L4 · Chapter 8

Reading the 2003 paper by Gore, Ritort and Bustamante on the bias of free energies from few pulls

Sandbox Physics · 2026-10-29

Open PNAS volume 100, pages 12564–12569 (2003), PubMed Central PMC240657. The paper asks one question: with only N pulls, how wrong is the Jarzynski estimate? After its three figures you will be able to take the dissipated work and the number of pulls in an experimental paper and estimate how far its free energy is likely to sit too high.

The question the paper answers

The authors write the Jarzynski average of $N$ pulls as an estimator $\widehat{\Delta F}_N$ . Repeat the experiment many times; the mean of the estimator minus the true value is the bias $B(N)$ . Bias is a systematic error. Repeating the experiment does not remove it; only more pulls per experiment do.

The simplest case is $N=1$ . The estimator is then the work itself, whose mean is $\Delta F+\overline{W}_{\rm dis}$ , so the bias equals the mean dissipated work. The paper writes the mean dissipated work as $\overline{W}_{\rm dis}$ ; it is the $W_d$ of the earlier chapters.

\widehat{\Delta F}_N=-k_BT\ln\!\left(\frac1N\sum_{i=1}^{N}e^{-W_i/k_BT}\right),\qquad B(N)=\big\langle\widehat{\Delta F}_N\big\rangle-\Delta F

The outer brackets average over many sets of N pulls each.

Source: J. Gore

Figure 1: a family of Gaussians

The horizontal axis is the dissipated work of one pull, $W-\Delta F$ . The four curves have mean dissipated work of 0.5, 1, 2 and 4 times $k_BT$ . The authors take the work to be Gaussian with variance $2k_BT\,\overline{W}_{\rm dis}$ , so a larger mean gives a wider curve.

The easy misreading: the wider curve reaches further into negative values, yet the fraction of pulls with negative dissipated work falls. That fraction is $\Phi\!\left(-\sqrt{\overline{W}_{\rm dis}/2k_BT}\right)$ , about 31%, 24%, 16% and 7.9% for the four curves.

The paper also notes that a bead dragged through water by a harmonic spring has Gaussian work at any speed. That is the model of Chapter 3, so the paper’s Gaussian results can be tested directly on the dragged trap.

Source: J. Gore · O. Mazonka and C. Jarzynski

Figure 2: four units of dissipation and only twenty pulls

The histogram shows 100 000 estimates minus the true value, each computed from 20 Gaussian pulls. The solid line is the work distribution itself; the dashed line is that distribution times the Jarzynski weight, peaking at $-4\,k_BT$ , where fewer than 1% of pulls fall. The dotted line marks the mean of the histogram: a bias of about $1.07\,k_BT$ .

Three quantities must be kept apart. How far the centre of the histogram sits from zero is the bias. The width of the histogram is the random error of the estimate. Their squares add to the mean squared error, the paper’s Eq. (6). Repeating experiments averages away the width but not the bias.

We recomputed the figure with our own Gaussian sampler: 100 000 sets of 20 pulls with random seed 2003 give a bias of $1.064\pm0.004\,k_BT$ , in agreement with the published 1.07.

Gaussian dissipated work with mean

4\,k_BT

, the case of Gore, Ritort and Bustamante (2003) Figure 2 (solid), and the same curve multiplied by the Jarzynski weight (dashed). The weighted curve peaks at

-4\,k_BT

, where pulls are rare. Recomputed analytic curves.

Source: J. Gore

Worked example: how slowly the bias falls at four units

Figure 3A plots the bias against $N$ on log–log axes for mean dissipated work from 0.5 to 64 times $k_BT$ ; each circle averages 150 000 sets. For few pulls the circles lie close to a straight line, a power law $B\approx\overline{W}_{\rm dis}N^{-\alpha}$ (the paper’s Eq. 10). For many pulls the logarithm can be expanded linearly and the bias falls as $1/N$ (Eq. 11).

The authors join the two at $N_C=C\big(e^{2\overline{W}_{\rm dis}/k_BT}-1\big)$ with $C=15$ (Eq. 12). Setting the two expressions equal there gives the exponent $\alpha$ . The equations in the open-access text are images; the formula below is our derivation from the description in the text.

For $\overline{W}_{\rm dis}=4\,k_BT$ : $N_C\approx44\,700$ and $\alpha\approx0.447$ , so $B(20)\approx4/20^{0.447}\approx1.05\,k_BT$ , within 2% of the 1.07 in Figure 2. For $5\,k_BT$ it gives $\alpha\approx0.39$ , matching the paper’s value of about 0.4. At $64\,k_BT$ , $N_C$ is about $10^{57}$ : almost every real experiment has far fewer pulls than that.

B(N)\approx\frac{k_BT\left(e^{2\overline{W}_{\rm dis}/k_BT}-1\right)}{2N}\quad(N\gg N_C)

For a Gaussian the variance of

e^{-W_{\rm dis}/k_BT}

e^{2\overline{W}_{\rm dis}/k_BT}-1

; insert it into the linear expansion of the logarithm.

\alpha=\frac{\ln\!\left(2C\,\overline{W}_{\rm dis}/k_BT\right)}{\ln N_C}

Take logarithms of

\overline{W}_{\rm dis}N_C^{-\alpha}=k_BT/2C

. More dissipation gives a smaller

\alpha

and a more slowly falling bias.

Source: J. Gore

Back to the dragged trap with twenty pulls

The Lab’s Only twenty pulls preset drags the trap 100 nm at $21\,\mu\mathrm{m/s}$ . The mean dissipated work is $3.99\,k_BT$ and there are 20 pulls in each direction, almost exactly the case of Figure 2. The default record gives a forward estimate of $3.71\,k_BT$ , $1.30\,k_BT$ above the true $2.41\,k_BT$ and of the same size as the paper’s mean bias. At the same speed, 20 000 pulls bring the estimate back to $2.40\,k_BT$ .

For the next paper that extracts a free energy from nonequilibrium pulls, note in turn: whether the work distribution is close to Gaussian and whether its low-work side was sampled; the mean dissipated work in thermal energy units; how the number of pulls compares with the exponential of the dissipated work; whether bias and random error are reported separately; and whether reverse pulls were used as a check.

On the experimental side, Blickle and Bechinger built a stochastic heat engine in 2012 from one colloidal bead in an optical trap. The work of single cycles fluctuates strongly; slow cycles approach an efficiency of 0.14, and at maximum power the efficiency is about 0.075. Reading that paper, ask the same question: what does a single cycle tell you, and what does the average tell you?

Source: J. Gore · V. Blickle and C. Bechinger

Try it in the Lab

Repeat the case of Figure 2

Choose Only twenty pulls. Predict where the forward Jarzynski estimate lands on average when you resample many times. Hint: the true value plus about $1\,k_BT$ .
Resample ten times, write down the forward estimate each time, average them and compare with your prediction. Check whether the crossing estimate is shifted the same way.
Raise the number of pulls to 200 and repeat a few times. Estimate how much the bias should shrink with the exponent formula of this chapter and compare.

Exercises with explained answers

Show that the bias at $N=1$ equals the mean dissipated work.
Show the answer
With one pull $\widehat{\Delta F}_1=-k_BT\ln e^{-W/k_BT}=W$ , whose mean is $\langle W\rangle=\Delta F+\overline{W}_{\rm dis}$ . Subtracting $\Delta F$ leaves the mean dissipated work.
For a mean dissipated work of $2\,k_BT$ and $C=15$ , how many pulls are needed before the large- $N$ formula applies, and how large is the bias there?
Show the answer
$N_C=15(e^4-1)\approx804$ . The large- $N$ formula then gives $B=k_BT/2C\approx0.033\,k_BT$ . A few hundred pulls are enough, which is the slow-pull situation of Chapter 3.
A paper estimates a free energy from 10 pulls, and its mean work exceeds the estimate by $6\,k_BT$ . Following Figure 3, how large is the bias likely to be?
Show the answer
The mean work minus the estimate underestimates the true dissipation, so the dissipated work is at least about $6\,k_BT$ . Then $\alpha\approx0.35$ and $B(10)\approx6/10^{0.35}\approx2.7\,k_BT$ . The estimate may be several thermal energy units too high; more pulls or reverse data are needed before drawing a conclusion.

Sources and what we recomputed

O. Mazonka and C. Jarzynski, Exactly solvable model illustrating far-from-equilibrium predictions, arXiv:cond-mat/9912121v1 (1999)
Version 1, 7 December 1999. A bead dragged through a viscous fluid by a harmonic spring at constant speed. Eqs. (4)–(7) set up the model and the work rate; Eq. (26) defines the finite-time factor; Eqs. (43)–(53) add a constant force, give the free-energy change in Eq. (49) and the Gaussian mean and variance of the work in Eqs. (51)–(53). We recompute these results; the paper does not give our numerical examples.
J. Gore, F. Ritort and C. Bustamante, Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements, PNAS 100, 12564–12569 (2003)
Open-access text on PubMed Central. Figure 1: Gaussian dissipated-work distributions with means of 0.5, 1, 2 and 4 thermal energy units. Figure 2: 100 000 estimates, each from 20 work values, at a mean dissipated work of 4 units; mean bias about 1.07 units. Figure 3: bias against sample size and the fall-off exponent for the constant C equal to 15. We reproduce the Figure 2 bias by our own sampling.
V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nature Physics 8, 143–146 (2012)
A single 2.94 μm colloidal bead in an optical trap whose stiffness cycles between 281 and 1180 fN/μm, with the bath switched between 22 °C and 86 °C by laser heating. In the 76 °C data the work of single cycles fluctuates strongly; the measured efficiency approaches 0.14 for slow cycles, about 0.91 of the Carnot value, and the efficiency at maximum power, about 0.075, is close to the Curzon–Ahlborn value 0.081. Cited as an experimental reading example only.