Using bijective maps to improve free-energy estimates

We derive a fluctuation theorem for generalized work distributions, related to bijective mappings of the phase spaces of two physical systems, and use it to derive a two-sided constraint maximum likelihood estimator of their free-energy difference which uses samples from the equilibrium configurations of both systems. As an application, we evaluate the chemical potential of a dense Lennard-Jones fluid and study the construction and performance of suitable maps.

Figure 1

(Color online) Targeted work probability distributions for the expansion of a cavity in an ideal gas and the associated overlap distribution. The up (down) triangles display the normalized histogram of a sample of forward (reverse) work values. The smooth solid curves are the exact analytic work distributions $p(W\mid 0;\mathcal{M})$ (right) and $p(W\mid 1;\mathcal{M})$ (left), and the dashed curve shows their overlap distribution $p_{\mathrm{ol}}(W\mid \mathcal{M})$. The straight vertical lines show the values of the targeted estimates of $\Delta F$ on the abscissa. From left to right: the reverse, the two-sided (which is indistinguishable from the exact analytic value), and the forward estimate.

Figure 2

The geometric setup.

Figure 3

(Color online) Free-energy estimates for the expansion of a cavity in a Lennard-Jones fluid. Shown are the average values of traditional and targeted estimates of $\Delta F$ in dependence of the sample size $N$, with an errorbar of one standard deviation. In order to distinguish the data points those corresponding to targeted estimates are shifted to the right and are spread, whereas those corresponding to traditional estimates are shifted to the left. For example, all four data points in the vicinity of $N=10000$ refer to $N=10000$. The dashed horizontal line represents a targeted two-sided estimate with $N={10}^6$, see Table .

Figure 4

(Color online) The radial densities $g_1\left(r\right)$ and $g_0\left(r\right)$ for a dense Lennard-Jones fluid (${\rho }^{*}=0.9$ and $T^{*}=1.2$), estimated from simulated data. The ratio $g_1\left(r\right)∕g_0\left(r\right)$ equals the radial distribution function.

Figure 5

(Color online) Simulated radial mapping function ${\psi }^{*}\left(r\right)$ for a dense Lennard-Jones fluid (solid). ${\psi }^{*}$ maps the radial density $g_0\left(r\right)$ to $g_1\left(r\right)$, cf. Fig. 4. For the ideal gas, ${\psi }^{*}$ is the identity map (dashed).

Figure 6

(Color online) Targeted estimates of the excess chemical potential ${\mu }^{\mathrm{ex}}$ of a dense Lennard-Jones fluid (${\rho }^{*}=0.9$, $T^{*}=1.2$) compared to traditional estimates.

Figure 7

(Color online) Members of the family of radial mapping functions ${\psi }_m$ for the Lennard-Jones potential. For $m\to 0$, ${\psi }_m$ converges to the identity map ${\psi }_0\left(r\right)=r$.

Figure 8

(Color online) Forward (right) and reverse (left) work distributions of a Lennard-Jones fluid (${\rho }^{*}=0.9$, $T^{*}=1.2$) for different radial mapping functions ${\psi }_m$. ($m=0$ results in the traditional work distributions.)

Figure 9

(Color online) A detailed enlargement of forward (ascending lines) and reverse (descending lines) work distributions of the Lennard-Jones fluid (${\rho }^{*}=0.9$, $T^{*}=1.2$) for two different radial mapping functions ${\psi }_m$. Notice the enhancement of overlap for $m=0.0005$. The vertical dashed line displays the estimated two-sided targeted value, $\beta {\widehat{{\mu }^{\mathrm{ex}}}}_{01}=1.91$.

Figure 10

(Color online) The average generalized work ${⟨W⟩}_0$ and ${⟨W⟩}_1$ in forward and reverse direction, respectively, and the two-sided overlap measure $A_{\mathrm{ol}}$ in dependence of the mapping parameter $m$. The forward dissipation is reduced up to 18 orders of magnitude if compared with the traditional dissipation, cf. Table . Among the one-sided estimators the best is found for $m=0$ in forward direction. The optimal two-sided estimator results from using the $m$ that maximizes $A_{\mathrm{ol}}$.

Figure 11

(Color online) Two-sided estimates of ${\mu }^{\mathrm{ex}}$ as function of the mapping parameter $m$ out of $n_0=n_1=\frac{N}{2}={10}^4$ work values per direction for each $m$. The value of the traditional estimate $(m=0)$ is ${\widehat{{\mu }^{\mathrm{ex}}}}_{01}=4.0\pm 2.0$. The error bars show the square root of the estimated mean square errors $X_{01}$. For comparison, the dashed line represents a two-sided estimate with $\frac{N}{2}={10}^6$ and $m=0.0005$ (standard deviation 0.03).

Figure 12

(Color online) Convergence measure $a\left(N\right)$ of two-sided estimates for some parameter values $m$, depending on the sample size $N$. A faster decrease of $a$ towards the value 0 indicates a faster convergence of the two-sided estimator.