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{{Distinguish|Landau principle}}

{{Distinguish|Landau principle}}

'''Landauer's principle''' is a [[Principle#Principle as scientific law|physical principle]] pertaining to the lower [[Theoretical physics|theoretical]] limit of [[Energy conservation|energy consumption]] of [[computation]]. It holds that "any logically irreversible manipulation of [[Information#As a property in physics|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 (physics and chemistry)|degrees of freedom]] of the information-processing apparatus or its environment".<ref name = bennett>{{Citation |arxiv=physics/0210005 |title=Notes on Landauer's principle, Reversible Computation and Maxwell's Demon |authorlink=Charles H. Bennett (computer scientist) |author=Charles H. Bennett |journal=Studies in History and Philosophy of Modern Physics |volume=34 |issue=3 |pages=501–510 |year=2003 |url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf |accessdate=2015-02-18 |doi=10.1016/S1355-2198(03)00039-X|bibcode=2003SHPMP..34..501B |s2cid=9648186 }}</ref>

'''Landauer's principle''' is a [[Principle#Principle as scientific law|physical principle]] pertaining to the lower [[Theoretical physics|theoretical]] limit of [[Energy conservation|energy consumption]] of [[computation]]. It holds that "any logically irreversible manipulation of [[Information#As a property in physics|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 (physics and chemistry)|degrees of freedom]] of the information-processing apparatus or its environment".<ref name = bennett>{{Citation |arxiv=physics/0210005 |title=Notes on Landauer's principle, Reversible Computation and Maxwell's Demon |authorlink=Charles H. Bennett (computer scientist) |author=Charles H. Bennett |journal=Studies in History and Philosophy of Modern Physics |volume=34 |issue=3 |pages=501–510 |year=2003 |url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf |accessdate=2015-02-18 |doi=10.1016/S1355-2198(03)00039-X|bibcode=2003SHPMP..34..501B |s2cid=9648186 }}</ref>

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".

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".

 

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.

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.  

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&nbsp;°C (room temperature, or 293.15&nbsp;K), the Landauer limit represents an energy of approximately 0.0175&nbsp;eV, or 2.805&nbsp;zJ. Theoretically, roomtemperature 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.

At 20&nbsp;°C (room temperature, or 293.15&nbsp;K), the Landauer limit represents an energy of approximately 0.0175&nbsp;eV, or 2.805&nbsp;zJ. Theoretically, roomtemperature 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.

在20 ° c (室温，或293.15 k)时，兰道尔极限表示大约0.0175 eV，或2.805 zJ 的能量。理论上，在兰道尔极限下工作的房间温度计算机存储器可以以每秒10亿比特(1gbps)的速度改变，能量在存储介质中以仅2.805万亿分之一瓦特的速度转化为热量(也就是说，只以2.805 pJ/s 的速度)。现代计算机每秒钟要消耗数百万倍的能量。

在20 ° c (室温，或293.15 k)时，兰道尔极限表示大约0.0175 eV，或2.805 zJ 的能量。理论上，在兰道尔极限下工作的房间温度计算机存储器可以以每秒10亿比特(1gbps)的速度改变，能量在存储介质中以仅2.805万亿分之一瓦特的速度转化为热量(也就是说，只以2.805 pJ/s 的速度)。现代计算机每秒消耗的能量是其数百万倍。

 

==History==

==History==

[[Rolf Landauer]] first proposed the principle in 1961 while working at [[IBM]].<ref name="landauer">{{Citation |author=Rolf Landauer |url=http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf |title=Irreversibility and heat generation in the computing process |journal=IBM Journal of Research and Development |volume=5 |issue=3 |pages=183–191 |year=1961 |accessdate=2015-02-18 |doi=10.1147/rd.53.0183 }}</ref> 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.

[[Rolf Landauer]] first proposed the principle in 1961 while working at [[IBM]].<ref name="landauer">{{Citation |author=Rolf Landauer |url=http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf |title=Irreversibility and heat generation in the computing process |journal=IBM Journal of Research and Development |volume=5 |issue=3 |pages=183–191 |year=1961 |accessdate=2015-02-18 |doi=10.1147/rd.53.0183 }}</ref> 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.

Rolf Landauer first proposed the principle in 1961 while working at IBM. 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.

Rolf Landauer first proposed the principle in 1961 while working at IBM. 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. Instead, the cost can be taken in another conserved quantity, such as angular momentum.

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. 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.

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.

 

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

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

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).

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).

2016年，研究人员使用激光探测器测量了纳米磁位从关闭到打开时所产生的能量耗散量。翻转钻头需要26毫电子伏(4.2 zeptojoules)。

2016年，研究人员使用激光探针测量了纳米磁性位从关到开时产生的能量耗散量。翻转位子需要26毫电子伏特（4.2泽普焦耳）。

 

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.

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.

2018年发表在《自然物理学》上的一篇文章描述了在低温(t = 1K)下对一排高自旋(s = 10)量子分子磁体进行的兰道尔擦除(Landauer erasure)。该阵列作为一个自旋寄存器，每个纳米粒子网编码一位信息。

2018年发表在《自然物理学》上的一篇文章描述了在低温(T = 1K)下对一排高自旋(S = 10)量子分子磁体进行的兰道尔擦除(Landauer erasure)。该阵列作为自旋寄存器，每个纳米粒子网编码一位信息。

==Rationale==

==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).

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).

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).

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).

的原理可以理解为热力学第二定律的一个简单蕴涵---- 该定律指出孤立系统的熵不能减少---- 与热力学温度的定义一起。因为，如果一个计算的可能逻辑状态的数量随着计算的进行而减少(逻辑不可逆性) ，这将构成熵的禁止减少，除非每个逻辑状态对应的可能物理状态的数量同时增加至少一个补偿量，以使可能的物理状态的总数不小于原来的数量(即:。总熵没有减少)。

兰道尔的原则可以理解为热力学第二定律的一个简单的逻辑后果--该定律指出，一个孤立系统的熵不能与热力学温度的定义一起减少。因为，如果一个计算的可能逻辑状态的数目随着计算的进行而减少(逻辑的不可逆性)，这将构成熵的被禁止的减少，除非与每个逻辑状态相对应的可能物理状态的数目同时增加至少一个补偿量，从而使可能物理状态的总数不比原来少(即总熵没有减少)。

 

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.

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.

有界物理系统的最大熵是有限的。(如果全息原理理论是正确的，那么有限表面积的物理系统有一个有限的最大熵; 但是不管全息原理理论是否正确，量子场理论指出，有限半径和能量的系统的熵是有限的，这是由于贝肯斯坦上限的存在。)为了避免在扩展计算过程中达到这个最大值，熵最终必须被驱逐到外部环境中。

有界物理系统的最大熵是有限的。(如果全息原理理论是正确的，那么有限表面积的物理系统的最大熵是有限的; 但是不管全息原理理论是否正确，量子场理论指出，由于贝肯斯坦约束，半径和能量有限的系统的熵是有限的。）为了避免在扩展计算过程中达到这个最大值，熵最终必须被驱逐到外部环境中。

==Equation==

==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'':

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'':

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:

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:

 

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

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

其中，k text { b } </math > 是波兹曼常数(大约1.38 × 10 < sup >-23 </sup > J/K) ，t </math > 是科尔文散热器的温度，而 < math > ln 2 </math > 是2的自然对数(大约0.69315)。设置 t 为室温20 ° c (293.15 k)后，可以得到每位擦除0.0175 eV (2.805 zJ)的朗道尔极限。

其中，K text { b } </math > 是波兹曼常数(大约1.38 × 10 < sup >-23 </sup > J/K) ，T </math > 是散热器的温度，单位为开尔文，而 < math > ln 2 </math > 是2的自然对数(大约0.69315)。设 T 为室温20 ° c (293.15 k)后，可以得到每位擦除0.0175 eV (2.805 zJ)的朗道尔极限。

For an environment at temperature T, energy E = ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least k_B ln&#8239;2, and so, the energy that must eventually be emitted to the environment is E ≥ k_BT ln&#8239;2.

For an environment at temperature T, energy E = ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least k_B ln&#8239;2, and so, the energy that must eventually be emitted to the environment is E ≥ k_BT ln&#8239;2.

对于一个温度为 t 的环境，如果附加熵的量为 s，则能量 e = ST 必须被发射到该环境中。对于一个计算操作，其中丢失了1位逻辑信息，所产生的熵至少为 k < sub > b </sub > ln & # 8239; 2，因此，最终发射到环境中的能量为 e ≥ k < sub > b </sub > t ln & # 8239; 2。

对于温度为T的环境，如果增加的熵量为S，则必须向该环境放出能量E=ST，对于丢失1位逻辑信息的计算操作，产生的熵量至少为k_Bln&#8239;2，所以，最终必须向环境放出的能量为E≥k_BT ln&#8239;2。

 

==Challenges==

==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)<ref name="shenker">[http://philsci-archive.pitt.edu/archive/00000115/ Logic and Entropy] Critique by Orly Shenker (2000)</ref> and Norton (2004,<ref name="norton">[http://philsci-archive.pitt.edu/archive/00001729/ Eaters of the Lotus] Critique by John Norton (2004)</ref> 2011<ref name="norton2">[http://www.pitt.edu/~jdnorton/papers/Waiting_SHPMP.pdf Waiting for Landauer] Response by Norton (2011)</ref>), and defended by Bennett (2003),<ref name="bennett" /> Ladyman et al. (2007),<ref name="short">[http://philsci-archive.pitt.edu/archive/00002689/ The Connection between Logical and Thermodynamic Irreversibility] Defense by Ladyman et al. (2007)</ref> and by Jordan and Manikandan (2019).<ref name="jordan">[https://inference-review.com/letter/some-like-it-hot Some Like It Hot], Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)</ref>

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)<ref name="shenker">[http://philsci-archive.pitt.edu/archive/00000115/ Logic and Entropy] Critique by Orly Shenker (2000)</ref> and Norton (2004,<ref name="norton">[http://philsci-archive.pitt.edu/archive/00001729/ Eaters of the Lotus] Critique by John Norton (2004)</ref> 2011<ref name="norton2">[http://www.pitt.edu/~jdnorton/papers/Waiting_SHPMP.pdf Waiting for Landauer] Response by Norton (2011)</ref>), and defended by Bennett (2003),<ref name="bennett" /> Ladyman et al. (2007),<ref name="short">[http://philsci-archive.pitt.edu/archive/00002689/ The Connection between Logical and Thermodynamic Irreversibility] Defense by Ladyman et al. (2007)</ref> and by Jordan and Manikandan (2019).<ref name="jordan">[https://inference-review.com/letter/some-like-it-hot Some Like It Hot], Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)</ref>

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) and Norton (2004, 2011), and defended by Bennett (2003), and by Jordan and Manikandan (2019).

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) and Norton (2004, 2011), and defended by Bennett (2003), and by Jordan and Manikandan (2019).

这一原则被广泛接受为物理定律，但近年来，它因使用循环推理和错误假设而受到挑战，尤其是在厄尔曼和诺顿(1998年) ，随后在申克(2000年)和诺顿(2004年，2011年) ，贝内特(2003年)和约旦和马尼坎达(2019年)为之辩护。

这一原则被广泛接受为物理定律，但近年来，它因使用循环推理和错误假设而受到挑战，尤其是厄尔曼和诺顿(1998年) ，随后是申克(2000年)和诺顿(2004年，2011年) ，贝内特(2003年)和约旦和马尼坎达(2019年)为之辩护。

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. 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.

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. 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.

 

==See also==

==See also==

* [[Margolus–Levitin theorem]]

* [[Margolus–Levitin theorem]]

* [[Bremermann's limit]]

* [[Bremermann's limit]]

* [[Bekenstein bound]]

* [[Bekenstein bound]]

* [[Kolmogorov complexity]]

* [[Kolmogorov complexity]]

* [[Entropy in thermodynamics and information theory]]

* [[Entropy in thermodynamics and information theory]]

* [[Information theory]]

* [[Information theory]]

* [[Jarzynski equality]]

* [[Jarzynski equality]]

* [[Limits to computation]]

* [[Limits to computation]]

* [[Extended mind thesis]]

* [[Extended mind thesis]]

* [[Maxwell's demon]]

* [[Maxwell's demon]]

*[[Koomey's law|Koomey's Law]]

*[[Koomey's law|Koomey's Law]]

 

==References==

==References==

{{reflist}}

{{reflist}}

==Further reading==

==Further reading==

*{{citation| author1-first= Mikhail | author1-last= Prokopenko | author2-first=Joseph T. | author2-last= Lizier | title= Transfer entropy and transient limits of computation | journal= [[Scientific Reports]] | date= 2014 | volume=4  | page= 5394 | doi= 10.1038/srep05394| pmid= 24953547 | bibcode= 2014NatSR...4E5394P | pmc= 4066251 }}

*{{citation| author1-first= Mikhail | author1-last= Prokopenko | author2-first=Joseph T. | author2-last= Lizier | title= Transfer entropy and transient limits of computation | journal= [[Scientific Reports]] | date= 2014 | volume=4  | page= 5394 | doi= 10.1038/srep05394| pmid= 24953547 | bibcode= 2014NatSR...4E5394P | pmc= 4066251 }}

==External links==

==External links==

{{Library resources box}}

{{Library resources box}}