Langevin Dynamics

Conducting hybrid molecular dynamics – Monte Carlo (MDMC) schemes is possible by addition of the Langevin (pseudo) move in the moves section. Example:

moves:
    - langevin_dynamics:
        nsteps: 25
        integrator: {time_step: 0.005, friction: 5}
    - ...

langevin	Description
nsteps	Number of time iterations for each Langevin dynamics event
integrator	Object with the following two keywords:
time_step	Time step for integration $\Delta t$ (ps)
friction	Friction coefficient $\gamma$ (1/ps)

This move will solve the Langevin equation for the particles in the system on the form

$$ d\begin{bmatrix} \mathbf{q} \ \mathbf{p} \end{bmatrix} = \underbrace{\begin{bmatrix} M^{-1}\mathbf{q} \ 0 \end{bmatrix} \mathrm{d}t}_{\text{A}} + \underbrace{\begin{bmatrix} 0 \ -\nabla U(\mathbf{q}) \end{bmatrix}\mathrm{d}t}_{\text{B}} + \underbrace{\begin{bmatrix} 0 \ -\gamma \mathbf{p} + \sqrt{2\text{k}_{\mathrm{b}}T\gamma} \sqrt{M} ~\mathrm{d}\mathbf{W} \end{bmatrix}}_{\text{O}} $$

Where A, B, and O makes up the terms for solving the Langevin equation, which can be individually solved to obtain a trajectory given by \begin{aligned} \varphi^{\mathrm{A}}(\mathbf{q}, \mathbf{p}) &= \left(\mathbf{q} + \Delta t \sqrt{M}~ \mathbf{p}, \mathbf{p}\right) \ \varphi^{\mathrm{B}}(\mathbf{q}, \mathbf{p}) &= \left(\mathbf{q}, \mathbf{p} - \Delta t \nabla U(\mathbf{q})\right) \ \varphi^{\mathrm{O}}(\mathbf{q}, \mathbf{p}) &= \left(\mathbf{q}, e^{-\gamma \Delta t}\mathbf{p} + \sqrt{\mathrm{k}_{\mathrm{B}}T (1 - e^{-2\gamma \Delta t})} \sqrt{M} \mathbf{R} \right) \end{aligned}

We currently use the splitting scheme “BAOAB” (Symmetric Langevin Velocity-Verlet) since it is less errorprone with increasing timestep Leimkuhler & Matthews, pp. 279-281.