A Leap-frog Algorithm for Stochastic Dynamics

TLDR

Stochastic dynamics algorithms allow integration time steps that are limited only by the systematic force, not by the friction coefficient. The authors propose a third‑order leap‑frog algorithm for stochastic dynamics that simplifies the earlier Verlet‑type algorithm while retaining third‑order accuracy. The algorithm is a third‑order leap‑frog scheme that reduces to the standard leap‑frog MD algorithm when the friction coefficient tends to zero and simplifies the earlier Verlet‑type SD algorithm. The leap‑frog SD algorithm incorporates constraints such as bond lengths and angles, is mathematically equivalent to the Verlet‑type SD algorithm, and is slightly more storage‑efficient than the Beeman‑type SD algorithm.

Abstract

Abstract A third-order algorithm for stochastic dynamics (SD) simulations is proposed, identical to the powerful molecular dynamics leap-frog algorithm in the limit of infinitely small friction coefficient γ. It belongs to the class of SD algorithms, in which the integration time step Δt is not limited by the condition Δt ≤ γ−1, but only by the properties of the systematic force. It is shown how constraints, such as bond length or bond angle constraints, can be incorporated in the computational scheme. It is argued that the third-order Verlet-type SD algorithm proposed earlier may be simplified without loosing its third-order accuracy. The leap-frog SD algorithm is proven to be equivalent to the verlet-type SD algorithm. Both these SD algorithms are slightly more economical on computer storage than the Beeman-type SD algorithm.