Stochastic position Verlet for Langevin dynamics

Other Langevin dynamics splitting schemes in common use utilize a different additive decomposition of the SDE vector field. For example, the Stochastic Position Verlet (SPV) method of [265] relies on the splitting strategy 

Similar behavior is exhibited using the splitting strategy in (7.5), when interposing the kick term B between two O steps, such as in the schemes [AOBOA or BOAOB. By avoiding OBO updates, we can derive schemes that are consistent even at infinite friction (the BAOAB and ABOBA schemes are examples of such methods). 

The Briinger-Brooks-Karplus (BBK) Langevin integrator [55] is one of the most popular and widely-implemented schemes. 

Often R is chosen such that R = R/,+i, so the random vector would be reused at the start of the following timestep, however there are many variants of this scheme that incorporate the redraw of random numbers differently. In other variants of BBK we take R = R , reusing the random number within the step. We may also consider independent R and R , however when using independent noise processes we must modify the algorithm as it will not sample canonically (see Exercise 1). 

As such, we return now to the harmonic oscillator, which as well as being the simplest molecular model is also one of the most relevant for molecular dynamics applications, as many issues of stability and timestep in molecular dynamics simulations arise due to harmonic potentials used to model covalent bonds (such as in crystalline solids and biomolecules). The canonical distributions of position and momentum are also of a simple form (Gaussians), making such oscillators particularly amenable to analysis. 

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