physics//thermodynamics//Landauer's principle

Landauer's principle states that erasing one bit of information dissipates at least \(kT\ln 2\) of heat, where \(k\) is Boltzmann's constant and \(T\) the temperature, and it is the thermodynamic floor under the energy cost of computing. At room temperature (300 K) that is about \(2.9 \times 10^{-21}\) J per bit.

Landauer's principle states that erasing one bit of information dissipates at least kTln⁡2kT\ln 2kTln2 of heat, where kkk is Boltzmann's constant and TTT the temperature, and it is the thermodynamic floor under the energy cost of computing. At room temperature (300 K) that is about 2.9×10−212.9 \times 10^{-21}2.9×10−21 J per bit.

Erasing a bit merges two possible states into one, which lowers the entropy of the memory; the second law requires that entropy to reappear in the surroundings as heat. Where Shannon's entropy counts the bits a message carries, the principle prices in joules the forgetting of one of them. The bound was measured in 2012 by erasing the state of a single colloidal particle held in a laser trap.

Real hardware is very far from it. A data-centre GPU spends roughly 10−1310^{-13}10−13 to 10−1210^{-12}10−12 J per floating-point operation, about eight orders of magnitude above the bound per operation, and still five or six if each operation erases a few hundred bits.

The principle limits erasure, never computation as such. A computation that erases nothing has no minimum cost in principle (reversible computing).

Like the Carnot efficiency, which follows from the same second law and bounds how much heat can become work, it is a bound on efficiency: it says how cheap a bit erasure can be, never how many can be done in total.