# Landauer's principle

Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat to its surroundings.[1]

The principle was first proposed by Rolf Landauer in 1961.

## Statement

Landauer's principle states that the minimum energy needed to erase one bit of information is proportional to the temperature at which the system is operating. More specifically, the energy needed for this computational task is given by

${\displaystyle E\geq k_{\text{B}}T\ln 2,}$

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant.[2] At room temperature, the Landauer limit represents an energy of approximately 0.018 eV (2.9×10−21 J). Modern computers use about a billion times as much energy per operation.[3][4]

## History

Rolf Landauer first proposed the principle in 1961 while working at IBM.[5] He justified and stated important limits to an earlier conjecture by John von Neumann. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.

In 2008 and 2009, researchers showed that Landauer's principle can be derived from the second law of thermodynamics and the entropy change associated with information gain, developing the thermodynamics of quantum and classical feedback-controlled systems.[6][7]

In 2011, the principle was generalized to show that while information erasure requires an increase in entropy, this increase could theoretically occur at no energy cost.[8] Instead, the cost can be taken in another conserved quantity, such as angular momentum.

In a 2012 article published in Nature, a team of physicists from the École normale supérieure de Lyon, University of Augsburg and the University of Kaiserslautern described that for the first time they have measured the tiny amount of heat released when an individual bit of data is erased.[9]

In 2014, physical experiments tested Landauer's principle and confirmed its predictions.[10]

In 2016, researchers used a laser probe to measure the amount of energy dissipation that resulted when a nanomagnetic bit flipped from off to on. Flipping the bit required 26 millielectronvolts (4.2 zeptojoules).[11]

A 2018 article published in Nature Physics features a Landauer erasure performed at cryogenic temperatures (T = 1 K) on an array of high-spin (S = 10) quantum molecular magnets. The array is made to act as a spin register where each nanomagnet encodes a single bit of information.[12] The experiment has laid the foundations for the extension of the validity of the Landauer principle to the quantum realm. Owing to the fast dynamics and low "inertia" of the single spins used in the experiment, the researchers also showed how an erasure operation can be carried out at the lowest possible thermodynamic cost—that imposed by the Landauer principle—and at a high speed.[12][1]

## Challenges

The principle is widely accepted as physical law, but in recent years it has been challenged for using circular reasoning and faulty assumptions, notably in Earman and Norton (1998), and subsequently in Shenker (2000)[13] and Norton (2004,[14] 2011[15]), and defended by Bennett (2003),[1] Ladyman et al. (2007),[16] and by Jordan and Manikandan (2019).[17] Sagawa and Ueda (2008) and Cao and Feito (2009) have shown that Landauer's principle is a consequence of the second law of Thermodynamics and the entropy reduction associated with information gain.[6][7]

On the other hand, recent advances in non-equilibrium statistical physics have established that there is no a priori relationship between logical and thermodynamic reversibility.[18] It is possible that a physical process is logically reversible but thermodynamically irreversible. It is also possible that a physical process is logically irreversible but thermodynamically reversible. At best, the benefits of implementing a computation with a logically reversible system are nuanced.[19]

In 2016, researchers at the University of Perugia claimed to have demonstrated a violation of Landauer’s principle.[20] However, according to Laszlo Kish (2016),[21] their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".

## References

1. ^ a b c Charles H. Bennett (2003), "Notes on Landauer's principle, Reversible Computation and Maxwell's Demon" (PDF), Studies in History and Philosophy of Modern Physics, 34 (3): 501–510, arXiv:physics/0210005, Bibcode:2003SHPMP..34..501B, doi:10.1016/S1355-2198(03)00039-X, S2CID 9648186, retrieved 2015-02-18.
2. ^ Vitelli, M.B.; Plenio, V. (2001). "The physics of forgetting: Landauer's erasure principle and information theory" (PDF). Contemporary Physics. 42 (1): 25–60. arXiv:quant-ph/0103108. Bibcode:2001ConPh..42...25P. doi:10.1080/00107510010018916. eISSN 1366-5812. hdl:10044/1/435. ISSN 0010-7514. S2CID 9092795.
3. ^ Thomas J. Thompson. "Nanomagnet memories approach low-power limit". bloomfield knoble. Archived from the original on December 19, 2014. Retrieved May 5, 2013.
4. ^ Samuel K. Moore (14 March 2012). "Landauer Limit Demonstrated". IEEE Spectrum. Retrieved May 5, 2013.
5. ^ Rolf Landauer (1961), "Irreversibility and heat generation in the computing process" (PDF), IBM Journal of Research and Development, 5 (3): 183–191, doi:10.1147/rd.53.0183, retrieved 2015-02-18.
6. ^ a b Sagawa, Takahiro; Ueda, Masahito (2008-02-26). "Second Law of Thermodynamics with Discrete Quantum Feedback Control". Physical Review Letters. 100 (8): 080403. arXiv:0710.0956. Bibcode:2008PhRvL.100h0403S. doi:10.1103/PhysRevLett.100.080403. PMID 18352605. S2CID 41799543.
7. ^ a b Cao, F. J.; Feito, M. (2009-04-10). "Thermodynamics of feedback controlled systems". Physical Review E. 79 (4): 041118. arXiv:0805.4824. Bibcode:2009PhRvE..79d1118C. doi:10.1103/PhysRevE.79.041118. PMID 19518184. S2CID 30188109.
8. ^ Joan Vaccaro; Stephen Barnett (June 8, 2011), "Information Erasure Without an Energy Cost", Proc. R. Soc. A, 467 (2130): 1770–1778, arXiv:1004.5330, Bibcode:2011RSPSA.467.1770V, doi:10.1098/rspa.2010.0577, S2CID 11768197.
9. ^ Antoine Bérut; Artak Arakelyan; Artyom Petrosyan; Sergio Ciliberto; Raoul Dillenschneider; Eric Lutz (8 March 2012), "Experimental verification of Landauer's principle linking information and thermodynamics" (PDF), Nature, 483 (7388): 187–190, arXiv:1503.06537, Bibcode:2012Natur.483..187B, doi:10.1038/nature10872, PMID 22398556, S2CID 9415026.
10. ^ Yonggun Jun; Momčilo Gavrilov; John Bechhoefer (4 November 2014), "High-Precision Test of Landauer's Principle in a Feedback Trap", Physical Review Letters, 113 (19): 190601, arXiv:1408.5089, Bibcode:2014PhRvL.113s0601J, doi:10.1103/PhysRevLett.113.190601, PMID 25415891, S2CID 10164946.
11. ^ Hong, Jeongmin; Lambson, Brian; Dhuey, Scott; Bokor, Jeffrey (2016-03-01). "Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits". Science Advances. 2 (3): e1501492. Bibcode:2016SciA....2E1492H. doi:10.1126/sciadv.1501492. ISSN 2375-2548. PMC 4795654. PMID 26998519..
12. ^ a b Rocco Gaudenzi; Enrique Burzuri; Satoru Maegawa; Herre van der Zant; Fernando Luis (19 March 2018), "Quantum Landauer erasure with a molecular nanomagnet", Nature Physics, 14 (6): 565–568, Bibcode:2018NatPh..14..565G, doi:10.1038/s41567-018-0070-7, hdl:10261/181265, S2CID 125321195.
13. ^ Shenker, Orly R. (June 2000). "Logic and Entropy [preprint]". PhilSci Archive. Archived from the original on 15 November 2023. Retrieved 20 December 2023.
14. ^ Norton, John D. (June 2005). "Eaters of the lotus: Landauer's principle and the return of Maxwell's demon". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 36 (2): 375–411. Bibcode:2005SHPMP..36..375N. doi:10.1016/j.shpsb.2004.12.002. S2CID 21104635. Archived from the original on 5 June 2023. Retrieved 20 December 2023.
15. ^ Norton, John D. (August 2011). "Waiting for Landauer" (PDF). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 42 (3): 184–198. Bibcode:2011SHPMP..42..184N. doi:10.1016/j.shpsb.2011.05.002. Retrieved 20 December 2023.
16. ^ Ladyman, James; Presnell, Stuart; Short, Anthony J.; Groisman, Berry (March 2007). "The connection between logical and thermodynamic irreversibility". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 38 (1): 58–79. Bibcode:2007SHPMP..38...58L. doi:10.1016/j.shpsb.2006.03.007. Retrieved 20 December 2023.
17. ^ Jordan, Andrew; Manikandan, Sreenath (12 December 2019). "Some Like It Hot". Inference: International Review of Science. 5 (1). doi:10.37282/991819.19.82. S2CID 241470079.
18. ^ Takahiro Sagawa (2014), "Thermodynamic and logical reversibilities revisited", Journal of Statistical Mechanics: Theory and Experiment, 2014 (3): 03025, arXiv:1311.1886, Bibcode:2014JSMTE..03..025S, doi:10.1088/1742-5468/2014/03/P03025, S2CID 119247579.
19. ^ David H. Wolpert (2019), "Stochastic thermodynamics of computation", Journal of Physics A: Mathematical and Theoretical, 52 (19): 193001, arXiv:1905.05669, Bibcode:2019JPhA...52s3001W, doi:10.1088/1751-8121/ab0850, S2CID 126715753.
20. ^
21. ^ Laszlo Bela Kish (2016). "Comments on 'Sub-kBT Micro-Electromechanical Irreversible Logic Gate'". Fluctuation and Noise Letters. 14 (4): 1620001–1620194. arXiv:1606.09493. Bibcode:2016FNL....1520001K. doi:10.1142/S0219477516200017. S2CID 12110986. Retrieved 2020-03-08.