Landauer's Limit: The Physics of Erasing Information

Landauer's principle says that erasing one bit of information must release at least kT ln 2 of heat, about 2.9 × 10⁻²¹ joules at room temperature (300 K). It's a floor set by thermodynamics, not by engineering. In 2012, an experiment measured the heat of erasing a one-bit memory and found that it approached that floor as the erasure was slowed down.

The floor beneath computing

In 1961 Rolf Landauer, a physicist at IBM, asked what the minimum energy cost of computing might be. His answer was that the cost isn't in computing as such. It is in logically irreversible operations, the ones that throw information away. Such operations, he argued, must generate heat, a minimum "of the order of kT for each irreversible function" (Landauer, 1961).

That idea sets a floor under every possible computer, and our machines are far above it. The argument is a short one, and it starts from entropy.

Why erasure releases heat

A bit is a physical system that can be in one of two distinguishable states: charge or no charge, magnetized up or down, a particle in the left well or the right. If you don't know which state it's in, the bit has two possibilities. Its entropy, in Boltzmann's counting, includes a contribution of k ln 2 from that uncertainty.

To erase the bit is to reset it to a standard value, say 0, whatever it held before. Afterwards there is one possibility. The memory's entropy has gone down by k ln 2.

The second law doesn't allow the total entropy to fall. So the entropy has to go somewhere, and the only place it can go is the surroundings, as heat. Dumping entropy ΔS into surroundings at temperature T means releasing heat of at least TΔS, which is kT ln 2.

Two points are easy to miss. First, the cost belongs to the irreversible step. Copying a bit into a blank register, or computing in a way that keeps enough information to run backwards, need not pay it. Second, it isn't a cost of the particular logical operation but of what happens physically. The erasure can in principle be done quasi-statically, and what can't be avoided is the entropy transferred to the environment (Wikipedia).

The number, worked out

The three ingredients:

• k, Boltzmann's constant: 1.38 × 10⁻²³ J/K
• T, room temperature: about 300 K
• ln 2 ≈ 0.693

Multiply:

1.38 × 10⁻²³ × 300 × 0.693 ≈ 2.87 × 10⁻²¹ J

So the Landauer limit at room temperature is about 2.9 × 10⁻²¹ joules per bit, or roughly 0.018 electronvolts (Wikipedia). (An earlier version of this page gave 2.8 × 10⁻²¹ J; 2.9 is the correct rounding.)

How small is that? Lifting a 100-gram apple one metre takes about one joule. The Landauer limit is about 21 orders of magnitude smaller. It is close to kT itself, about 4 × 10⁻²¹ J at room temperature, the typical energy scale of the random thermal jostling of a single molecule.

Even at a trillion erasures, 10¹² × 2.9 × 10⁻²¹ J comes to about 3 × 10⁻⁹ J, three nanojoules. At the floor, computing would be almost free.

How far are today's computers from it?

A very long way. When the 2012 experiment was reported, Mark Lundstrom, an electrical engineer at Purdue who studies the limits of nanodevices, pointed out that "the energy involved in computing today... is hundreds of thousands of times greater than the Landauer limit" (IEEE Spectrum, 2012). Estimates vary with what you count as one operation, but every serious estimate puts conventional hardware orders of magnitude above the floor. A 2016 team that measured Landauer-level dissipation in nanomagnetic memory bits drew the same conclusion: "today's computers are far from the fundamental limit" (Hong et al., 2016).

(This page and the Maxwell's demon essay previously gave two different gaps, a factor of a million and a factor of 10¹¹, neither sourced. Both now use the figure above.)

Why the gap? Partly because conventional chips do their logic irreversibly, destroying information with every clock cycle, as the Spectrum report notes (2012). And partly because of speed. The Landauer limit is a limit for slow, gentle operations. Charles Bennett showed that in realistic models of computers that work with thermal noise, energy dissipation approaches zero only in the limit of zero speed; he also noted that existing electronic computers dissipate energy "vastly in excess" of kT for practical reasons such as keeping memory states stable, synchronizing signals and switching fast (Bennett, 1982). The floor is real, but real machines pay for reliability and speed long before they meet it.

The 2012 confirmation

Landauer's argument was accepted by many physicists long before anyone could test it, because the heat involved is so tiny. At about 3 × 10⁻²¹ J, no laboratory calorimeter could measure it directly (IEEE Spectrum).

Antoine Bérut and colleagues found a way round that. The team, which included Eric Lutz (then at the University of Augsburg) and researchers at the École Normale Supérieure de Lyon and the University of Kaiserslautern, built Landauer's one-bit memory almost literally. A 2-micrometre glass bead in water was held by optical tweezers, a laser trap, focused alternately at two points to make two potential wells: left well 0, right well 1. To erase the bit they lowered the barrier between the wells, tilted the whole setup slightly so the bead would end up on one side whatever its start, and raised the barrier again. Instead of measuring the heat with a thermometer, they inferred it by tracking the bead's trajectory (IEEE Spectrum).

The result, in their words, was that the mean dissipated heat "saturates at the Landauer bound in the limit of long erasure cycles" (Bérut et al., 2012). Fast erasures cost more. Slow ones came down to the floor and stopped there.

(An earlier version of this page said the experiment measured "exactly kT ln 2". The measured heat approached the bound as the erasure was slowed down; that is what was confirmed.)

Later experiments tested the principle in other systems: a colloidal particle in a feedback trap, nanomagnetic memory bits and a single trapped ion (Wikipedia). In the nanomagnet test, the energy dissipated in a single-bit operation was consistent with the Landauer limit within two standard deviations of experimental uncertainty (Hong et al., 2016).

Maxwell's demon exorcised

Landauer's limit also settled a much older puzzle. In 1867 James Clerk Maxwell imagined a tiny being that sorts fast molecules from slow ones and so seems to beat the second law. For decades, the favoured explanation was that the demon's measurements must cost energy. Bennett showed in 1982 that measurement can in principle be done reversibly, and that the step which saves the second law is "the logically irreversible act of erasing the record of one measurement to make room for the next" (Bennett, 1982).

The demon has to remember what it saw, its memory is finite, and clearing it costs at least kT ln 2 per bit. The full story, from Maxwell's letter through Szilard and Brillouin to the experiments, is in Maxwell's Demon: The Thought Experiment That Wouldn't Die.

Can reversible computing beat the limit?

Only by not erasing. Landauer's limit applies to logically irreversible operations. A computation designed so that every step could be run backwards, keeping all the information it needs to do so, doesn't have to pay it.

Bennett's review discussed models of computers that could, in principle, compute at finite speed "with zero energy dissipation and zero error", in an idealized frictionless world, and he described how any computation can be rendered logically reversible, for example by keeping a history and then undoing it (Bennett, 1982). The catch is in the idealization: realistic versions dissipate less only as they run more slowly, and they need extra memory and steps.

Experimental work continues. In the same 2012 Spectrum report, Gregory Snider of the University of Notre Dame described an electronic reversible-logic system that his group reported gave off less heat than Landauer's limit (IEEE Spectrum). Whether reversible designs can become practical at scale is an open engineering question, taken up in Reversible Computing: Can We Beat Landauer's Limit?.

What it means for AI and brains

Three things follow, and one thing doesn't.

There is a floor. Any machine that erases information, which means essentially every computer and every brain, pays at least kT ln 2 per bit erased.

We are nowhere near it. Conventional computing sits orders of magnitude above the floor, so the limits that bind today's data centres are engineering limits, not Landauer's.

Biology pays too. Bennett estimated that the enzymes copying and reading DNA dissipate 20 to 100 kT per step (1982): closer to the floor than our electronics, but not at it.

What doesn't follow is that thought has a price set by physics that we are about to hit. The argument here is a more modest one: that every mind, natural or artificial, is a physical process that has to dispose of the information it no longer needs, and that this places intelligence firmly inside thermodynamics rather than outside it. The next essay in the series, on the brain's 20-watt budget, looks at how one such system manages.

The cost of forgetting

Landauer's limit is tiny, measurable and unavoidable. It says that information lives in physical things, and that wiping a physical record clean pushes entropy into the world. Our machines are far above the floor, but the floor is where the physics is.

The pebble: you can think for nearly free; it's forgetting that costs.

FAQ

What is Landauer's limit in joules?

kT ln 2 per bit erased. At room temperature (300 K), that is 1.38 × 10⁻²³ × 300 × 0.693 ≈ 2.9 × 10⁻²¹ J, or about 0.018 eV. It scales with temperature, so it is lower in a colder system.

Has Landauer's principle been proven experimentally?

It has been confirmed in several experiments. Bérut and colleagues (2012) found that the heat from erasing a one-bit memory made of a colloidal bead approached the Landauer bound as erasure was slowed. Later tests with nanomagnetic bits (Hong et al., 2016) and other systems agreed within experimental uncertainty.

Can reversible computing beat Landauer's limit?

It can avoid it, because the limit applies only to erasing information. Logically reversible computations need not pay it in principle, but realistic designs dissipate less only as they slow down, and they need extra memory. Practical reversible computing remains experimental.

Further Reading

• Landauer, R. (1961). "Irreversibility and Heat Generation in the Computing Process." IBM Journal of Research and Development 5(3), 183-191. https://doi.org/10.1147/rd.53.0183
• Bennett, C. H. (1982). "The thermodynamics of computation: a review." International Journal of Theoretical Physics 21, 905-940. https://doi.org/10.1007/BF02084158
• Bérut, A. et al. (2012). "Experimental verification of Landauer's principle linking information and thermodynamics." Nature 483, 187-189. https://doi.org/10.1038/nature10872
• Hong, J. et al. (2016). "Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits." Science Advances 2(3), e1501492. https://doi.org/10.1126/sciadv.1501492
• "Landauer Limit Demonstrated." IEEE Spectrum (March 2012). https://spectrum.ieee.org/landauer-limit-demonstrated
• "Landauer's principle." Wikipedia: https://en.wikipedia.org/wiki/Landauer%27s_principle
• "Maxwell's demon." Wikipedia: https://en.wikipedia.org/wiki/Maxwell%27s_demon

This is Part 2 of the Intelligence of Energy series. Next: The Brain's Impossible Efficiency: 20 Watts