Minimizing thermodynamic length to select intermediate states for free-energy calculations and replica-exchange simulations

In computational thermodynamics, a sequence of intermediate states is commonly introduced to connect two equilibrium states. We consider two cases where the choice of intermediate states is particularly important: minimizing statistical error in free-energy difference calculations and maximizing average acceptance probabilities in replica-exchange simulations. We derive bounds for these quantities in terms of the thermodynamic distance between the intermediates, and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states.

Figure 1

Left: geodesics for an ideal gas in the $\left(\beta ,\beta P\right)$ coordinates. Right: in the $\left[\text{ln}\beta ,\text{ln}\left(\beta P\right)\right]$ coordinates, the metric is $g\mathbf{\left(}\text{ln}\beta ,\text{ln}\left(\beta P\right)\mathbf{\right)}=\text{diag}\left(C_V{k}_B^{-1},n\right)$, and the geodesics are Euclidean straight lines. Quasistatic adiabatic expansions of an ideal gas follow geodesics (thick lines). Equidistant points on the geodesics are shown as circles.