Maxwell's Demon and Information

Equivalent forms

\Delta S_{\text{memory}} \geq k_B \ln 2

W_{\text{extract}} \leq k_B T \ln 2 \text{ (Szilard)}

The thought experiment that revealed information IS physical — a bit of knowledge has a thermodynamic price of k_B T ln2.

Symbol	Name	SI unit	Dimension	Range
$W$	Erasure workoutput Minimum work to erase the demon's memory	J	M L² T⁻²	0 – 1e-18
$N$	Bits erased Number of information bits the demon must erase	dimensionless	1	0 – 200
$T$	Temperature Temperature of the demon's environment	K	Θ	100 – 600

Unit systems

SI:: W in J
natural:: in units of k_B T
CGS:: W in erg

Where it holds

Resolution holds for any physical implementation of memory; Landauer's bound has been verified experimentally in colloidal and electronic single-bit systems.

Discovery

James Clerk Maxwell; resolved by Szilard, Landauer & Bennett · 1867

Maxwell posed the demon in 1867; Szilard (1929) linked it to information, Landauer (1961) showed erasing a bit costs kT ln2, and Bennett (1982) closed the case using the demon's memory.

Try this

Can a clever gatekeeper break the Second Law?

Maxwell imagined a tiny demon sorting fast molecules from slow ones to create a temperature difference for free. It took 150 years and the physics of information to exorcise it.

Research status: active

Common misconceptions

The demon is not defeated by the energy of measurement — measurement can in principle be reversible; the irreducible cost is ERASING the accumulated information to reset the demon.

Derivation

A Szilard engine extracts

W = k_B T \ln 2

from a single molecule once the demon knows which half it is in.

But storing that one bit and later erasing it (to reset for the next cycle) compresses the memory's phase space by a factor 2, decreasing its entropy by k_B ln2 — which by the Second Law requires dumping at least k_B T ln2 of heat.

The books balance exactly: no net work.

Limiting cases

Demon never erases memory⟶ Memory fills up; once finite, it must be reset, paying k_B T ln2 per bit

T \to 0

⟶ Erasure cost

\to 0

, but the Third Law makes reaching

T = 0

impossible

One-molecule Szilard engine⟶ Extracts exactly k_B T ln2 of work per cycle — precisely cancelled by the measurement/erasure cost

Second Law of Thermodynamics→Boltzmann Entropy→Entropy Change→Canonical Partition Function→