PYTHON · NUMPY

The physical price of a bit: Landauer

Find the exchange rate between information and heat.

01Challenge

Erase a bit at four different speeds and discover the floor you cannot get under.

Predict firstIs there a fundamental minimum energy to ERASE one bit of information?

02Model

So far a bit has been an abstraction. Landauer's insight is that it is not: erasing one bit at temperature T dumps at least kT·ln2 of heat into the world, no matter what device you use. Computation is free; forgetting is not. This is where the information spine touches the physical one — the same place every chip's power budget lives, and the reason the energy cost of intelligence is a physics question and not only an engineering one. The bound is thermodynamic, so it does not care about your transistor.

03Guided practice

1. The bit. Two states, equally likely: you do not know which. 2. The erasure. Raise one state's energy until everything ends up in the other, regardless of where it started. 3. The bill. Work is the level moving under its population, summed. 4. The floor. Do it slowly and the cost approaches kT·ln2 from above. Do it quickly and you overpay. Nothing gets under the line.

04Feedback

PASS WHEN PASS when no protocol costs less than kT·ln2 and the slow ones approach it.

If your run fails, check:

A cell errored — run it again.
Work should never fall below the bound, and should fall toward it as you slow down.

05Retrieve & space

What costs energy — computing, or forgetting? (forgetting; erasure is the irreversible step).
What is the bound? (kT·ln2 per bit erased).
Why does erasing quickly cost more? (you drive the system out of equilibrium and dissipate the difference).

06Mastery & project

You can state Landauer's bound and explain why it is a floor rather than an estimate.

The habit: every bit you throw away is heat you paid for.

← Bandwidth is a budget: Bode–Fano Learned matching: when AI designs the interface →

PYTHON · NUMPY

The physical price of a bit: Landauer

Erase a bit slowly and quickly, and read the thermodynamic bill.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you