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Common algorithms for computationally simulating Langevin dynamics mustdiscretize the stochastic differential equations of motion. These resultingfinite time step integrators necessarily have several practical issues incommon: Microscopic reversibility is violated, the sampled stationarydistribution differs from the desired equilibrium distribution, and the workaccumulated in nonequilibrium simulations is not directly usable in estimatorsbased on nonequilibrium work theorems. Here, we show that even with atime-independent Hamiltonian, finite time step Langevin integrators can bethought of as a driven, nonequilibrium physical process. Once an appropriatework-like quantity is defined -- here called the shadow work -- recentlydeveloped nonequilibrium fluctuation theorems can be used to measure or correctfor the errors introduced by the use of finite time steps. In particular, wedemonstrate that amending estimators based on nonequilibrium work theorems toinclude this shadow work removes the time step dependent error from estimatesof free energies. We also quantify, for the first time, the magnitude ofdeviations between the sampled stationary distribution and the desiredequilibrium distribution for equilibrium Langevin simulations of solvatedsystems of varying size. While these deviations can be large, they can beeliminated altogether by Metropolization or greatly diminished by smallreductions in the time step. Through this connection with driven processes,further developments in nonequilibrium fluctuation theorems can provideadditional analytical tools for dealing with errors in finite time stepintegrators.

Using nonequilibrium fluctuation theorems to understand and correct errors in equilibrium and nonequilibrium discrete Langevin dynamics simulations