  Common algorithms for computationally simulating Langevin dynamics must
discretize the stochastic differential equations of motion. These resulting
finite time step integrators necessarily have several practical issues in
common: Microscopic reversibility is violated, the sampled stationary
distribution differs from the desired equilibrium distribution, and the work
accumulated in nonequilibrium simulations is not directly usable in estimators
based on nonequilibrium work theorems. Here, we show that even with a
time-independent Hamiltonian, finite time step Langevin integrators can be
thought of as a driven, nonequilibrium physical process. Once an appropriate
work-like quantity is defined -- here called the shadow work -- recently
developed nonequilibrium fluctuation theorems can be used to measure 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. While these deviations can be large, they can be
eliminated altogether by Metropolization or greatly diminished by small
reductions in the time step. Through this connection with driven processes,
further developments in nonequilibrium fluctuation theorems can provide
additional analytical tools for dealing with errors in finite time step
integrators.
