Number of trials required to estimate a free-energy difference, using fluctuation relations

The difference $\mathrm{\Delta }F$ between free energies has applications in biology, chemistry, and pharmacology. The value of $\mathrm{\Delta }F$ can be estimated from experiments or simulations, via fluctuation theorems developed in statistical mechanics. Calculating the error in a $\mathrm{\Delta }F$ estimate is difficult. Worse, atypical trials dominate estimates. How many trials one should perform was estimated roughly by Jarzynski [Phys. Rev. E 73, 046105 (2006)]. We enhance the approximation with the following information-theoretic strategies. We quantify “dominance” with a tolerance parameter chosen by the experimenter or simulator. We bound the number of trials one should expect to perform, using the order-$\infty$ Rényi entropy. The bound can be estimated if one implements the “good practice” of bidirectionality, known to improve estimates of $\mathrm{\Delta }F$. Estimating $\mathrm{\Delta }F$ from this number of trials leads to an error that we bound approximately. Numerical experiments on a weakly interacting dilute classical gas support our analytical calculations.

Figure 1

Dominant values of work extractable from reverse-protocol trials. Large values $W$ of work contribute the most to the integral in the nonequilibrium fluctuation relation (2). An amount $W$ of extracted work is called $w^{\delta }$-dominant if it is at least as great as the threshold $w^{\delta }$ specified by the experimenter: $W\ge w^{\delta }$. The probability that any particular reverse trial will output a $w^{\delta }$-dominant amount of work is ${\int }_{w^{\delta }}^{\infty }dW{P}_{\mathrm{rev}}(-W)=1-\delta$. This probability equals the area of the region under the distribution's right-hand tail.

Figure 2

Dominant values of work invested in forward-protocol trials. Small values $W$ of work dominate the nonequilibrium work relation (17). An amount $W$ of invested work is called $W^{\varepsilon }$-dominant if it lies below or on the threshold $W^{\varepsilon }$ chosen by the experimenter: $W\le W^{\varepsilon }$. The probability that any particular forward trial will require a $W^{\varepsilon }$-dominant amount of work is $1-\varepsilon ={\int }_{-\infty }^{W^{\varepsilon }}dW{P}_{\mathrm{fwd}}\left(W\right)$. This probability equals the area under the distribution's left-hand tail.

Figure 3

Probability densities and numerical data for a weakly interacting dilute gas. We considered a gas undergoing compression (a forward protocol) and expansion (a reverse protocol). The probability per unit energy that any particular trial will involve an amount $W$ of work [Eq. (33)] was calculated in [15]. The short, right-hand, brown curve represents $P_{\mathrm{fwd}}\left(W\right)$. The tall, left-hand, dark-blue curve represents $P_{\mathrm{rev}}(-W)$. By sampling work values from these distributions, we effectively simulated each protocol ${10}^5$ times. The cyan bars (under the left-hand curve) depict the data gathered from the forward-protocol samples. The orange bars (under the right-hand curve) depict the data from the reverse-protocol samples.

Figure 4

Three number-of-required-trial measures. The abscissa shows possible choices of the threshold $w^{\delta }$ for $w^{\delta }$-dominant work values. The blue (gently sloping) curve, calculated from ${10}^5$ forward-trial samples, represents the bound on the number $N_{\delta }$ of reverse trials expected to be performed before any trial outputs a $w^{\delta }$-dominant amount $W\ge w^{\delta }$ of work (Theorem t2). The red (staggered) curve, calculated from ${10}^5$ reverse-trial samples, depicts the actual number $N_{\mathrm{true}}$ of trials performed before $W\ge w^{\delta }$ is extracted. The green curve (flat, nearly coincident with the abscissa) was calculated from forward-trial samples. This green curve represents relation (6): an estimate $N_{\mathrm{est}}$ of the number of trials required to extract a dominant amount of work, wherein the meaning of “dominant” is unspecified. The blue (gently sloping) curve follows the red (staggered) curve's shape more faithfully than the green (flat) does, illustrating the precision of Theorem t2. As expected, the blue (gently sloping) curve lower bounds the red (staggered) at most $w^{\delta }$ values.

Figure 5

Three number-of-required-trial measures at low threshold work values $w^{\delta }$. At most threshold values $w^{\delta }$, the $N_{\delta }$ bound (blue, gently sloping) lower bounds the actual number $N_{\mathrm{true}}$ (red, staggered) of reverse trials performed before any trial outputs a $w^{\delta }$-dominant amount $W\ge w^{\delta }$ of work. At low $w^{\delta }$ values, the red curve zigzags across the blue (gently sloping). This zigzagging stems from the technical definition of $N_{\delta }$.