---
title: "Information Engine: Work From Measuring a Bead Interactive Lab | Sandbox Physics"
description: "How much work can a measurement of a trapped bead buy, and what limits it?"
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last_modified: "2026-10-29"
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FR25 · Feedback and information

# Information Engine: Work From Measuring a Bead

**How much work can a measurement of a trapped bead buy, and what limits it?** Measuring where a trapped bead has wandered and moving the trap toward it lets the bead drop in energy, so the water heat pays work out. On average that work cannot exceed the thermal energy times the mutual information the measurement carried. Moving the trap only captures part of it: a sharp reading wastes most of its information, while a blurry reading used with a small gain converts most.

Measure where a trapped bead sits, shift the trap toward it and extract work from the water heat. The mutual information caps the average gain.

> 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 10 pN/µm harmonic trap. Each cycle measures its position with Gaussian error, shifts the trap instantly by the gain times the measured offset, then lets the bead relax exactly for the wait time. The cost of the measurement and of erasing the record is not included. Every cycle is a model simulation; no measured data are shown.

Work and information against the gain

Single cycles can beat the bound; the mean cannot

Cumulative work of 64 engines

More numbers and how they are computed

- **Exact mean work extracted per cycle**: $0$
- **Standard error of the simulated mean work**: $0$
- **Exact mutual information per measurement**: $0$
- **Information about the end state left unused**: $0$
- **Bead variance before measuring, relative to thermal**: $0$
- **Signal-to-noise ratio of the reading**: $0$
- **Best gain after a long wait**: $0$
- **Efficiency at that gain**: $0$
- **Sample average of the exact equality, expected 1**: $0$
- **Share of that average from the largest cycle**: $0$
- **Mean total entropy production plus information**: $0$
- **Relaxation time**: $0$
- **Engines times cycles**: $0$

Exact values come from the steady-state variance of the cycle in closed form. The standard error uses the spread of the 64 engine means. The sample average of the exact equality is pulled down by rare cycles whose reading points the wrong way; it reaches 1 only with far more cycles, as the Jarzynski average does.

Physics tutorial

## A measurement can pay for work

**Background** A bead in a trap jiggles in the warm water. If you know which way it has wandered, you can slide the trap under it and let it fall, so the trap does negative work and the water pays.

**Why it matters** A closed cycle at one temperature cannot deliver work on average, unless something learns about the bead. The question is how much a measurement is worth in work, and why real feedback falls short of it.

Start with the essentials

- **Focus question**: How much work can a measurement of a trapped bead buy, and what limits it?
- **One-sentence intuition**: The average work extracted per cycle never exceeds the thermal energy times the mutual information between the bead and the reading. A sharp measurement carries many nats but a trap shift can use only about half a thermal energy of them, so efficiency is highest for blurry readings used with a small gain.

### Core mathematical model

#### Work of one instant shift

$$
W=\tfrac{k}{2}\left[(z-gy)^2-z^2\right],\qquad y=z+\varepsilon
$$

Here $z$ is the bead offset from the trap centre, $y$ the reading with Gaussian error $\varepsilon$ and $g$ the gain. The bead does not move during the shift, so the work is the change of potential energy.

#### Mutual information of a Gaussian reading

$$
\langle I\rangle=\tfrac12\ln\!\left(1+\frac{\sigma_z^2}{\sigma_m^2}\right)
$$

It compares the spread of the bead, $\sigma_z$, with the measurement error $\sigma_m$, and is measured in nats.

#### Sagawa–Ueda bound

$$
-\langle W\rangle\le k_{\mathrm B}T\,\langle I\rangle,\qquad \left\langle e^{-(\Delta s_{\mathrm{tot}}+I)}\right\rangle=1
$$

Over a steady cycle the average extracted work is capped by the information. The exact equality behind it holds cycle by cycle, but its average is dominated by rare misleading readings.

#### Best gain after a long wait

$$
g^\ast=\frac{\mathrm{SNR}}{1+\mathrm{SNR}},\qquad \eta^\ast=\frac{\mathrm{SNR}}{(1+\mathrm{SNR})\ln(1+\mathrm{SNR})}
$$

With $\mathrm{SNR}=k_{\mathrm B}T/(k\sigma_m^2)$, the efficiency at the best gain approaches one only for very noisy readings.

### Common difficulties

#### Work from a single bath

**Typical misconception** Extracting work from water at one temperature breaks the second law.

**Better mental model** The engine is fed by information. Recording the reading and later erasing it costs at least the thermal energy times the same information, so the total stays within the second law.

### Run the experiment

1. 01

   #### Watch one cycle

   Watch the large trap: the pink line is the reading, the arrow is the shift.

   **What to observe:** When the reading is right, the bead ends up lower after the shift and the cycle extracts work. Below, each small trap is one of 64 engines.

2. 02

   #### Overshoot

   Choose Overshoot costs work.

   **What to observe:** The trap jumps past the bead, the beads turn red and the mean work turns negative, although the information is the same as before.

3. 03

   #### Be the demon

   Drag the trap handle toward the bead whenever it wanders off centre.

   **What to observe:** Quick, well-aimed moves extract work; random moves pay it back to the water.

Physical conclusion

Information is a thermodynamic resource: it can be spent as work, but never more than the thermal energy per nat.

## 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 much work can be extracted by measuring a trapped bead and moving the trap**](https://sandboxphysics.com/stories/stochastic-thermodynamics/feedback-and-information/)

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)