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.

Work extracted per cycle $0.409\,k_{\mathrm B}T$

Information per measurement $0.834\,\mathrm{nat}$

Share of information used $49.1\,\%$

Power $0.136\,k_{\mathrm B}T/\mathrm{ms}$

Turn measurements into work

Measurement error

10

\sigma_m\;(\mathrm{nm})

Feedback gain

0.8

g

Wait between measurements

3

t_w\;(\mathrm{ms})

Drag the cyan handle to move the big trap yourself.

Sampling seed

Seed of the 64 engines

2511

Model, units and sources

$k_{\mathrm B}T=4.142\,\mathrm{pN\,nm},\quad k=10\,\mathrm{pN}/\mu\mathrm{m},\quad \sqrt{k_{\mathrm B}T/k}=20.4\,\mathrm{nm},\quad \tau_r=\gamma/k=0.942\,\mathrm{ms}$

Each cycle starts in the steady state of the feedback loop. Work on the bead and heat released to the water are positive; information is in nats. The 64 engines run 60 cycles each with independent noise.

Sagawa & Ueda (2010), Eqs. 3–4 · Seifert (2012), Eqs. 151–163 · Sagawa & Ueda (2009), Eq. 4

Derive the information bound

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.

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 misconceptionExtracting work from water at one temperature breaks the second law.

Better mental modelThe 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

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.
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.
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.

Information Engine: Work From Measuring a Bead

A measurement can pay for work

Core mathematical model

Work of one instant shift

Mutual information of a Gaussian reading

Sagawa–Ueda bound

Best gain after a long wait

Common difficulties

Work from a single bath

Run the experiment

Watch one cycle

Overshoot

Be the demon

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