  When simulating molecular systems using deterministic equations of motion
(e.g., Newtonian dynamics), such equations are generally numerically integrated
according to a well-developed set of algorithms that share commonly agreed-upon
desirable properties. However, for stochastic equations of motion (e.g.,
Langevin dynamics), there is still broad disagreement over which integration
algorithms are most appropriate. While multiple desiderata have been proposed
throughout the literature, consensus on which criteria are important is absent,
and no published integration scheme satisfies all desiderata simultaneously.
Additional nontrivial complications stem from simulating systems driven out of
equilibrium using existing stochastic integration schemes in conjunction with
recently-developed nonequilibrium fluctuation theorems. Here, we examine a
family of discrete time integration schemes for Langevin dynamics, assessing
how each member satisfies a variety of desiderata that have been enumerated in
prior efforts to construct suitable Langevin integrators. We show that the
incorporation of a novel time step rescaling in the deterministic updates of
position and velocity can correct a number of dynamical defects in these
integrators. Finally, we identify a particular splitting that has essentially
universally appropriate properties for the simulation of Langevin dynamics for
molecular systems in equilibrium, nonequilibrium, and path sampling contexts.
