FIG. 1: Comparison of logarithmic partition function estimates: the exact Kaufman formula (dashed line), the original replica gas identity, Eq. 8 with α = 1 (upward triangles), and the optimized form, Eq. 12 (downward triangles). Markers and error bars indicate the mean and standard deviation of 1000 independent simulations. Inset: magnification around the critical tem- perature. 12) is more complex, the increased difficulty of implementation is no greater than using the Bennett Acceptance Ratio [4] relative to unidirectional estimates of free energy differences , and does not require sampling from more ensembles. Expected statistical efficiency gains should make this investment worthwhile. D.M. thanks Artur Adib for sharing and explaining his manuscript, and Lee Makowski for sponsoring a Director’s Postdoctoral Fellowship. ∗ Electronic Address: daveminh@gmail.com ACKNOWLEDGMENTS [1] A. Gelman and X.-L. Meng, Stat. Sci. 13 , 163 (1998). [2] C. Jarzynski, in Computational Methods for Macromolecules, Lecture Notes , edited by T. Schlick and H. H. Gan (Springer, Berlin, 2002), vol. 24. 7 0.5 1 1.5 2 2.5 3 1000 2000 3000 4000 5000  = ln Z $^{−1}$ (= k$_{B}$ T) 2 2.5 900 1000 1100 1200