A simple and effective Verlet-type algorithm for simulating Langevin dynamics

Abstract

We present a revision to the well known Störmer–Verlet algorithm for simulating second order differential equations. The revision addresses the inclusion of linear friction with associated stochastic noise, and we analytically demonstrate that the new algorithm correctly reproduces diffusive behaviour of a particle in a flat potential. For a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency and time step for both underdamped and overdamped behaviour within the usual stability limit of the Verlet algorithm. Given the structure and simplicity of the method, we conclude that this approach can trivially be adapted for contemporary applications, including molecular dynamics with extensions such as molecular constraints.

Keywords:

• molecular dynamics
• simulating Langevin equations
• Störmer–Verlet algorithms
• computational statistical mechanics

Acknowledgements

NGJ acknowledges a very useful discussion with Zhaojun Bai. This project was supported in part by the US Department of Energy Project No. DE-NE0000536 000.