Measuring the convergence of Monte Carlo free-energy calculations

The nonequilibrium work fluctuation theorem provides the way for calculations of (equilibrium) free-energy based on work measurements of nonequilibrium, finite-time processes, and their reversed counterparts by applying Bennett’s acceptance ratio method. A nice property of this method is that each free-energy estimate readily yields an estimate of the asymptotic mean square error. Assuming convergence, it is easy to specify the uncertainty of the results. However, sample sizes have often to be balanced with respect to experimental or computational limitations and the question arises whether available samples of work values are sufficiently large in order to ensure convergence. Here, we propose a convergence measure for the two-sided free-energy estimator and characterize some of its properties, explain how it works, and test its statistical behavior. In total, we derive a convergence criterion for Bennett’s acceptance ratio method.

Figure 1

Displayed are free-energy estimates $\widehat{\Delta f}$ in dependence of the sample size $N$, reaching a seemingly stable plateau if $N$ is restricted to $N=1000$ (top panel). Another stable plateau is reached if the sample size is increased up to $N=100000$ (bottom panel). Has the estimate finally converged? The answer is given by the corresponding graph of the convergence measure $a$ which is shown in the inset. The fluctuations around zero indicate convergence. The exact value of the free-energy difference is visualized by the dashed horizontal line.

Figure 2

Schematic diagram of reverse $p_1$, overlap $p_{\alpha }$, and forward $p_0$ work densities (top). Schematic histograms of finite samples from $p_0$ and $p_1$, where in particular the latter is imperfectly sampled, resulting in a biased estimate $\widehat{\Delta f}$ of the free-energy difference (bottom).

Figure 3

Schematic plot which shows that the forward work density, $p_0\left(w\right)$, samples the Fermi function $b_{\Delta f}\left(w\right)=1/\left(\beta +\alpha e^{w-\Delta f}\right)$ somewhat earlier than its square.

Figure 4

Exponential (left panel) and Gaussian (right panel) work densities.

Figure 5

Statistics of two-sided free-energy estimation (exponential work densities): shown are averaged estimates of $\Delta f$ in dependence of the total sample size $N$. The error bars reflect the standard deviation. The dashed line shows the exact value of $\Delta f$ and the inset the details for large $N$ (top). Statistics of the convergence measure $a$ corresponding to the estimates of the top panel: shown are the average values of $a$ together with their standard deviation in dependence of the sample size $N$. Note the characteristic convergence of $a$ towards zero in the large $N$ limit (bottom).

Figure 6

Mean values of overlap estimates ${\widehat{U}}_{\alpha }$ and ${\widehat{U}}_{\alpha }^{\left(II\right)}$ of first and second order, respectively. The slightly slower convergence of ${\widehat{U}}_{\alpha }^{\left(II\right)}$ towards $U_{\alpha }$ results in the characteristic properties of the convergence measure $a$. To enhance clarity, data points belonging to the same value of $N$ are spread.

Figure 7

(Color online) Double-logarithmic scatter plot of ${\widehat{U}}_{\alpha }^{\left(II\right)}$ versus ${\widehat{U}}_{\alpha }$ for many individual estimates in dependence of the sample size $N$. The dotted lines mark the exact value of $U_{\alpha }$ on the axes and the dashed line is the bisectrix ${\widehat{U}}_{\alpha }^{\left(II\right)}={\widehat{U}}_{\alpha }$. The approximatively linear relation between the logarithms of ${\widehat{U}}_{\alpha }^{\left(II\right)}$ and ${\widehat{U}}_{\alpha }$ is continued up to the smallest observed values ($<{10}^{-100}$, not shown here).

Figure 8

The average convergence measure $⟨a⟩$ plotted against the corresponding mean-square error $⟨{\left(\widehat{\Delta f}-\Delta f\right)}^2⟩$ of the two-sided free-energy estimator. The inset shows an enlargement for small values of $⟨a⟩$.

Figure 9

(Color online) A scatter plot of the deviation $\widehat{\Delta f}-\Delta f$ versus the convergence measure $a$ for many individual estimates in dependence of the sample size $N$. Note that the majority of estimates belonging to $N=32$ and 100 have large values of $\widehat{\Delta f}-\Delta f$ well outside the displayed range with $a$ being close to one.

Figure 10

Estimated constrained probability densities $p\left(\widehat{\Delta f}\mid a<0.9\right)$ (black) and $p\left(\widehat{\Delta f}\mid a\ge 0.9\right)$ (grayscale) for two different sample sizes $N$, plotted versus the deviation $\widehat{\Delta f}-\Delta f$. The inset shows averaged estimates of $\Delta f$ over the total sample size $N$ subject to the constraints $a\ge 0.9$ and $a<0.9$, respectively.

Figure 11

Gaussian work densities result in the displayed averaged estimates of $\Delta f$. For comparison, three different fractions $\alpha$ of forward work values are used (top). Average values of the convergence measure $a$ corresponding to the estimates of the top panel (bottom).

Figure 12

Mean values of overlap estimates ${\widehat{U}}_{\alpha }$ and ${\widehat{U}}_{\alpha }^{\left(II\right)}$ of first and second order $\left(\alpha =0.5\right)$.

Figure 13

The average convergence measure $⟨a⟩$ plotted against the corresponding mean square error $⟨{\left(\widehat{\Delta f}-\Delta f\right)}^2⟩$ of free-energy estimates in dependence of $N$.

Figure 14

(Color online) A scatter plot of the deviation $\widehat{\Delta f}-\Delta f$ versus the convergence measure $a$ for many individual estimates in dependence of the sample size $N$ $\left(\alpha =0.5\right)$.

Figure 15

Averaged two-sided estimates of $\Delta f$ in dependence of the total sample size $N$ for the constraints $a\ge 0.9$, $a$ unconstrained, and $a<0.9$ $\left(\alpha =0.99999\right)$.

Figure 16

Running estimates of the excess chemical potential ${\mu }^{ex}$ in dependence of the sample size $N$ $\left(\alpha =0.9\right)$. The inset displays the corresponding values of the convergence measure $a$.

Figure 17

Statistics of estimates of the excess chemical potential shown are the average value and the standard deviation (as error bars) in dependence of the sample size $N$. The statistics of the corresponding values of the convergence measure is shown in the inset.