Quantifying configuration-sampling error in Langevin simulations of complex molecular systems

While Langevin integrators are popular in the study of equilibrium properties of complex systems, it is challenging to estimate the timestep-induced discretization error: the degree to which the sampled phase-space or configuration-space probability density departs from the desired target density due to the use of a finite integration timestep.
In \cite{Sivak:Phys.Rev.X:2013}, Sivak \textit{et al.}\ introduced a convenient approach to approximating a natural measure of error between the sampled density and the target equilibrium density, the KL divergence, in \emph{phase space}, but did not specifically address the issue of \emph{configuration-space properties}, which are much more commonly of interest in molecular simulations.
Here, we introduce a variant of this near-equilibrium estimator capable of measuring the error in the configuration-space marginal density, validating it against a complex but exact nested Monte Carlo estimator to show that it reproduces the KL divergence with high fidelity. To illustrate its utility, we employ this new near-equilibrium estimator to assess a claim that a recently proposed Langevin integrator introduces extremely small configuration-space density errors up to the stability limit at no extra computational expense. Finally, we show how this approach to quantifying sampling bias can be applied to a wide variety of stochastic integrators by following a straightforward procedure to compute the appropriate shadow work, and describe how it can be extended to quantify the error in arbitrary marginal or conditional distributions of interest.