By contrast, the w shadow (X) depends on internal details of the propagation scheme used. While neglect of the shadow work can lead to large errors in general, we select a specific Langevin integrator that preserves the configurational distribution to very high accuracy, the BAOAB integrator of Matthews and Leimkuhler, allowing us to neglect this contribution without introducing large error. We justify this approximation by observing that the sequence of Langevin propagation kernels are nearly exact Markov kernels, each preserving the distribution ⇡ t (x) / e ut(x) with high fidelity. Recall that, due to discretization error, the invariant distribution ⇢ t (x) sampled by a numerical algorithm for Langevin dynamics will differ slightly from the target (i.e. ⇢ t ⇡ ⇡ t ), and the magnitude of this difference increases with the integrator step size. This may introduce bias. We neglect this bias, since the specific integrator employed here is thought to preserve the configurational distribution to very high accuracy . This conclusion is based on analytical results showing that the integrator approximates configurational averages to fourthorder in the timestep (as opposed to second-order for competing integrators), and extensive numerical evidence examining particular biomolecular observables. Note that we would also use this criterion if each propagation step were a reversible MCMC move.