---
title: "Jarzynski Equality: Dragging a Trapped Bead Interactive Lab | Sandbox Physics"
description: "Can repeated irreversible pulls reveal the free-energy change of a reversible one?"
canonical_url: "https://sandboxphysics.com/labs/dragged-trap-work/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/dragged-trap-work/"
last_modified: "2026-10-29"
---

FR25 · Work along one path

# Jarzynski Equality: Dragging a Trapped Bead

**Can repeated irreversible pulls reveal the free-energy change of a reversible one?** One pull of a trapped bead can cost more or less work than the free-energy change, because the surrounding water kicks it at random. On average a pull always costs more. Averaging the exponential of minus the work over many pulls, or finding where the forward and reversed work distributions cross, still recovers the free-energy change, as long as the rare low-work pulls are sampled.

Drag a trapped bead uphill hundreds of times. Each pull costs different work, yet the work distributions still reveal the free-energy change.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A one-dimensional overdamped bead of radius 0.5 µm in water at 300 K sits in a harmonic optical trap with a constant load force. Each time step first moves the trap with the bead held still, which is work, then lets the bead relax exactly in the fixed trap, which is heat released to the water. Inertia, wall corrections to the drag, trap nonlinearity, detector noise and laser heating are omitted. Every pull is a model simulation; no measured data are shown.

Forward and mirrored reverse work

Estimate as pulls accumulate

Work, heat and potential energy of one typical pull

More numbers and how they are computed

- **Exact mean work of this protocol**: $0$
- **Exact dissipated work**: $0$
- **Continuous-time mean work**: $0$
- **Estimate from reverse pulls**: $0$
- **Bennett acceptance ratio**: $0$
- **Bootstrap error of the Jarzynski estimate**: $0$
- **Expected mean bias at this pull count**: $0$
- **Share of the largest single weight**: $0$
- **Effective number of pulls**: $0$
- **Largest mismatch of work minus heat minus potential change**: $0$
- **Relaxation time**: $0$
- **Time steps per pull**: $0$

Exact values come from the mean and covariance recursions of the discrete protocol; their ratio checks the fluctuation–dissipation relation. The expected band uses 600 seeded Gaussian replicates at each pull count. Pulls are in simulation order, so jumps in the running curves are single rare pulls.

Physics tutorial

## Work fluctuates, free energy does not

**Background** A bead in an optical trap feels random kicks from the water. Moving the trap does work on the bead, and the bead hands energy back to the water as heat.

**Why it matters** Each pull follows a different path, so its work differs. The free-energy change between the start and end traps is a single number. The question is how to read that number from many noisy, irreversible pulls.

Start with the essentials

- **Focus question**: Can repeated irreversible pulls reveal the free-energy change of a reversible one?
- **One-sentence intuition**: The mean work overshoots the free-energy change, but the exponential average and the crossing of forward and reversed work distributions both land on it. The exponential average leans on rare low-work pulls, so too few pulls bias it upward.

### Core mathematical model

#### Where the work of one pull goes

$$
W=Q+\Delta V,\qquad V(x,\lambda)=\tfrac{k}{2}(x-\lambda)^2+fx
$$

Work done by the moving trap either raises the bead potential energy or leaves as heat into the water. Work equals heat plus the potential-energy change for every single pull.

#### Jarzynski equality

$$
\left\langle e^{-W/k_{\mathrm B}T}\right\rangle=e^{-\Delta F/k_{\mathrm B}T},\qquad \Delta F=fL
$$

The average runs over pulls that start in equilibrium. Here the free-energy change equals the load force times the drag distance, so the estimate can be checked exactly.

#### Crooks fluctuation theorem

$$
\frac{P_{\mathrm F}(W)}{P_{\mathrm R}(-W)}=e^{(W-\Delta F)/k_{\mathrm B}T}
$$

Forward pulls and the same path run backwards have work distributions that cross exactly at the free-energy change.

#### Mean dissipated work at constant speed

$$
\langle W\rangle-\Delta F=\gamma v^2\left[\tau-\tau_r\left(1-e^{-\tau/\tau_r}\right)\right]=\frac{\operatorname{Var}W}{2k_{\mathrm B}T}
$$

Here $\gamma$ is the bead drag coefficient, $v$ the trap speed, $\tau$ the pull time and $\tau_r=\gamma/k$ the relaxation time. The work is Gaussian, and its spread is fixed by the extra work.

### Common difficulties

#### A pull below the free-energy change

**Typical misconception** A pull that costs less than the free-energy change breaks the second law.

**Better mental model** The second law limits the average over many pulls. A single small system can borrow heat from the water now and then; such pulls are rare, and they are exactly what the exponential average needs.

### Run the experiment

1. 01

   #### Watch the green beads

   Play Uphill drag and watch the upper band.

   **What to observe:** A few beads turn green: so far they have cost less work than the bowl has risen. The histogram shows them as the left tail.

2. 02

   #### Pull faster

   Choose Rare pulls decide.

   **What to observe:** Mean work rises well above the free-energy change, and the forward exponential average stays high while the crossing still finds it.

3. 03

   #### Drag the trap yourself

   Drag the forward trap handle to any new position, then let go.

   **What to observe:** Whatever path you draw, the estimate approaches the load force times your net displacement, and every pull still splits its work exactly into heat plus potential-energy change.

Physical conclusion

Fluctuating work carries exact information about free energy, but extracting it needs the rare pulls the average hides.

## Related experiments and articles

Stochastic thermodynamics · Reading path

### From a trapped bead to free energy from irreversible pulls

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How far a trapped bead wanders at room temperature**](https://sandboxphysics.com/stories/stochastic-thermodynamics/bead-in-a-trap/)
2. [Lab **Optical Tweezers: Brownian Calibration** Turn it around: measure stiffness from the wandering](https://sandboxphysics.com/labs/optical-tweezers-calibration/)
3. [Article **Work and heat for a single pull**](https://sandboxphysics.com/stories/stochastic-thermodynamics/work-and-heat-along-one-path/)
4. [Article **The mean work and its spread when a trap moves at constant speed**](https://sandboxphysics.com/stories/stochastic-thermodynamics/dragging-the-trap/)
5. [Article **How the Jarzynski equality and the Crooks relation recover free energy from irreversible pulls**](https://sandboxphysics.com/stories/stochastic-thermodynamics/jarzynski-and-crooks/)
6. [Article **Reading the 2003 paper by Gore, Ritort and Bustamante on the bias of free energies from few pulls**](https://sandboxphysics.com/stories/stochastic-thermodynamics/reading-the-jarzynski-bias-paper/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)