In Figure , we compare numerical results of the ranked-weight distributions to analytical results obtained by assuming Gaussian distributions of ∆u, parameterized by ⟨∆u⟩ and var(∆u) calculated from the weights (see Appendix B). For the largest value of θ and thus the smallest values of S KL , the agreement is excellent. However, for smaller values of θ, the agreement deteriorates somewhat. For the double-well system (Figure , left column), the analytical approximation captures the trends qualitatively but fails quantitatively because the underlying distribution of ∆u is clearly non-Gaussian. For the polymer model, the distribution of ∆u is skewed, but unimodal for all θ values considered (see Figure ), resulting in near-quantitative agreement of the Gaussianbased approximations and the actual weight distributions after refinement. For the most realistic case presented here, Ala 5 , the ∆u distribution is well approximated by a Gaussian. Consequently, the approximations of the weight distributions are quite accurate, even for the smallest value of θ = 1 with S KL ≈ 1.43.