Landauer's principle

Landauer's principle states that erasing one bit of information from a system in contact with a heat reservoir at temperature T must release at least k_BT ln 2 of heat into the reservoir. At 300 K that is 2.87 × 10⁻²¹ J, or 0.0179 eV. The physicist Rolf Landauer of IBM argued the point in 1961 for computers, and a series of laboratory experiments since 2012 has measured the bound with beads, nanomagnets and single ions.^[1]

The principle matters because it puts the cost of computation in an unexpected place. Reading, copying and many logical steps can in principle be done with no dissipation at all. Only the step that throws information away has a minimum price, and the price is paid in the surroundings. This is how the principle disposes of Maxwell's demon: the demon can sort molecules for free, but its memory fills up, and clearing it costs at least as much as the sorting gained.^[2]

Statement

Three drawings of a double-well potential: a ball in either well before erasure, the barrier lowered and the curve tilted during it, and the ball in the right well after it, with the entropy and heat balance written below — Erasing one bit stored as the position of a particle in a double-well potential. Both starting states end in one, so the entropy of the memory falls by k_B ln 2 and, by the second law, the surroundings must receive at least k_BT ln 2 of heat. The drawing is schematic.

Take a bit stored in a physical system with two distinguishable states, such as a particle in one of two wells. Erasure, or "reset to zero", sends both starting states to one final state. The memory's entropy falls by k_B ln 2, the entropy of a choice between two equally likely states. The second law forbids a fall in the total, so the entropy of the environment must rise by at least that much, and heat Q delivered at temperature T raises it by Q/T. The inequality is

Q \ge k_B T \ln 2.

Landauer's term for the property that triggers the cost was logical irreversibility. A step that maps several inputs onto one output, as erasure and an ordinary AND gate do, cannot be run backward, because the output does not say which input produced it. A step that can be run backward, such as swapping two bits, carries no such cost.^[1] Landauer's later summary was the slogan "information is physical".^[3]

The bound is small. Thermal energy at 300 K is k_BT = 4.14 × 10⁻²¹ J, and k_BT ln 2 is 69% of that. Erasing every bit of a gigabyte would release 2.3 × 10⁻¹¹ J. A logic gate that charges 1 fF through 0.7 V dissipates about ½CV² = 0.25 fJ in one transition, some 85,000 times the bound for one bit.^[a]

History

Landauer (1927 to 1999) was born in Stuttgart, left Germany with his family in 1938, took his doctorate at Harvard in 1950 and spent most of his career at IBM.^[4] His 1961 paper asked how small the energy of a computing step could be made. He concluded that the logically irreversible steps were the ones that dissipated, at least k_BT ln 2 for each bit erased.

Charles Bennett, also at IBM, showed in 1973 that any computation can be made logically reversible by keeping a record of every step, copying the answer, and running the steps backward to clear the record.^[5] Fredkin and Toffoli gave reversible gates a systematic form in 1982.^[6] Bennett's reading of the demon followed in 1982, and he returned to the subject in 2003.^[2]^[7] The story of the demon itself, from James Clerk Maxwell to Szilard's one-molecule engine, is told in the article on Maxwell's demon.

Landauer's name is attached to a second, unrelated result. A 1957 paper of his on current through a scatterer is the origin of the Landauer formula for the conductance of a nanoscale conductor.^[8]

Experiments

The bound resisted direct measurement for fifty years, because the experiments need a one-bit memory small enough to be pushed around by thermal motion and gentle enough that the heat can be seen.

Tests of Landauer's bound
Year	System	Result
2012	A silica bead about 2 µm across in a double-well optical trap	The mean heat released in erasing the bead's position saturated at the Landauer bound as the erasure cycle was lengthened^[9]
2014	A colloidal particle in a virtual potential made by a feedback trap	Erasure required work of at least k_BT ln 2 on average; individual cycles could fall below it, consistent with the Jarzynski equality^[10]
2014	A single-electron box run as a Szilard engine	k_BT ln 2 of heat extracted from the reservoir for each bit of information created^[11]
2016	Lithographic nanomagnets used as memory bits	The energy dissipated in a single-bit reset was consistent, within two standard deviations, with the bound^[12]
2018	A trapped ultracold ion	An equality linking the energy cost of erasure to the entropy change of the quantized environment, the quantum version of the principle^[13]

None of these overturns the second law, and none could. In each, the information is paid for somewhere in the apparatus.

Criticism

The philosophers John Earman and John D. Norton argued in 1998 and 1999 that the information-theoretic exorcisms of the demon either assume the second law they set out to defend or rest on principles with no independent support.^[14] Norton returned to Landauer's principle itself in 2005. He accepted that it can be stated precisely in statistical mechanics, but only for a restricted class of erasure processes, and argued that the general justifications fail because they treat a memory holding random data as if it were a canonical ensemble.^[15] The experiments above test the bound in the systems they used, and the philosophical dispute is about what such tests can establish.

At Molecular Reality

The company's educational material applies the bound to a pore. A pore that measures a molecule and then lets it pass, sends it back or holds it is, in the company's words, "a demon of exactly this kind", and it prices a decision made over 10 µs at 200 mV and 3.1 nA at about 6 fJ, roughly two million times the Landauer limit.^[16]^[b] The same material adds that the difficulty lies elsewhere: "The real question is the decision itself." The company's original name, demonpore, inc., came from the same thought experiment, and the story is told in the article on Maxwell's demon.

Notes

^ Calculated: ½ × (10⁻¹⁵ F) × (0.7 V)² = 2.45 × 10⁻¹⁶ J, divided by 2.87 × 10⁻²¹ J. The gate values are round numbers chosen for the example and describe no particular chip.
^ The arithmetic is 0.2 V × 3.1 nA × 10 µs = 6.2 × 10⁻¹⁵ J, divided by 2.87 × 10⁻²¹ J. The 3.1 nA is the current that a model of the pore with access resistance predicts, not a measurement, and the 6 fJ is the electrical energy passing through the pore in the window. It is a comparison of scales; the bound itself concerns the erasure of the memory that holds the result.

References

^ ^a ^b Landauer, R. (1961). "Irreversibility and heat generation in the computing process". IBM Journal of Research and Development 5 (3): 183-191. doi:10.1147/rd.53.0183.
^ ^a ^b Bennett, C. H. (1982). "The thermodynamics of computation: a review". International Journal of Theoretical Physics 21 (12): 905-940. doi:10.1007/BF02084158.
^ Landauer, R. (1991). "Information is physical". Physics Today 44 (5): 23-29. doi:10.1063/1.881299.
^ Fowler, A.; et al. (1999). "Rolf William Landauer". Physics Today 52 (10): 104-105. doi:10.1063/1.882874.
^ Bennett, C. H. (1973). "Logical reversibility of computation". IBM Journal of Research and Development 17 (6): 525-532. doi:10.1147/rd.176.0525.
^ Fredkin, E.; Toffoli, T. (1982). "Conservative logic". International Journal of Theoretical Physics 21 (3-4): 219-253. doi:10.1007/BF01857727.
^ Bennett, C. H. (2003). "Notes on Landauer's principle, reversible computation, and Maxwell's Demon". Studies in History and Philosophy of Science Part B 34 (3): 501-510. doi:10.1016/S1355-2198(03)00039-X.
^ Landauer, R. (1957). "Spatial variation of currents and fields due to localized scatterers in metallic conduction". IBM Journal of Research and Development 1 (3): 223-231. doi:10.1147/rd.13.0223.
^ Bérut, A.; et al. (2012). "Experimental verification of Landauer's principle linking information and thermodynamics". Nature 483 (7388): 187-189. doi:10.1038/nature10872.
^ Jun, Y.; Gavrilov, M.; Bechhoefer, J. (2014). "High-precision test of Landauer's principle in a feedback trap". Physical Review Letters 113 (19): 190601. doi:10.1103/PhysRevLett.113.190601.
^ Koski, J. V.; et al. (2014). "Experimental realization of a Szilard engine with a single electron". Proceedings of the National Academy of Sciences 111 (38): 13786-13789. doi:10.1073/pnas.1406966111.
^ Hong, J.; et al. (2016). "Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits". Science Advances 2 (3): e1501492. doi:10.1126/sciadv.1501492.
^ Yan, L. L.; et al. (2018). "Single-atom demonstration of the quantum Landauer principle". Physical Review Letters 120 (21): 210601. doi:10.1103/PhysRevLett.120.210601.
^ Earman, J.; Norton, J. D. (1998). "Exorcist XIV: The wrath of Maxwell's demon. Part I. From Maxwell to Szilard". Studies in History and Philosophy of Modern Physics 29 (4): 435-471. doi:10.1016/S1355-2198(98)00023-9.
^ Norton, J. D. (2005). "Eaters of the lotus: Landauer's principle and the return of Maxwell's demon". Studies in History and Philosophy of Science Part B 36 (2): 375-411. doi:10.1016/j.shpsb.2004.12.002.
^ "Company records". Molecular Reality Corporation. October 2026.