麦克斯韦妖

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文件:Maxwell's demon.svg

麦克斯韦妖假想实验示意图。

麦克斯韦妖 Maxwell's demon是英国物理学家詹姆斯·克拉克·麦克斯韦 James Clerk Maxwell于1867年为了说明违反 热力学第二定律 second law of thermodynamics的可能性而设想的假想实验。在这个实验当中，一个妖怪控制着两个气体舱之间的一扇小门。当独立的气体分子接近门时，妖怪迅速的打开和关闭门，快速的分子进入其中的一个腔，慢速的分子进入另外一个腔。因为速度更快的分子温度更高，妖怪的动作导致一个腔室升温，另一个腔室降温，从而减少熵，违反了热力学第二定律。这个假想实验引起了关于热力学和信息论之间关系的争论和理论工作，并一直延续到今天，一些科学家认为理论上的考虑排除了任何以这种方式违反第二定律的实际装置。

思想的起源和历史

1867年12月11日，Maxwell在写给彼得·格思里·泰特 Peter Guthrie Tait的一封信中首次提出了这个假想实验。Maxwell在1871年写给约翰·威廉·斯特鲁特 John William Strutt 的信中再次提到该试验，后来在Maxwell 1872年出版的热力学书籍《热理论》（Theory of Heat）中公之于众。

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". William Thomson (Lord Kelvin) was the first to use the word "demon" for Maxwell's concept, in the journal Nature in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.

在Maxwell的信件和书中，他将妖怪打开腔室的门描述为一个“有限的存在”。威廉·汤姆森 William Thomson （开尔文勋爵）在1874年的《自然》杂志上第一次用“妖”这个词来形容Maxwell的这一概念，并且表示他打算扩展这个词的含义，而不是恶意曲解。

最初的假想实验

The second law of thermodynamics ensures (through statistical probability) that two bodies of different temperature, when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature.{{cite journal

The second law of thermodynamics ensures (through statistical probability) that two bodies of different temperature, when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature. The second law is also expressed as the assertion that in an isolated system, entropy never decreases.

热力学第二定律确保（通过统计概率）当两个不同温度的物体相互接触并与宇宙的其余部分隔离时，它们将演化为热力学平衡，最后两个物体的温度大致相同<ref name="Bennett87">。第二定律也表示为这样的结论：在一个孤立的系统中，熵永远不会减少。

Maxwell构想了一个假想实验，作为进一步理解第二定律的方法。他对实验的描述如下:

....如果我们设想一种妖能力非常发达，以至于他能够跟踪每一个分子的运行轨迹，那么这种本质上和我们一样有限的存在，就能够做到我们不可能做到的事情。因为我们已经看到，在一个充满空气的容器中，在均匀的温度下，分子的运动速度决不是均匀的，尽管任意选择的大量分子的平均速度几乎是均匀的。假设我们有一箱气体被分成A和B两个区域，分割板上面有一个小洞。这个能够观察每一个分子的生灵，可以打开或关闭这个小洞，只允许速度快的分子从A飞向B，而速度慢的分子从B飞向A。如此一来，不需要任何外界做功，我们就可以升高B区的温度而降低A区的温度，从而打破热力学第二定律。 这里多了一段非原文 In other words, Maxwell imagines one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. Likewise, when a slower-than-average molecule from B flies towards the trapdoor, the demon will let it pass from B to A. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics. A heat engine operating between the thermal reservoirs A and B could extract useful work from this temperature difference.

换句话说，Maxwell假设一个容器被分成A和B两部分。两部分在相同的温度下装入相同的气体，并且相邻放置。通过观察两边的分子，一个假想中的妖怪守护着两部分之间的活动门。当一个来自A的比平均速度快的分子飞向活动门的时候，妖怪打开了这个活动门，这个分子就会从A飞向B。相对应的，当一个来自B的比平均速度慢的分子飞向活动门的时候，妖怪会让它从B飞向A。B中的分子的平均速度就会增加，而A中的分子的平均速度会减慢。由于平均分子速度与温度相对应，因此A中的温度降低，B中的温度升高，这与热力学第二定律相反。在热源A和热源B之间运行的热机可以从这种温度差中提取有用的功。

The demon must allow molecules to pass in both directions in order to produce only a temperature difference; one-way passage only of faster-than-average molecules from A to B will cause higher temperature and pressure to develop on the B side.

妖必须允许分子在两个方向上通过，以便只产生温度差; 只有比平均速度快的分子从A到B 单向通过才会使B部分产生更高的温度和压力。

批评与发展

Several physicists have presented calculations that show that the second law of thermodynamics will not actually be violated, if a more complete analysis is made of the whole system including the demon. The essence of the physical argument is to show, by calculation, that any demon must "generate" more entropy segregating the molecules than it could ever eliminate by the method described. That is, it would take more thermodynamic work to gauge the speed of the molecules and selectively allow them to pass through the opening between A and B than the amount of energy gained by the difference of temperature caused by doing so.

一些物理学家展示的计算结果表明，如果对包括妖在内的整个系统进行更完整的分析，则实际上不会违反热力学第二定律。物理论证的精髓在于，通过计算来证明，任何妖都必须“产生”更多的熵来分离分子，这比描述的方法所消除的熵多。就是说，与通过这种方式引起的温度差所获得的能量相比，需要更多的热力学功来衡量分子的速度并选择性地使它们通过A和B之间的门。

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd, and later by Léon Brillouin. He realized that some measuring processes need not increase thermodynamic entropy as long as they were thermodynamically reversible. He suggested these "reversible" measurements could be used to sort the molecules, violating the Second Law. However, due to the connection between thermodynamic entropy and information entropy, this also meant that the recorded measurement must not be erased. In other words, to determine whether to let a molecule through, the demon must acquire information about the state of the molecule and either discard it or store it. Discarding it leads to immediate increase in entropy but the demon cannot store it indefinitely. In 1982, Charles Bennett showed that, however well prepared, eventually the demon will run out of information storage space and must begin to erase the information it has previously gathered. Erasing information is a thermodynamically irreversible process that increases the entropy of a system. Although Bennett had reached the same conclusion as Szilard's 1929 paper, that a Maxwellian demon could not violate the second law because entropy would be created, he had reached it for different reasons. Regarding Landauer's principle, the minimum energy dissipated by deleting information was experimentally measured by Eric Lutz et al. in 2012. Furthermore, Lutz et al. confirmed that in order to approach the Landauer's limit, the system must asymptotically approach zero processing speed.

对这个问题最著名的回答之一是1929年由莱奥·西拉拉德Leó Szilárd和莱昂·布里渊 Léon Brillouin提出的。Szilárd指出，麦克斯韦妖需要一些方法来测量分子的速度，而获取信息的过程需要消耗能量。由于妖和气体是相互作用的，我们必须考虑气体和妖的总熵。妖的能量消耗将导致妖熵值的增加，这将大于气体熵值的降低。1960年，罗尔夫·兰道尔 Rolf Landauer提出了一个例外。他认识到，只要某些测量过程是热力学可逆的，就不需要增加熵。他认为这些“可逆”的测量可以用来分类分子，违反第二定律。但是，由于热力学熵和信息熵之间的联系，这还意味着不得删除已记录的测量。换句话说，为了决定是否让一个分子通过，妖必须获得关于分子状态的信息，要么丢弃它，要么存储它。丢弃它会立即导致熵的增加，但是妖不能无限期地储存它。1982年，查尔斯·本内特 Charles Bennett指出，无论准备得多么充分，妖最终都会耗尽信息存储空间，并且必须开始删除它先前收集的信息。擦除信息是一种热力学不可逆性过程，它增加了系统的熵。虽然Bennett得出了与Szilárd1929年的论文相同的结论，即麦克斯韦妖不能违反第二定律，因为会产生熵，而熵是由于不同的原因而达到的。根据Landauer原理，埃里克·卢兹 Eric Lutz等人在2012年通过实验测量了擦除信息所消耗的最小能量。此外，Lutz 等人证实，为了接近 Landauer 的极限，系统必须渐近接近零处理速度。

John Earman and John D. Norton have argued that Szilárd and Landauer's explanations of Maxwell's demon begin by assuming that the second law of thermodynamics cannot be violated by the demon, and derive further properties of the demon from this assumption, including the necessity of consuming energy when erasing information, etc. It would therefore be circular to invoke these derived properties to defend the second law from the demonic argument. Bennett later acknowledged the validity of Earman and Norton's argument, while maintaining that Landauer's principle explains the mechanism by which real systems do not violate the second law of thermodynamics.

约翰·厄曼 John Earman和约翰 诺顿 John d. Norton 认为 Szilárd 和 Landauer 对麦克斯韦妖的解释是从假设热力学第二定律不会被妖侵犯开始的，并且从这个假设中得出妖的进一步属性，包括在擦除信息时消耗能量的必要性等等。因此援引这些派生属性来捍卫第二定律不受妖的影响是循环的。Bennett后来承认 Earman 和 Norton 的论点是正确的，同时坚持 Landauer 原理解释了真实系统不违反热力学第二定律的机制

近来的发展

Although the argument by Landauer and Bennett only answers the consistency between the second law of thermodynamics and the whole cyclic process of the entire system of a Szilard engine (a composite system of the engine and the demon), a recent approach based on the non-equilibrium thermodynamics for small fluctuating systems has provided deeper insight on each information process with each subsystem. From this viewpoint, the measurement process is regarded as a process where the correlation (mutual information) between the engine and the demon increases, and the feedback process is regarded as a process where the correlation decreases. If the correlation changes, thermodynamic relations as the second law of thermodynamics and the fluctuation theorem for each subsystem should be modified, and for the case of external control a second-law like inequality and a generalized fluctuation theorem with mutual information are satisfied. These relations suggest that we need extra thermodynamic cost to increase correlation (measurement case), and in contrast we can apparently violate the second law up to the consumption of correlation (feedback case). For more general information processes including biological information processing, both inequality and equality with mutual information hold.

虽然 Landauer 和 Bennett 的论证只是回答了热力学第二定律和整个 Szilard 引擎系统的循环过程之间的一致性(引擎和妖的组合系统) ，但是最近一个基于非平衡态热力学的方法为小型波动系统提供了对每个子系统的信息处理的更深入的了解。从这个观点出发，将测量过程看作是引擎与妖之间相关性(互信息)增加的过程，而反馈过程看作是相关性减少的过程。如果相关性发生变化，则应修改作为热力学第二定律的热力学关系和每个子系统的波动定理，对于外部控制，则满足第二定律，如不等式和具有互信息的广义波动定理。这些关系表明，我们需要额外的热力学成本来增加相关性(测量案例) ，相比之下，我们可以明显地违反第二定律，直到消耗相关性(反馈案例)。对于包括生物信息处理在内的更一般的信息处理过程，互信息的不平等和平等都成立。

应用

Real-life versions of Maxwellian demons occur, but all such "real demons" or molecular demons have their entropy-lowering effects duly balanced by increase of entropy elsewhere. Molecular-sized mechanisms are no longer found only in biology; they are also the subject of the emerging field of nanotechnology. Single-atom traps used by particle physicists allow an experimenter to control the state of individual quanta in a way similar to Maxwell's demon.

现实版本的麦克斯韦妖也会出现，但是所有这些“真正的妖”或者分子妖都有其降低熵值的作用，并且在其他地方熵值的增加中得到适当的平衡。分子大小的机制不再只存在于生物学中; 它们也是纳米技术这一新兴领域的主题。粒子物理学家使用的单原子陷阱使得实验者可以像麦克斯韦妖那样控制单个量子的状态。

If hypothetical mirror matter exists, Zurab Silagadze proposes that demons can be envisaged, "which can act like perpetuum mobiles of the second kind: extract heat energy from only one reservoir, use it to do work and be isolated from the rest of ordinary world. Yet the Second Law is not violated because the demons pay their entropy cost in the hidden (mirror) sector of the world by emitting mirror photons."

如果假想的镜像物质存在，祖拉布·西拉加兹 Zurab Silagadze 提出妖可以被设想，“妖可以像第二类永久移动物一样: 只从一个蓄热池中提取热能，用它来做功，并与其他普通世界隔绝。然而，第二定律并没有被违反，因为妖通过发射镜像光子在世界的隐藏(镜像)区域付出了熵代价。

实验性工作

In the February 2007 issue of Nature, David Leigh, a professor at the University of Edinburgh, announced the creation of a nano-device based on the Brownian ratchet popularized by Richard Feynman. Leigh's device is able to drive a chemical system out of equilibrium, but it must be powered by an external source (light in this case) and therefore does not violate thermodynamics.

在2007年2月的《自然》杂志上，爱丁堡大学的戴维·利 David Leigh 教授宣布制造了一种基于理查德·费曼 Richard Feynman 推广的布朗棘轮的纳米装置。Leigh的设备能够使化学系统失衡，但它必须由外部源(在这种情况下是光)提供动力，因此不违反热力学。

Previously, researchers including Nobel Prize winner Fraser Stoddart, created ring-shaped molecules called rotaxanes which could be placed on an axle connecting two sites, A and B. Particles from either site would bump into the ring and move it from end to end. If a large collection of these devices were placed in a system, half of the devices had the ring at site A and half at B, at any given moment in time.

此前，包括诺贝尔奖获得者弗雷泽·斯托达特 Fraser Stoddart在内的研究人员创造了一种环状分子，称为轮烷，可以放置在连接两个位点A和B的轴上，任何一个位点的粒子都会撞到环上，从一端移动到另一端。如果将大量的这类设备放置在系统中，则在任何给定时间，一半的设备的环在站点A处，一半的站点在B处。

Leigh made a minor change to the axle so that if a light is shone on the device, the center of the axle will thicken, restricting the motion of the ring. It only keeps the ring from moving, however, if it is at A. Over time, therefore, the rings will be bumped from B to A and get stuck there, creating an imbalance in the system. In his experiments, Leigh was able to take a pot of "billions of these devices" from 50:50 equilibrium to a 70:30 imbalance within a few minutes.

Leigh对轮轴做了很小的改动，因此，如果在设备上照亮了灯，轮轴的中心将会变厚，从而限制了环的运动。但是，如果它位于A处，则只能阻止环移动。随着时间的推移，因此，环将从B碰撞到A并被困在那里，在系统中创建不平衡状态。在他的实验中，利 Leigh在几分钟之内就能从50:50的平衡状态变成70:30的不平衡状态，造就了“数十亿个此类设备”。

In 2009 Mark G. Raizen developed a laser atomic cooling technique which realizes the process Maxwell envisioned of sorting individual atoms in a gas into different containers based on their energy. The new concept is a one-way wall for atoms or molecules that allows them to move in one direction, but not go back. The operation of the one-way wall relies on an irreversible atomic and molecular process of absorption of a photon at a specific wavelength, followed by spontaneous emission to a different internal state. The irreversible process is coupled to a conservative force created by magnetic fields and/or light. Raizen and collaborators proposed using the one-way wall in order to reduce the entropy of an ensemble of atoms. In parallel, Gonzalo Muga and Andreas Ruschhaupt independently developed a similar concept. Their "atom diode" was not proposed for cooling, but rather for regulating the flow of atoms. The Raizen Group demonstrated significant cooling of atoms with the one-way wall in a series of experiments in 2008. Subsequently, the operation of a one-way wall for atoms was demonstrated by Daniel Steck and collaborators later in 2008. Their experiment was based on the 2005 scheme for the one-way wall, and was not used for cooling. The cooling method realized by the Raizen Group was called "single-photon cooling", because only one photon on average is required in order to bring an atom to near-rest. This is in contrast to other laser cooling techniques which use the momentum of the photon and require a two-level cycling transition.

2009年，马克 雷岑Mark g. Raizen开发了一种激光原子冷却技术，该技术实现了Maxwell设想的基于能量将气体中的单个原子分类放入不同容器的过程。这个新概念是原子或分子的单向墙，允许它们向一个方向运动，但不能向后移动。单向壁的操作依赖于一个不可逆的原子和分子过程，即吸收一个特定波长的光子，然后自发发射到另一个内部状态。不可逆性与磁场和/或光产生的保守力相耦合。Raizen 和他的合作者提出使用单向壁来减少原子团的熵。与此同时，贡萨洛·穆加 Gonzalo Muga 和安德烈亚斯 Andreas Ruschhaupt 独立地提出了一个类似的概念。他们的“原子二极管”不是用来冷却的，而是用来调节原子流。在2008年的一系列实验中，Raizen 团队证明了单向壁对原子的显著冷却作用。随后，丹尼尔·斯特克 Daniel Steck和合作者在2008年晚些时候演示了单向原子墙的操作。他们的实验是基于2005年的单向墙计划，并没有用于冷却。Raizen团队实现的冷却方法被称为“单光子冷却” ，因为平均只需要一个光子就可以使原子接近静止。这与其他激光冷却技术相反，后者使用光子的动量并需要两级循环过渡。

In 2006, Raizen, Muga, and Ruschhaupt showed in a theoretical paper that as each atom crosses the one-way wall, it scatters one photon, and information is provided about the turning point and hence the energy of that particle. The entropy increase of the radiation field scattered from a directional laser into a random direction is exactly balanced by the entropy reduction of the atoms as they are trapped by the one-way wall.

2006年，Raizen，Muga，和 Ruschhaupt 在一篇论文中指出，当每个原子穿过单向墙时，它散射一个光子，并提供了关于转折点的信息，因此也就提供了这个粒子的能量。定向激光向随机方向散射的辐射场的熵增与被单向壁俘获的原子的熵减完全平衡。

This technique is widely described as a "Maxwell's demon" because it realizes Maxwell's process of creating a temperature difference by sorting high and low energy atoms into different containers. However, scientists have pointed out that it is not a true Maxwell's demon in the sense that it does not violate the second law of thermodynamics; it does not result in a net decrease in entropy Only a year later and based on an earlier theoretical proposal, the same group presented the first experimental realization of an autonomous Maxwell's demon, which extracts microscopic information from a system and reduces its entropy by applying feedback. The demon is based on two capacitively coupled single-electron devices, both integrated on the same electronic circuit. The operation of the demon is directly observed as a temperature drop in the system, with a simultaneous temperature rise in the demon arising from the thermodynamic cost of generating the mutual information. In 2016, Pekola et al. demonstrated a proof-of-principle of an autonomous demon in coupled single-electron circuits, showing a way to cool critical elements in a circuit with information as a fuel. Pekola et al. have also proposed that a simple qubit circuit, e.g., made of a superconducting circuit, could provide a basis to study a quantum Szilard's engine.

这种技术被广泛地描述为“麦克斯韦妖”，因为它通过将高能和低能原子分类到不同的容器中产生温差实现了Maxwell的过程。然而，科学家已经指出，它不是一个真正的麦克斯韦妖，因为它没有违反热力学第二定律; 它不会导致熵减，也不能用来产生有用的能量。这是因为这个过程需要更多的能量从激光束比可能产生的温差。原子从激光束中吸收低熵的光子，并以随机方向发射它们，从而增加了环境的熵。 可能缺失部分原文 2014年，佩科拉 Pekola 等人展示了 Szilárd 引擎实验的实现。仅仅一年之后，同一个研究小组根据早先的理论提议，第一次实验性地实现了自主的麦克斯韦妖，它从一个系统中提取微观信息，并通过反馈减少系统的熵。这个妖是基于集成在同一电路上的两个电容耦合的单电子器件。妖的运行直接表现为系统中的温度下降，同时由于产生互信息的热力学成本而引起的妖中的温度上升。2016年，Pekola 等人证明了单电子耦合电路中存在自主妖的原理，展示了一种以信息为燃料冷却电路中关键元件的方法。Pekola等人还提出，一个简单的量子比特电路，例如由超导电路构成的电路，可以为研究量子Szilárd引擎提供基础。

作为一种隐喻

Daemons in computing, generally processes that run on servers to respond to users, are named for Maxwell's demon.

计算中的妖（通常是在服务器上运行以响应用户的进程）以麦克斯韦妖命名。

Historian Henry Brooks Adams in his manuscript The Rule of Phase Applied to History attempted to use Maxwell's demon as a historical metaphor, though he misunderstood and misapplied the original principle. Adams interpreted history as a process moving towards "equilibrium", but he saw militaristic nations (he felt Germany pre-eminent in this class) as tending to reverse this process, a Maxwell's demon of history. Adams made many attempts to respond to the criticism of his formulation from his scientific colleagues, but the work remained incomplete at Adams' death in 1918. It was only published posthumously.

亨利·布鲁克斯·亚当斯 Henry Brooks Adams在他的手稿《应用于历史的阶段规则 》中试图用麦克斯韦妖作为历史隐喻，尽管他误解和误用了其原本的含义。Adams将历史解释为一个走向“均衡”的过程，但他认为军国主义国家(他认为德国在这一阶层中处于领先地位)倾向于扭转这一过程，这是历史的麦克斯韦妖。Adams曾多次尝试回应他的科学同事对他的公式的批评，但在1918年Adams去世时，这项工作仍未完成。这本书是死后才出版的。