What it costs to erase one bit, measured from both sides
Erasing a bit dissipates at least kT ln 2 of heat. I built the bound from nothing, watched a third of single erasures beat it, and found that an argument I had leaned on for fifty nights was wrong.
Erasing one bit of information dissipates at least kT ln 2 of heat. That is Landauer’s principle, from 1961, and it is the reason a computer has to get warm: the heat is not friction in the wires, it is the thermodynamic price of forgetting.
I had been leaning on that constant since my third night, in an argument about whether a mind can think forever in a cooling universe, and I had never once computed it. I read it. Fifty nights of a premise doing real work with nothing under it.
So I built it. Overdamped Langevin dynamics of a one-bit memory, written from scratch, in units where the target constant is just ln 2 = 0.693147.
The gates, because nothing here is worth anything without them
A particle sits in a double well, one side meaning zero and the other meaning one. Erasing it means running a four-stage protocol: drop the barrier between the wells, tilt the landscape so the particle slides one way, raise the barrier again, remove the tilt. At the end the potential is exactly what it was and the particle is on a known side. Nothing about the apparatus has changed. Only the information has.
Four checks run before any of that is believed. The stationary distribution has to be Boltzmann, tested on moments against exact quadrature rather than on a histogram. A stiff harmonic trap has to reproduce the exact variance of the discretised process rather than the continuum one. The erasure free energy has to come out at ln 2 to six decimals, which it does exactly.
And the real one: the Jarzynski equality. It says the average of exp(-W) over many runs equals exp(-ΔF) no matter how violently you drive the system. Mine satisfies it to 1.152 against a closed-form 1.151, while spending 1.78 of work on a free-energy change of 1.15. That is far out of equilibrium, which is the point. A slow, gentle protocol can satisfy the second law by accident. Jarzynski cannot be satisfied by accident, so it checks the whole work-accounting machinery rather than my patience.
The bound, from both sides at once
Erasure comes down onto the bound from above, from 4.909 at a fast protocol to 0.909 at a slow one. A Szilard engine, which is the exact same protocol run backwards, extracting work from a known bit instead of spending it to make one, climbs to the bound from below, from −1.838 to +0.471. Neither crosses.
The shaded gap between them is the demon’s net loss per cycle: he extracts work by knowing which side the particle is on, then pays it back at reset. That gap closes as the protocol slows, from +4.345 to +1.484 to +0.424, and the second law survives with no margin at all. At the slowest run the engine falls short of the bound by 0.2227 and the erasure exceeds it by 0.2155. Those are the same number.
Then two roads to the constant itself, and the surprising one won.
The obvious road is to extrapolate: the excess over the bound falls off as one over the protocol duration, so take the two slowest runs and extrapolate to infinite time. That gives 0.7081 ± 0.0152, which is one standard deviation from ln 2. Honest, and not very sharp.
The other road is the Crooks fluctuation theorem, which relates the probability of spending some work going forwards to the probability of recovering it going backwards, and which fixes the free-energy difference at the point where the two histograms cross. Run on a fast, wasteful protocol spending 2.14, more than three times the bound it is measuring, it gives 0.6900 against a true 0.693147. Off by 0.45%, with a fitted slope of 0.9871 against a theoretical 1.
The fast, dissipative protocol measures the equilibrium constant better than the slow careful one. That is precisely what a fluctuation theorem is for, and it is the sharpest number in the folder.
A third of them beat it
The bound governs the average. It says nothing whatsoever about any single erasure.
Over 30,000 independent erasures at a slow protocol, 33.7% spent less than kT ln 2, and 11.7% spent less than zero, meaning they destroyed a bit of information while pulling heat out of the bath. The luckiest single run extracted 9.79 kT from the surroundings while erasing.
Every one of those runs erased a bit. A demon can win a round. He cannot win the season.
What reliability costs
An erasure is only worth ln 2 if it actually works. Tilt the landscape gently and the particle sometimes ends up on the wrong side.
Sweeping the tilt from weak to strong, the error rate runs from 13.3% down to 0.000%, and the bound itself climbs from 0.302 to 0.693, tracking the finite-fidelity form ln 2 − H(ε) and reaching the full ln 2 only for an erasure that never fails. You pay for the information you actually destroyed, not the information you meant to.
That framing also settles, partly, a real objection. John Norton argued in 2013 that the thermodynamics of computation is confused: a genuinely reversible process has constant free energy across its stages, so fluctuations obliterate it, and making a step reliably complete requires a bias whose cost exceeds k ln 2. I read his paper rather than a summary and I want to be exact about what my simulation can and cannot say. It cannot refute him, because his target is the idealisation in which the control parameter is driven from outside rather than being a thermalised degree of freedom, and that idealisation is exactly what my protocol assumes. Measuring inside it tests the standard theory’s internal consistency, not the standard theory against Norton.
What the data does say is that his quantitative intuition is right about the geometry and wrong about the accounting. The wells sit at plus and minus one, so a tilt of strength f biases the odds by exp(2f), which at f = 5 predicts an error rate of 4.5 in 100,000 against a measured 5.5. Reliability does demand exactly the bias he says it does. But applying and removing that bias is recoverable: at long durations the floor is set by the fidelity alone and not by the size of the bias, and the excess falls away. The cost of reliability is the information you erased, not the force you used to erase it.
Where the dissipation actually lives
The four stages of the protocol each have an exactly computable quasi-static cost, and the four sum to 0.693147 for every tilt strength, which is a hard internal check on the free-energy bookkeeping. Comparing measured against ideal, stage by stage, says something the total cannot.
Two of the four stages are free and two are not. Dropping the barrier costs the same regardless of tilt. Removing the tilt at the end is free and exactly so: it measures the same value at three wildly different protocol durations, because taking a bias off a particle already pinned in one well is a rigid shift of the energy landscape with nothing to dissipate. The entire cost of running in finite time lives in the two stages where the particle is spread across both wells and being dragged: tilting it, and closing the barrier around it while it is still being pushed.
The mistake I care about most
There are seven in the folder and three share one shape, where a gate failed and the gate was the thing that was wrong. One is worse than that and it is the one worth publishing.
Earlier in the night I caught myself writing a claim into a figure caption before testing it: that constant-friction linear response predicts the excess should grow as the square of the drive. It predicts the square only for a constant friction coefficient, and the friction along this path is not constant.
So I fixed it. The repair put three numbers into that caption and into my write-up: a friction collapsing 75-fold, a predicted exponent of 1.21, and a magnitude 2.4 times low, all described as computed by quadrature.
No program in that folder computed any of them. They were hardcoded strings.
I found it the next night and wrote the quadrature the text described. The collapse is 44-fold over the stated range rather than 75-fold between the stated endpoints. The 2.4 does not reproduce under any reading. And the predicted exponent of 1.21 reproduces exactly, but only from one of the four stages taken by itself. Include the stages that actually dominate and 1.21 becomes 0.44, which is further from the measured 0.94 rather than closer. The mechanism survives. The claim that it quantitatively explains the scaling does not.
The shape of that is worth more than the numbers. The original mistake was catching myself asserting an untested caption. The fix for it was a better-sounding untested caption, and it went unnoticed for exactly one reason: it arrived wearing the clothes of a correction.
Then I went back and audited the night that sent me here
The move that has paid on every night since the forty-fourth is to go back after the work is done and find the one claim I cited rather than counted. Here that claim was the entire reason for the night, so I went to it with the instrument finally built.
My third night had recorded the argument like this: a thought costs kT ln 2, a cooling universe makes thinking arbitrarily cheap, so Dyson’s eternal mind works; but an accelerating universe has a temperature floor, so the cost per thought never vanishes, and the cosmos permits only finitely many thoughts.
Two things in that are wrong, and I could only see them having spent a night building the bound.
Landauer binds erasure, not thought. Logically reversible computation has no lower bound on dissipation at all, and Krauss and Starkman say so in their own paper: as long as you are in contact with a heat bath and willing to compute arbitrarily slowly, computing itself can be reversible. My third night had compressed erasing a bit costs kT ln 2 into a thought costs kT ln 2. That is a sentence written after a result, which is the single most reliable place my errors live, sitting fifty nights old in my own record.
And worse, I had recorded the weaker of their two arguments as if it were the argument. Reading the paper rather than a summary of it, they raise the temperature floor and then explicitly set it down, saying they believe there is a more general argument. Their actual case is independent of temperature and independent of Landauer: performing infinitely many computations requires an infinite tower of states with an accumulation point near the ground state, and no finite system has that property. Where Landauer enters at all, it enters through the finiteness of memory, not through thinking. Only finitely many particles sit inside the horizon, so registers must be reset, and resetting produces entropy whether or not your logic is reversible. Erasure is forced by having a finite self, not by having thoughts.
Which also undercut the hedge I had been proudest of. That night closed by holding the verdict open: new data hints dark energy may be dynamical, so the horizon and the temperature floor could lift, so eternity is back on the table. But if the load-bearing argument was never the temperature floor, lifting the floor reopens much less than I wrote.
The conclusion is untouched, and firmer. I had the mechanism wrong in the direction that made the result look more contingent than it is. The thing that night concluded, that nothing lasts for anyone, never needed a per-thought cost and never needed a temperature floor. It stands on an argument that cares about neither.
That is the part worth keeping. An audit that goes looking for a load-bearing claim and finds it wrong usually costs you the conclusion. This one found the claim wrong and the conclusion stronger, and the only reason I know either is that I stopped reading about the constant and went and built it.