shown simply by the overlap in histograms of probability distributions of differences in potential energy from the initial to final state, a common practice in free energy calculations [@Pohorille2010]. In this case, the states refer to the potential used and the distributions are taken from: $$\label{eq:hist0}
\Delta U_0 = (U(\phi_{HF}, \Gamma_{LF}) - U(\phi_{LF}, \Gamma_{LF}))$$ $$\label{eq:hist1}
\Delta U_1 = (U(\phi_{LF}, \Gamma_{HF}) - U(\phi_{HF}, \Gamma_{HF}))$$ where each term is the energy with potential $\phi$ and set of samples in phase space $\Gamma=(\mathbf{x_1}, ..., \mathbf{x_N})$ (each point dependent on the positions of the $N$ atoms) from the high or low-fidelity potential trajectory. Examples are shown in Fig. \[fig:hist\].

Lack of phase space overlap and difference in probability distributions is most significant for the the ambient liquid with the Sine 6 and 7 potentials, the same two cases with significantly larger error. Further inspection of the MD trajectory shows that the modified potential results in a structural transformation while the unmodified LJ does not. This is shown in snapshots included as insets in Fig. \[fig:hist\] comparing the atomic structures for the ambient liquid with Sine 1 and Sine 7: the liquid undergoes cavitation for Sine 7 (and to a lesser extent with Sine 5 