Hence, to determine the optimal spacing, we need an estimate of √ t λ σ ∂H ∂λ λ or σ ∂H ∂λ as a function of λ. One way to obtain this is from an initial short set of simulations with non-optimal spacing. The windows can then be spaced according to Equation 19 or 18 by ensuring that the areas under the σ ∂H ∂λ or √ t λ σ ∂H ∂λ λ curves between each value of λ are equal. We can either specify N Wind. , or the entire right-hand sides of Equations 19 or 18. We term these "thermodynamic speeds" following Minh, who discussed the spacing of windows according to Equation . We specify a constant thermodynamic speed, ensuring that equivalent spacing is obtained between different transformations of varying thermodynamic length. This algorithm is illustrated in Figure for the case where σ ∂H ∂λ is used. A common approach to spacing λ windows is to aim for consistent and sufficient overlap between adjacent windows. The uncertainty of the BAR estimator can be expressed in terms of overlap, 51 but also in terms of the variance of the gradient (under our assumption of infinitely close window spacing, see Section S1). Hence, these spacing approaches are closely related and the variance of the gradient must predict the overlap in the limit of infinitely closely spaced windows. Intuitively, this is because the dimensionless variance of the gradient is the average squared relative change in probability with λ