# Article: Time step rescaling recovers continuous-time dynamical properties for discrete-time Langevin integration of nonequilibrium systems

David A. Sivak, John D. Chodera, Gavin E. Crooks, arXiv:1301.3800

## Abstract

While the numerical integration of deterministic equations of motion for molecular systems now has a well-developed set of algorithms with commonly agreed-upon desirable properties, the simulation of stochastic equations of motion lacks algorithms with a similar degree of universal acceptance. Part of the difficulty is in determining which of many properties should be satisfied by such a discrete time integration scheme, with additional difficulties in satisfying many properties simultaneously with a single scheme. The desire to use these integration schemes for nonequilibrium simulations and in conjunction with recent nonequilibrium fluctuation theorems adds additional complications. Here, we compare a number of discrete time integration schemes for Langevin dynamics, moreassessing how well they satisfy several desiderata that have been enumerated in prior attempts at constructing good Langevin integrators. We show that the addition of a simple time step rescaling in the position update can correct a number of defects in these integrators, and we also identify a particular resulting splitting that has nearly universally good properties for the simulation of Langevin dynamics for molecular systems in equilibrium, nonequilibrium, and path sampling contexts.

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