In 1867 James Clerk Maxwell wrote a letter that has haunted physics ever since. Imagine, he said, a being small enough to see individual molecules, standing at a tiny frictionless door between two halves of a gas-filled box. It opens the door only to let fast molecules drift one way and slow ones the other. No work is done — the door is frictionless, the being only watches and flicks. Yet one side grows hot and the other cold, all by itself. You could run a heat engine on that difference, forever, drawing useful work from a single warm room. The Second Law of Thermodynamics — the iron rule that heat does not spontaneously flow from cold to hot, that entropy does not fall on its own — would simply break.
Maxwell called his creature a "finite being." Kelvin gave it the name that stuck: the demon. And for a long time no one could say exactly what was wrong with it, because nothing is wrong with the stipulated ideal mechanism. Within that thought experiment, the molecules separate and the door is free. In the Landauer-Bennett account illustrated here, the missing cost is in the demon's memory.
IThe simplest demon
In 1929 Leó Szilárd stripped the demon down to one moving part. Put a single molecule in a box at temperature T. Slide a partition into the middle. The molecule is now trapped on one side — but which? The demon measures: left or right. One bit of information. Knowing the side, it hooks a tiny piston to the partition on the molecule's side and lets the lone molecule's batterings push the piston outward. The gas expands from half the box back to the whole box, isothermally, the heat bath topping up the energy the molecule spends. Work comes out.
How much? For one ideal-gas molecule the pressure is P = kT/V, so an isothermal expansion from V/2 to V yields W = ∫P dV = kT·ln2. One measured bit, converted into kT·ln2 of work, drawn from a single heat reservoir. That is precisely the forbidden thing. Below, run the engine — and notice what happens if the demon inserts the partition off-centre.
The mean work you can extract is exactly kT·H(f), where H(f) = −f·ln f − (1−f)·ln(1−f) is the Shannon entropy of the measurement — the very same quantity that, in bits, measures how much you learned by looking. Push the partition to the wall and you learn almost nothing (the molecule is nearly always on the big side) and win almost nothing. The maximum sits dead centre, at a fair bit, f = 1/2: one full bit of information, W = kT·ln2. Information is not a metaphor here. It is the fuel, and you can read the exchange rate straight off the curve: one bit buys kT·ln2 of work.
So the demon works. A century of physicists checked the mechanism and it holds. Which means the Second Law is saved, if it is saved at all, somewhere we have not yet looked — in the part of the cycle we keep forgetting to count.
IIThe bill nobody counted
Here is the step everyone skips. To run the engine again, the demon must take its next measurement — but its memory already holds the last answer. Before it can write "left or right?" afresh, it has to clear the old bit. It has to forget. In 1961 Rolf Landauer derived the erasure bound used in this account.
The argument is pure bookkeeping of states. The illustrated one-bit memory starts with two equally probable possibilities, 0 and 1. Erasing means forcing it to a known value — say 0 — no matter where it started. Two possible states collapse into one: the memory's phase space halves, its entropy falls by k·ln2. Entropy cannot simply vanish, so by the Second Law itself an equal amount must appear in the surroundings as heat: at least kT·ln2 dumped into the environment on average, for complete erasure of this initially unbiased bit. This is Landauer's principle, and its slogan is that information is physical — a logical act of forgetting has an irreducible thermodynamic price.
Below is that erasure, modelled as an idealised physical process: an initially unbiased one-bit memory represented by a particle in a double well. The left well is 0, the right well is 1, and the protocol resets the memory almost entirely into the left well, well 0. The mean work it costs depends on how fast you push. Drag the speed and watch the cost fall toward kT·ln2 over the range shown. The finite ramp leaves a small residual probability in well 1; the check below accounts for that incomplete reset.
Across the speeds shown, the mean work to erase the initially unbiased bit stays at or above kT·ln2; the gold floor is never broken. Go quasistatically (large τ) and you pay almost exactly the floor; the excess cost of haste falls off as 1/τ, the price of dissipation. This is not just theory: in 2012 Bérut and colleagues built a physical double-well memory (a micron-sized glass bead in a double laser-trap well), erased it, and measured the mean heat creeping down to the kT·ln2 bound. For complete erasure of an initially unbiased bit in this memory, the average heat floor at 300 K is 2.871 zeptojoules.
IIIThe demon's books balance
Now put the two halves together, as Charles Bennett did in his 1982 resolution of the demon. The measurement itself, Bennett showed, can in principle be done reversibly, at no thermodynamic cost. So the demon's +kT·ln2 of winnings looks like pure profit. But the cycle is not closed until the demon is ready to measure again, and that means erasing the bit it just used, paying Landauer's ≥ kT·ln2 on average for complete erasure of the initially unbiased bit. In the reversible limit, the lunch and the bill are the same size. Per full cycle, the demon nets zero, or less. The Second Law was never in danger; we had just forgotten to charge the demon for forgetting.
Below are the demon's books. Choose a fair bit or a lopsided one, then flip the fatal switch: let the demon forget for free, the counterfactual assumption in this illustrated cycle with a memory, and watch the ledger tip into the impossible. For bias f, both the mean extracted work and the ideal mean erasure floor are kT·H(f); only the fair bit reaches kT·ln2.
With erasure honoured, the bars cancel: every cent the engine wins from a measured bit is paid straight back to clear that bit, and the net is never positive. Switch erasure off and the green bar stands alone — a machine that turns ambient heat into endless work, a perpetuum mobile of the second kind. In this ledger, removing the cost of forgetting creates the apparent net gain. The demon is not defeated by being unable to measure. It is defeated by being unable to remember forever — and by being unable to forget for free.
IVWhat is settled, and what is not
It would be tidy to stop there, and most popular accounts do. The Wasteland's rule is to show where the floor of a story is still being argued over — and this story has a live argument running underneath it.
What is settled: the bound is real. Complete erasure of an initially unbiased bit in the illustrated thermal memory costs at least kT·ln2 of heat on average. The state-counting is sound, and the Bérut experiment and its successors measured the floor directly. No one expects to build the demon. On that, everyone agrees.
What is open is subtler, and it is a question in the philosophy of physics, not the lab. Does Landauer's principle actually do the work of exorcising the demon? John Earman and John Norton argued (1998–99) that the standard erasure-based exorcism faces a dilemma: if you have already assumed the demon obeys the Second Law, you do not need the erasure argument; and if you have not, the erasure argument is not sufficient to force it. Norton went further (2005, 2013): the usual proofs of Landauer's principle, he contends, are not rigorous: they quietly assume the very thing they set out to establish, and lean on idealisations that thermal fluctuations destroy. On his view the demon is genuinely impossible, but for a different reason (a fluctuation argument descending from Smoluchowski (1912), generalised into a no-go theorem), and he accepts restricted erasure results while rejecting a general proof of Landauer's principle or its use to save the Second Law.
He has not had the last word. Ladyman, Presnell, Short and Groisman (2007) gave a fresh derivation of Landauer's principle from the bare Kelvin statement of the Second Law, and Ladyman and Robertson (2014) answered the circularity charge head-on; Bennett (2003) defended the picture too. In their joint 2023 clarification, Myrvold and Norton say their analyses are compatible under different background assumptions. So the honest shape of this page is: the engine runs, the bound is real and measured, and the question of why the demon ultimately fails is a genuine, unfinished dispute between two camps of careful people. We have shown you the parts that compute. The part that is still being written, we have left visibly open.
The Landauer-Bennett account leaves a strange little moral: observing, deducing and deciding can in principle be reversible, while clearing the memory carries the bill in this cycle. Norton disputes that exclusive role for erasure. For complete erasure of an initially unbiased bit in the illustrated thermal memory, the average heat cost is at least kT·ln2. Individual realizations can fall below that bound.
Show the check
Live instrument readouts use the deterministic engines checked offline in /research/maxwell-demon/verify.mjs (run it: node research/maxwell-demon/verify.mjs → 39 / 39 checks pass). Offline, the mean Szilard work W = kT·H(f) is cross-checked against the isothermal integral ∫P dV by quadrature. The two-state erasure starts with an unbiased bit and approaches kT·ln2 with finite-ramp and incomplete-reset corrections: the ramp free-energy change is ΔF = kT[ln2 − ln(1 + exp(−20))], while the full protocol also includes the restoration work −20kT·presidual. The tested speeds give mean work above kT·ln2, with excess approximately proportional to 1/τ in the tested slow range. An independent stochastic (Markov-jump) simulation and the molar conversion are checked offline. The laboratory figures use exact SI constants, including k = 1.380649×10⁻²³ J/K. What the check does not cover: the Second Law itself (assumed, as everywhere); the claim that real measurement can be made reversible (Bennett's argument, cited not simulated); and the philosophical dispute of §IV, which is reported, with both sides named, not adjudicated.
Sources & provenance
J. C. Maxwell, Theory of Heat (Longmans, Green & Co., 1871), §"Limitation of the Second Law" — the demon (first sketched in his 11 Dec 1867 letter to P. G. Tait).
W. Thomson (Lord Kelvin), "Kinetic Theory of the Dissipation of Energy," Nature 9 (1874) 441–444 — coins "demon."
L. Szilárd, "Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen," Zeitschrift für Physik 53 (1929) 840–856 — the single-molecule engine; information ↔ entropy.
R. Landauer, "Irreversibility and Heat Generation in the Computing Process," IBM J. Res. Dev. 5 (1961) 183–191 — the kT·ln2 cost of erasure.
C. H. Bennett, "The Thermodynamics of Computation — a Review," Int. J. Theor. Phys. 21 (1982) 905–940 — measurement reversible, erasure pays the debt.
A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, E. Lutz, "Experimental verification of Landauer's principle…," Nature 483 (2012) 187–189 — the floor, measured (colloidal bead, double-well optical trap).
J. Earman & J. D. Norton, "Exorcist XIV: The Wrath of Maxwell's Demon," Stud. Hist. Phil. Mod. Phys. 29 (1998) 435–471 & 30 (1999) 1–40 — the "unnecessary-or-insufficient" dilemma.
J. D. Norton, "Eaters of the Lotus," Stud. Hist. Phil. Mod. Phys. 36 (2005) 375–411; "All Shook Up," Entropy 15(10) (2013) 4432–4483 — the circularity charge and the fluctuation no-go.
J. Ladyman, S. Presnell, A. J. Short, B. Groisman, Stud. Hist. Phil. Mod. Phys. 38 (2007) 58–79; J. Ladyman & K. Robertson, "Going Round in Circles," Entropy 16 (2014) 2278–2290; C. H. Bennett, Stud. Hist. Phil. Mod. Phys. 34 (2003) 501–510: the defence; Myrvold & Norton's joint 2023 clarification says their analyses are compatible under different background assumptions.