Numerical Langevin Simulations: Equilibrium Dynamics and Nonequilibrium Thermodynamics

Abstract

Stochastic differential equations of motion (e.g., Langevin dynamics) provide a popular framework for simulating molecular systems. Any computational algorithm must discretize these equations, yet the resulting finite time step integration schemes suffer from several practical shortcomings: for example, equilibrium dynamical properties diverge from those of the continuous equations of motion; path-sampling applications are impeded by a path action that is either unknown or cumbersome; and thermodynamic statistics of driven systems are skewed, leading to biased inference using recently-developed nonequilibrium fluctuation theorems. We show how any finite time step Langevin integrator, even with a time-independent Hamiltonian, can be thought of as a driven, nonequilibrium physical process. Once an appropriate work-like quantity (the shadow work) is defined, nonequilibrium fluctuation theorems can characterize or correct for the errors introduced by the use of finite time steps. In particular, we demonstrate that amending estimators based on nonequilibrium work theorems to include this shadow work removes the time step dependent error from estimates of free energies. We also quantify, for the first time, the magnitude of deviations between the sampled stationary distribution and the desired equilibrium distribution for equilibrium Langevin simulations of solvated systems of varying size. What is more, 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.