The result is a set of statistically correct methods given by one free functional parameter, which can be interpreted as the one-time-step velocity attenuation factor. We define the statistical characteristics of both true on-site v n and true half-step u n+ 1 2 velocities, and use these definitions for each statistically correct Størmer-Verlet method to find a unique associated half-step velocity expression, which yields correct kinetic Maxwell-Boltzmann statistics for linear systems. It is shown that no other similar, statistically correct on-site velocity exists. We further discuss the use and features of finite-difference velocity definitions that are neither true on-site, nor true halfstep. The set of methods is written in convenient and conventional stochastic Verlet forms that lend themselves to direct implementation for, e.g., Molecular Dynamics applications. We highlight a few specific examples, and validate the algorithms through comprehensive Langevin simulations of both simple nonlinear oscillators and complex Molecular Dynamics.To Appear in Molecular Physics: https://doi.

Extended Lagrangian Born–Oppenheimer molecular dynamics: from density functional theory to charge relaxation models

“…The equations of motion can be integrated step by step using, for example, the frequently used velocity Verlet algorithm or other integrations schemes [8,[89][90][91][92][93][94][95][96] which generate the molecular trajectories.…”

“…The the equilibrated charges and the Born-Oppenheimer potential energy surface are then given from the solution of a quasi-diagonal, linear system of equations as in Eq. (95).…”

DPD Modelling of the Self- and Co-Assembly of Polymers and Polyelectrolytes in Aqueous Media: Impact on Polymer Science

“…However, their effect cannot be neglected and is accounted for by the dissipative and random forces linked with the fluctuation-dissipation theorem (FDT), which reflects the interplay of the ensembleaverage friction (which slows down the motion of DPD beads) versus the overall thermal agitation (which accelerates the beads). The balance of dissipative and random forces then serves as thermostat in DPD systems [116]. Analogously to other MD methods, the DPD computational machinery consists of a numerical solution of the Newtonian equations of motion for a system of DPD beads (i, j, k, ...) that mutually interact by conservative forces F c (r ij ) derived from the soft repulsive distance-dependent pair interaction potential U c (r ij ).…”

Coarse-Grained Simulation of Mechanical Properties of Single Microtubules With Micrometer Length

“…All the UCG MD simulations mentioned above were carried out with using the software LAMMPS ( Plimpton, 1995 ) in the NVT ensembles at 300K. The Langevin dynamics simulation ( Grønbech-Jensen, 2019 ) was performed under the nonperiodic condition with the damping factor being 10.0 ps. The time step of simulation is chosen to be 10.0 fs for the simulation of Young’s modulus.…”