# Landauer's principle

**Landauer's principle** is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that "any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment".^{[1]}

Another way of phrasing Landauer's principle is that if an observer loses information about a physical system, the observer loses the ability to extract work from that system.

A so-called logically-reversible computation, in which no information is erased, may in principle be carried out without releasing any heat. This has led to considerable interest in the study of reversible computing. Indeed, without reversible computing, increases in the number of computations-per-joule-of-energy-dissipated must come to a halt by about 2050: because the limit implied by Landauer's principle will be reached by then, according to Koomey's law.

At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0175 eV, or 2.805 zJ. Theoretically, room‑temperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second (1Gbps) with energy being converted to heat in the memory media at the rate of only 2.805 trillionths of a watt (that is, at a rate of only 2.805 pJ/s). Modern computers use millions of times as much energy per second.^{[2]}^{[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 2011, the principle was generalized to show that while information erasure requires an increase in entropy, that increase could theoretically occur at no energy cost.^{[6]} 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.^{[7]}

In 2014, physical experiments tested Landauer's principle and confirmed its predictions.^{[8]}

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 millielectron volts (4.2 zeptojoules).^{[9]}

A 2018 article published in *Nature Physics* features a Landauer erasure performed at cryogenic temperatures (T = 1K) 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.^{[10]} 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.^{[10]}

## Rationale

Landauer's principle can be understood to be a simple logical consequence of the second law of thermodynamics—which states that the entropy of an isolated system cannot decrease—together with the definition of thermodynamic temperature. For, if the number of possible logical states of a computation were to decrease as the computation proceeded forward (logical irreversibility), this would constitute a forbidden decrease of entropy, unless the number of possible physical states corresponding to each logical state were to simultaneously increase by at least a compensating amount, so that the total number of possible physical states was no smaller than it was originally (i.e. total entropy has not decreased).

Yet, an increase in the number of physical states corresponding to each logical state means that, for an observer who is keeping track of the logical state of the system but not the physical state (for example an "observer" consisting of the computer itself), the number of possible physical states has increased; in other words, entropy has increased from the point of view of this observer.

The maximum entropy of a bounded physical system is finite. (If the holographic principle is correct, then physical systems with finite surface area have a finite maximum entropy; but regardless of the truth of the holographic principle, quantum field theory dictates that the entropy of systems with finite radius and energy is finite due to the Bekenstein bound.) To avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment.

## Equation

Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the *Landauer limit*:

- [math]\displaystyle{ E = k_\text{B} T \ln 2 }[/math]

where [math]\displaystyle{ k_\text{B} }[/math] is the Boltzmann constant (approximately 1.38×10^{−23} J/K), [math]\displaystyle{ T }[/math] is the temperature of the heat sink in kelvins, and [math]\displaystyle{ \ln 2 }[/math] is the natural logarithm of 2 (approximately 0.69315). After setting *[math]\displaystyle{ T }[/math]* equal to room temperature 20 °C (293.15 K), we can get the Landauer limit of 0.0175 eV (2.805 zJ) per bit erased.

The equation can be deduced from the Boltzmann's entropy formula ([math]\displaystyle{ S = k_\text{B} \ln W }[/math]), considering that [math]\displaystyle{ W }[/math] is the number of states of the system, which in case of a bit is 2, and the entropy [math]\displaystyle{ S }[/math] is defined as [math]\displaystyle{ E/T }[/math]. So the operation of erasing a single bit increases the entropy of a value of at least [math]\displaystyle{ k_\text{B} \ln 2 }[/math], emitting in the environment a quantity of energy equals or greater than [math]\displaystyle{ k_\text{B} T \ln 2 }[/math].

## 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)^{[11]} and Norton (2004,^{[12]} 2011^{[13]}), and defended by Bennett (2003),^{[1]} Ladyman et al. (2007),^{[14]} and by Jordan and Manikandan (2019).^{[15]}

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.^{[16]} 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 systems are nuanced.^{[17]}

In 2016, researchers at the University of Perugia claimed to have demonstrated a violation of Landauer’s principle.^{[18]} However, according to Laszlo Kish (2016),^{[19]} their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".

## See also

- Margolus–Levitin theorem
- Bremermann's limit
- Bekenstein bound
- Kolmogorov complexity
- Entropy in thermodynamics and information theory
- Information theory
- Jarzynski equality
- Limits to computation
- Extended mind thesis
- Maxwell's demon
- Koomey's Law

## References

- ↑
^{1.0}^{1.1}Charles H. Bennett (2003), "Notes on Landauer's principle, Reversible Computation and Maxwell's Demon",*Studies in History and Philosophy of Modern Physics***34**(3): 501–510, doi:10.1016/S1355-2198(03)00039-X, Bibcode: 2003SHPMP..34..501B, http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf, retrieved 2015-02-18 - ↑ "Tikalon Blog by Dev Gualtieri". Tikalon.com. http://tikalon.com/blog/blog.php?article=2011/Landauer. Retrieved May 5, 2013.
- ↑ "Nanomagnet memories approach low-power limit | bloomfield knoble". Bloomweb.com. http://www.bloomweb.com/nanomagnet-memories-approach-low-power-limit/. Retrieved May 5, 2013.
- ↑ "Landauer Limit Demonstrated - IEEE Spectrum". Spectrum.ieee.org. https://spectrum.ieee.org/computing/hardware/landauer-limit-demonstrated. Retrieved May 5, 2013.
- ↑ Rolf Landauer (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, http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf, retrieved 2015-02-18 - ↑ Joan Vaccaro; Stephen Barnett (June 8, 2011), "Information Erasure Without an Energy Cost",
*Proc. R. Soc. A***467**(2130): 1770–1778, doi:10.1098/rspa.2010.0577, Bibcode: 2011RSPSA.467.1770V - ↑ 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",
*Nature***483**(7388): 187–190, doi:10.1038/nature10872, PMID 22398556, Bibcode: 2012Natur.483..187B, http://www.physik.uni-kl.de/eggert/papers/raoul.pdf - ↑ 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, doi:10.1103/PhysRevLett.113.190601, PMID 25415891, Bibcode: 2014PhRvL.113s0601J - ↑ 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" (in en).
*Science Advances***2**(3): e1501492. doi:10.1126/sciadv.1501492. ISSN 2375-2548. PMID 26998519. Bibcode: 2016SciA....2E1492H. - ↑
^{10.0}^{10.1}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, doi:10.1038/s41567-018-0070-7, Bibcode: 2018NatPh..14..565G, http://resolver.tudelft.nl/uuid:c3926045-6e1a-4dd7-a584-df4a5c6b51b6 - ↑ Logic and Entropy Critique by Orly Shenker (2000)
- ↑ Eaters of the Lotus Critique by John Norton (2004)
- ↑ Waiting for Landauer Response by Norton (2011)
- ↑ The Connection between Logical and Thermodynamic Irreversibility Defense by Ladyman et al. (2007)
- ↑ Some Like It Hot, Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)
- ↑ Takahiro Sagawa (2014), "Thermodynamic and logical reversibilities revisited",
*Journal of Statistical Mechanics: Theory and Experiment***2014**(3): 03025, doi:10.1088/1742-5468/2014/03/P03025, Bibcode: 2014JSMTE..03..025S - ↑ David H. Wolpert (2019), "Stochastic thermodynamics of computation",
*Journal of Physics A: Mathematical and Theoretical***52**(19): 193001, doi:10.1088/1751-8121/ab0850, Bibcode: 2019JPhA...52s3001W - ↑ "Computing study refutes famous claim that 'information is physical'". https://m.phys.org/news/2016-07-refutes-famous-physical.html.
- ↑ Laszlo Bela Kish42.27Texas A&M University. "Comments on "Sub-kBT Micro-Electromechanical Irreversible Logic Gate"". https://www.researchgate.net/publication/304582612. Retrieved 2020-03-08.

## Further reading

- "Transfer entropy and transient limits of computation",
*Scientific Reports***4**: 5394, 2014, doi:10.1038/srep05394, PMID 24953547, Bibcode: 2014NatSR...4E5394P

## External links

Library resources about Landauer's principle |

- Public debate on the validity of Landauer's principle (conference Hot Topics in Physical Informatics, November 12, 2013)
- Introductory article on Landauer's principle and reversible computing
- Maroney, O.J.E. "Information Processing and Thermodynamic Entropy" The Stanford Encyclopedia of Philosophy.
- Eurekalert.org: "Magnetic memory and logic could achieve ultimate energy efficiency", July 1, 2011

Original source: https://en.wikipedia.org/wiki/ Landauer's principle.
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