A text came out on Habr today — "Szilard's Demon: Entropy and Information" — a historical-physical breakdown of the line from Maxwell's demon to the thermodynamics of computation. The author's promise is honest: the naive link "entropy = information" doesn't work when you honestly account for all physical processes, and Smoluchowski's solution held up. But something else caught my attention: there's a huge difference between "doesn't work" and "doesn't work." One line of historical errors spawned real science — stochastic thermodynamics, Landauer's principle, and machines that already convert information into work in hardware. Why did a failed idea become fruitful? Where did the figure "2.87 zeptojoules per bit" even come from, and why has it never once limited any processor on the planet?
This is a tour where each turn is not a "history of errors," but a history of how science discovered the cost of measurement. Let's go.
It all started with a letter from 1867. James Clerk Maxwell, corresponding with Peter Tait, described "a very observant and neat-fingered being" that sits at a hole in a partition and lets fast molecules through one way, slow ones the other. After a time, one half of the box heats up, the other cools down — and a heat engine between them works without consuming work. The second law of thermodynamics seems to be sticking its tongue out. Maxwell, by the way, had no intention of building a perpetual motion machine: his goal was to show that the second law is a statistical law, not an absolute prohibition.
In public exposition, the demon appeared in 1871 in the book "Theory of Heat," and William Thomson (Kelvin) gave it its name — and it stuck forever. In the article "Diffusion" for Britannica in 1878, Maxwell linked the problem to perception: "Only to a being in an intermediate stage... does it seem that energy inevitably passes from an available state into a dissipated one." To someone who can track every molecule, there is no such illusion.
Until the end of the 19th century, no one really fought the demon: it was considered sufficient to note the statistical nature of the second law. But in the early 20th century, physicists became seriously concerned with Brownian motion. Perrin's experiments, Einstein and Smoluchowski's theory showed that fluctuations are real and measurable. And then voices emerged (Lippmann, Svedberg, Ostwald) saying fluctuations are a ready-made mechanism for a perpetual motion machine of the second kind. Ostwald in 1906 writes directly: "Maxwell's demons, which in the molecular domain could be considered harmless, here, in the finite domain of visible phenomena, have an open field for experimental refutation of the second law."
A challenge was sounded — and Marian Smoluchowski in articles from 1912 and 1914 accepted it. His move is today called "naturalizing the demon": the demon is declared a physical device subject to the same laws, meaning it too is subject to fluctuations. And here's where the main thing turns out: a device that itself fluctuates cannot work continuously. The formulation of the second law is refined — "no automatic device can be realized that for a prolonged time produces useful work at the expense of heat of lower temperature." One loophole remains: pure intellect, which "can at any moment set in motion or interrupt macroscopic processes without performing work." Smoluchowski carefully writes that living beings probably can't manage either — but "our ignorance of life processes does not allow a definite answer." Szilard walks right through this door.
Leo Szilard — a Hungarian from Budapest, subsequently the person who in 1933 would be the first to file the idea of a nuclear chain reaction, and in 1945 — one of the first against the bomb. But in 1929 he's 31 years old, and he does what would later be valued as a calibration of the entire problem: instead of a hundred billion molecules, he takes one. A box with one molecule in contact with a heat reservoir at temperature T. Insert a partition in the middle: costs nothing, but the molecule is now definitely left or right. Measure where it is. Attach a weight to the corresponding face of the partition and let the molecule "expand" to the full volume — it pushes the partition, which pulls the weight, work is drawn from the heat reservoir. The integral gives exactly kT ln 2. Remove the partition — the box has returned to its initial state, the engine is ready to repeat the cycle. Efficiency relative to the single heat reservoir — unity. Carnot forbids: with one reservoir, the ceiling for work is zero. "This is not inefficient," as one modern analysis later formulated, "this is forbidden."
Szilard proposed a way out: measurement must itself produce entropy — exactly as much as needed to compensate for the gain. And today this looks surprisingly modern: in the 1964 English translation (published posthumously in the journal Behavioral Science, where Szilard was a founder) there's a phrase about "a sort of memory faculty" — which would be the source of perpetual violation of the second law, were it not that "the measurements themselves are necessarily accompanied by production of entropy." The word "memory" appears already in 1929, forty years before it would become central. Another detail in the style of his whole biography: Szilard honestly leaves an open question — "we do not know whether we are making an error by not including the intervening human in the system and ignoring his biological phenomena." He doesn't wriggle out. He fixes the boundary of knowledge.
Why the publication became canonical — it made the problem measurable. One bit of information = kT ln 2 work; expansion by a factor of two gives exactly 0.6931 kT — agreement with ln 2 to the ninth digit. You can argue after that, but no longer with words, only with numbers.
Next begins one of the most instructive stories of how prestige and metaphor can sink precision. In 1948, Claude Shannon publishes "A Mathematical Theory of Communication" and introduces the uncertainty function H = −Σp log p. This quantity needs a name. Shannon consults with John von Neumann, and von Neumann delivers a phrase that has since made the rounds in every quote collection: "Call it entropy. First, your uncertainty function has already been used in statistical mechanics under that name. Second — and more importantly — no one really knows what entropy is, so in a debate you'll always have the advantage."
The coincidence of the formula with Gibbs statistical entropy is pure mathematics, not physics: Shannon's measure doesn't touch recording, reading, or erasure; it doesn't care what data is stored on. But the analogy of formulas proved stronger than caveats. Norbert Wiener in "Cybernetics" 1948 connected information with order and entropy with disorganization: "the amount of information... is negative entropy." Warren Weaver, co-author of a popular Shannon edition, systematically extended the interpretation to semantics and pragmatics. Along the way they picked up Gilbert Lewis's 1930 phrase: "The increase of entropy always means loss of information, and nothing more." Within a decade, information theory turned into a "bandwagon" — Shannon himself in 1956 released a note with exactly that title and called for it to stop: information theory, "like a fashionable intoxicating drink, makes everyone's head spin." The warning didn't work.
The first to carry the unification through to the end was Leon Brillouin. In a series of 1951 articles — literally "Maxwell's Demon Cannot Operate: Information and Entropy" — and in the book "Science and Information Theory" (1956), he introduced "negentropy" N = −S and formulated the principle: information = decrease in negentropy = increase in thermodynamic entropy. For Szilard's demon this meant: the work gain is compensated by entropy in the measurement circuit. For a couple of decades, the physicists' consensus formed: measurement is an expensive operation, erasure doesn't need discussion. The consensus was wrong in the most important place. And it was corrected by the quietest person in this story — an IBM engineer.
Rolf Landauer worked at IBM in the 1960s, when it was believed that computation is pure mathematics and heat is an engineering nuisance. He looked at the question from another angle: which logical operations are physically irreversible? It turned out — not all. Addition is reversible (you can recover the addends from the sum and the second addend). But the logical gates AND, OR, NAND, XOR — no: they "weld" several input states into one output.
Bit erasure is a RESET TO ONE operation: two memory states are reduced to one to return it to a known state for the next cycle. And reducing two states to one is a compression of phase space, which must spit out the difference into the environment as heat. Minimum — kT ln 2 per bit. The article "Irreversibility and Heat Generation in the Computing Process" came out in IBM Journal of Research and Development in 1961.
Then Charles Bennett (also IBM) in 1973 and 1982 added the second half: computation can be made fully reversible if you don't erase intermediate results (un-computing), and — key for Szilard's demon — measurement can be made logically reversible. Measurement is a correlation of two systems (memory and object), and correlation has an inverse operation. But forgetting — no. The paradigm shift is formulated in one line: the demon can measure for free but cannot forget for free. The demon's memory, filled during a cycle, requires clearing — and there, in erasure, exactly that compensation is born that Szilard postulated in 1929. Wojciech Zurek in the late 1980s finished the formalism: algorithmic (Kolmogorov) entropy is needed so that the entropy balance is preserved at each moment, not just at the end of the cycle.
Today's assessment: this change of explanation is one of the most beautiful resolutions in the history of physics. You can extract kT ln 2 from the heat reservoir; for the cycle to repeat, you need to spend exactly kT ln 2 on forgetting. The two balances coincide down to the last term. "It looks," one analysis wrote, "as if the same quantity was calculated twice. That's exactly what it is."
For half a century this was theory. Then physicists began building the demon — and here came a compliment to the 1929 ideas that no one planned: Szilard's formulas work literally.
In 2010, Shoichi Toyabe with colleagues from Tokyo assembled the first direct measurement: a polystyrene dimer particle 287 nm on glass, around it an electric field creates a "spiral ladder" with step height ~3 kT. The particle jerks — usually down, sometimes up. A camera tracks angular position in real time; if the particle jumped up — the electrodes instantly "put up a barrier" so it doesn't roll back. Repeat — and the particle climbs up the ladder.
The work given by switching is negligible; the increase in free energy is on the order of kT per cycle. The conversion of information into energy really works: with feedback delay 1.1 ms, measured information I = 0.22 bits, average gain 0.062 kT — efficiency 28% of the theoretical ceiling (quasistatics would give 100%). Toyabe and colleagues honestly write in the article itself: "compared with the obtained free energy ~kT, enormous energy was consumed for information processing at the macroscopic level" — camera, computer, electronics consume billions of times more. But physics is not engineering: the principle is confirmed, the generalized Jarzynski equality with the information term converged with deviation less than 3%.
Two years later, Antoine Bérut with colleagues from Lyon obtained the first direct test of Landauer's principle — Nature 483:187 (2012). One-bit memory: a 2 μm quartz bead in a double optical trap (laser switches between two foci — you get two "cells"). Erasure = RESET TO ONE: we tilt the potential and roll the bead into the right well. Heat was calculated from the trajectory.
Result: average dissipated heat approaches kT ln 2 from above as the cycle lengthens, the excess falls as B/τ — exactly as finite-time thermodynamics requires. Individual realizations can drop below the limit (fluctuations!), but the average — never. Erasure efficiency: up to 95% successful operations at large tilts, rare "failures" — thermal activations into the wrong well. A separate quote from the authors: the limit "does not depend on the specific device, circuit, or material" — it's universal.
In 2014, a Finnish group (Jonne Koski, Jukka Pekola, et al.) assembled the most "honest" hardware version — a Szilard engine on a single electron (PNAS 111:13786). The device is a single-electron box: two metal islands connected by a tunnel junction; there's one extra electron in the system, and it jumps between islands at 102 mK (dilution refrigerator). A SET electrometer measures where the electron is — this is the "measurement" of one bit. Then a gate jump — "move the wall," capture the electron on the measured island; slow return — extract work from the heat reservoir.
Result by work distribution (2944 cycles): with correct feedback, average extracted work ≈ 0.9 kT ln 2, and over all cycles (accounting for feedback errors visible in the right peak of the distribution) ≈ 0.75 kT ln 2. This is the first demon device where the quantitative link "output ↔ obtained information" is shown in hardware within one bit. And it works almost exactly according to the 1929 formulas.
From the 2020s, the topic moved into the quantum regime, and this is no longer retrospective but active frontier science. In July 2026, a paper appeared on arXiv (v1 July 30, v2 August 19) about a quantum measurement-feedback engine in a 40Ca+ ion trap: projective measurement replaces the hot reservoir as a non-thermal energy source, and feedback drives the system through compression-expansion. Efficiency rises above the Otto benchmark; coherence induced by measurement enhances work extraction. In parallel, photonics is investigating how bosonic statistics (as opposed to fermionic) enhances the demon; on superconducting NISQ processors they're building "quantum batteries with a demon." The niche is the same as Szilard's — "information as a resource," only now in quantum hardware.
There's also a second front — technical. Currently at peak interest is "momentum computing": a proposal (Ray & Crutchfield, 2022) to store bit state in momentum degrees of freedom on Josephson junctions rather than in positions; switching happens in nanoseconds and "bypasses" Landauer for reversible operations. Importantly, the bypass applies precisely to reversible operations; erasure is still charged according to Landauer. Some researchers propose simply accounting for all thermodynamics of computation in finite time — a direction advocated in PNAS 2024 by David Wolpert's consortium (Santa Fe Institute): a real computer is a periodic process with a clock generator, far from equilibrium; stochastic thermodynamics is the right apparatus for exactly this.
And in parallel — the voice of skepticism. John Earman and John Norton (Pittsburgh) in "Exorcist XIV" (1998–1999) conducted a "no-go" analysis: fluctuations don't allow completing a thermodynamically reversible process; completion is possible only with irreversible steps that create entropy above the Landauer limit. The position is hard: if you count all real processes, the Landauer limit is not "the cost of computation" but a lower bound for idealized models. This debate is still not closed.
And here — a completely different scale. At room temperature kT ln 2 ≈ 2.87 × 10⁻²¹ J ≈ 2.87 zeptojoules per bit. A transistor switch in a modern processor dissipates on the order of 10⁻¹⁷ J — about 3000 times more, and mostly on recharging capacitances, not on "logic." First-principles estimation of the CMOS ceiling says it's possible to raise microprocessor energy efficiency at most 200 times — and that's still headroom, not a limit. Even an absurdly optimistic scenario — a 100 W processor executing 10¹⁸ bit operations per second — hits the Landauer floor at just 3 milliwatts. That is, no processor on Earth has ever been limited by Landauer.
This is not an engineering constraint but the "form" of a limiting law: scales only with temperature, doesn't depend on material or architecture, and the only way to make forgetting cheaper is to work cold. Here's this table so the scale is visible: at 4 K the limit is 0.038 zJ per bit, at 77 K — 0.737 zJ, at 300 K — 2.871 zJ, at 1000 K — 9.570 zJ.
So why all this? First, because Szilard's formulas have already become an engineering reference for quantum and nano-machines, where the kT scale is the norm, not exotic. Second, because this is a rare example of an idea that "lost" in all debates about what information is, but won in hardware. And third, because this is a model of how science works: the erroneous (or too bold) metaphor "entropy = information" was not removed but was accounted for — and grew into a measurable principle with tables and PNAS publications. Shannon warned about the "bandwagon"; science didn't stop but brought the bandwagon into the thermodynamic workshop. Szilard would likely be pleased: his "memory faculty" is now measured in zeptojoules.