# 时间之箭中的熵

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**Entropy** is the only quantity in the physical sciences (apart from certain rare interactions in particle physics; see below) that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future. However, in thermodynamic systems that are not closed, entropy can decrease with time: many systems, including living systems, reduce local entropy at the expense of an environmental increase, resulting in a net increase in entropy. Examples of such systems and phenomena include the formation of typical crystals, the workings of a refrigerator and living organisms, used in thermodynamics.

Entropy is the only quantity in the physical sciences (apart from certain rare interactions in particle physics; see below) that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future. However, in thermodynamic systems that are not closed, entropy can decrease with time: many systems, including living systems, reduce local entropy at the expense of an environmental increase, resulting in a net increase in entropy. Examples of such systems and phenomena include the formation of typical crystals, the workings of a refrigerator and living organisms, used in thermodynamics.

熵是物理科学中 (除了粒子物理学中某些罕见的相互作用；见下文) 唯一需要特定的时间方向的量，时间的方向有时被称为时间之箭。热力学第二定律显示，随着时间的推移，孤立系统的熵可能增加，但不会减少。因此，从某个角度来看，熵的测量是区分过去和未来的一种方式。然而，在非封闭的热力学系统中，熵可能随时间减少：许多系统，包括生命系统，以环境熵的增加为代价减少局部熵，导致熵的净增加。在热力学中，这些系统和现象的例子包括典型晶体的形成、冰箱的工作原理以及生物体。

Much like temperature, despite being an abstract concept, everyone has an intuitive sense of the effects of entropy. For example, it is often very easy to tell the difference between a video being played forwards or backwards. A video may depict a wood fire that melts a nearby ice block, played in reverse it would show that a puddle of water turned a cloud of smoke into unburnt wood and froze itself in the process. Surprisingly, in either case the vast majority of the laws of physics are not broken by these processes, a notable exception being the second law of thermodynamics. When a law of physics applies equally when time is reversed, it is said to show T-symmetry, in this case entropy is what allows one to decide if the video described above is playing forwards or in reverse as intuitively we identify that only when played forwards the entropy of the scene is increasing. Because of the second law of thermodynamics, entropy prevents macroscopic processes showing T-symmetry.

Much like temperature, despite being an abstract concept, everyone has an intuitive sense of the effects of entropy. For example, it is often very easy to tell the difference between a video being played forwards or backwards. A video may depict a wood fire that melts a nearby ice block, played in reverse it would show that a puddle of water turned a cloud of smoke into unburnt wood and froze itself in the process. Surprisingly, in either case the vast majority of the laws of physics are not broken by these processes, a notable exception being the second law of thermodynamics. When a law of physics applies equally when time is reversed, it is said to show T-symmetry, in this case entropy is what allows one to decide if the video described above is playing forwards or in reverse as intuitively we identify that only when played forwards the entropy of the scene is increasing. Because of the second law of thermodynamics, entropy prevents macroscopic processes showing T-symmetry.

虽然熵像温度一样是抽象概念，但每个人对熵的影响都有直观感觉。例如，人们区分一条视频是正放还是倒放通常很容易。倒放一段关于木头燃烧融化附近冰块的视频，原过程就会变为一滩水把一团烟雾变成了未燃烧的木头，并在这个过程中冻结了自己。令人惊讶的是，在任何情况下，绝大多数物理定律都没有被这些过程打破，但热力学第二定律是一个值得注意的例外。如果物理定律在时间倒转时同样适用，那么它就表现出了T对称，在这种情况下，熵使人们有能力分辨上述视频是正放还是倒放的，因为我们可以直观地发现，只有在正放时，场景的熵才会增加。基于热力学第二定律，熵会阻止宏观过程呈现T对称。

When studying at a microscopic scale, the above judgements cannot be made. Watching a single smoke particle buffeted by air, it would not be clear if a video was playing forwards or in reverse, and, in fact, it would not be possible as the laws which apply show T-symmetry, as it drifts left or right *qualitatively* it looks no different. It is only when you study that gas at a macroscopic scale that the effects of entropy become noticeable. On average you would expect the smoke particles around a struck match to drift away from each other, diffusing throughout the available space. It would be an astronomically improbable event for all the particles to cluster together, yet you cannot comment on the movement of any one smoke particle.

When studying at a microscopic scale, the above judgements cannot be made. Watching a single smoke particle buffeted by air, it would not be clear if a video was playing forwards or in reverse, and, in fact, it would not be possible as the laws which apply show T-symmetry, as it drifts left or right qualitatively it looks no different. It is only when you study that gas at a macroscopic scale that the effects of entropy become noticeable. On average you would expect the smoke particles around a struck match to drift away from each other, diffusing throughout the available space. It would be an astronomically improbable event for all the particles to cluster together, yet you cannot comment on the movement of any one smoke particle.

在微观尺度上研究时不能做出上述判断。观察单个烟尘颗粒被空气反复冲击时，我们不清楚视频是正放的还是倒放的，其实也不可能分辨出来，因为适用的定律显示了t对称，也因为它向左或向右漂动看起来没有什么不同。只有当你在宏观尺度上研究气体时，熵的影响才会变得明显。平均而言，你会认为被擦过的火柴周围的烟尘颗粒会彼此远离，扩散到整个空间。所有的粒子都聚集在一起是一个极不可能的事件，你也不能解释任何一个烟雾粒子的运动。

By contrast, certain subatomic interactions involving the weak nuclear force violate the conservation of parity, but only very rarely.^{[citation needed]} According to the CPT theorem, this means they should also be time irreversible, and so establish an arrow of time. This, however, is neither linked to the thermodynamic arrow of time, nor has anything to do with the daily experience of time irreversibility.^{[1]}

By contrast, certain subatomic interactions involving the weak nuclear force violate the conservation of parity, but only very rarely. According to the CPT theorem, this means they should also be time irreversible, and so establish an arrow of time. This, however, is neither linked to the thermodynamic arrow of time, nor has anything to do with the daily experience of time irreversibility.

相比之下，某些涉及弱核力的亚原子相互作用却很少违反宇称守恒定律。根据CPT定理，它们也应该是时间不可逆的，从而建立了一个时间箭头。然而，这与时间的热力学箭头无关，也与具有时间不可逆性的日常经验无关。

## Overview

The Second Law of Thermodynamics allows for the entropy to *remain the same* regardless of the direction of time. If the entropy is constant in either direction of time, there would be no preferred direction. However, the entropy can only be a constant if the system is in the highest possible state of disorder, such as a gas that always was, and always will be, uniformly spread out in its container. The existence of a thermodynamic arrow of time implies that the system is highly ordered in one time direction only, which would by definition be the "past". Thus this law is about the boundary conditions rather than the equations of motion.

The Second Law of Thermodynamics allows for the entropy to remain the same regardless of the direction of time. If the entropy is constant in either direction of time, there would be no preferred direction. However, the entropy can only be a constant if the system is in the highest possible state of disorder, such as a gas that always was, and always will be, uniformly spread out in its container. The existence of a thermodynamic arrow of time implies that the system is highly ordered in one time direction only, which would by definition be the "past". Thus this law is about the boundary conditions rather than the equations of motion.

无论时间方向如何，热力学第二定律都允许熵保持不变。如果熵在时间的任何一个方向上都恒定，那么就不存在优选的方向。然而，只有当系统处于最高可能的无序状态时，熵才能是一个常数，例如，气体总是均匀地散布在容器中。热力学时间箭头的存在意味着系统在一个时间方向上是高度有序的，这个方向就是定义中的“过去”。因此，这个定律是关于边界条件而不是运动方程的。

The Second Law of Thermodynamics is statistical in nature, and therefore its reliability arises from the huge number of particles present in macroscopic systems. It is not impossible, in principle, for all 6 × 10^{23} atoms in a mole of a gas to spontaneously migrate to one half of a container; it is only *fantastically* unlikely—so unlikely that no macroscopic violation of the Second Law has ever been observed. T Symmetry is the symmetry of physical laws under a time reversal transformation. Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.

The Second Law of Thermodynamics is statistical in nature, and therefore its reliability arises from the huge number of particles present in macroscopic systems. It is not impossible, in principle, for all 6 × 10^{23} atoms in a mole of a gas to spontaneously migrate to one half of a container; it is only fantastically unlikely—so unlikely that no macroscopic violation of the Second Law has ever been observed. T Symmetry is the symmetry of physical laws under a time reversal transformation. Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.

热力学第二定律有统计学本质，因此它的可靠性来自于宏观系统中存在的大量粒子。原则上，1摩尔气体中6×10^{23}个原子全部自发迁移到容器的一半空间并不是不可能的，只是极不可能，以至于观察不到宏观上违反第二定律的情况。T对称性是物理定律在时间反演变换下的对称性。虽然在有限的情况下人们可能会发现这种对称性，但在时间反演下，可观测的宇宙本身并不表现出对称性，这主要归因于热力学第二定律。

The thermodynamic arrow is often linked to the cosmological arrow of time, because it is ultimately about the boundary conditions of the early universe. According to the Big Bang theory, the Universe was initially very hot with energy distributed uniformly. For a system in which gravity is important, such as the universe, this is a low-entropy state (compared to a high-entropy state of having all matter collapsed into black holes, a state to which the system may eventually evolve). As the Universe grows, its temperature drops, which leaves less energy available to perform work in the future than was available in the past. Additionally, perturbations in the energy density grow (eventually forming galaxies and stars). Thus the Universe itself has a well-defined thermodynamic arrow of time. But this does not address the question of why the initial state of the universe was that of low entropy. If cosmic expansion were to halt and reverse due to gravity, the temperature of the Universe would once again grow hotter, but its entropy would also continue to increase due to the continued growth of perturbations and the eventual black hole formation,^{[2]} until the latter stages of the Big Crunch when entropy would be lower than now.^{[citation needed]}

The thermodynamic arrow is often linked to the cosmological arrow of time, because it is ultimately about the boundary conditions of the early universe. According to the Big Bang theory, the Universe was initially very hot with energy distributed uniformly. For a system in which gravity is important, such as the universe, this is a low-entropy state (compared to a high-entropy state of having all matter collapsed into black holes, a state to which the system may eventually evolve). As the Universe grows, its temperature drops, which leaves less energy available to perform work in the future than was available in the past. Additionally, perturbations in the energy density grow (eventually forming galaxies and stars). Thus the Universe itself has a well-defined thermodynamic arrow of time. But this does not address the question of why the initial state of the universe was that of low entropy. If cosmic expansion were to halt and reverse due to gravity, the temperature of the Universe would once again grow hotter, but its entropy would also continue to increase due to the continued growth of perturbations and the eventual black hole formation, until the latter stages of the Big Crunch when entropy would be lower than now.

热力学箭头经常与时间的宇宙学箭头联系在一起，因为它最终与早期宇宙的边界条件有关。根据大爆炸理论，宇宙最初非常热，能量均匀分布。在重力很重要的系统（比如宇宙）中，这是一种低熵状态(相比之下，所有物质都坍缩成黑洞的高熵状态，系统最终可能会演化到这种状态)。宇宙温度随着发展而下降，未来可用于运转的能量比过去更少。此外，能量密度的扰动会增加(最终形成星系和星体)。因此，宇宙本身就有一个定义明确的热力学时间箭头。但这并没有解决宇宙的初始状态为什么是低熵的问题。如果宇宙膨胀由于重力而停止和逆转，宇宙的温度将再次变得更高，但由于扰动持续增长和最终黑洞形成，宇宙的熵也将继续增加，直到大破裂的后期，那时的熵将比现在更低。

## An example of apparent irreversibility

Consider the situation in which a large container is filled with two separated liquids, for example a dye on one side and water on the other. With no barrier between the two liquids, the random jostling of their molecules will result in them becoming more mixed as time passes. However, if the dye and water are mixed then one does not expect them to separate out again when left to themselves. A movie of the mixing would seem realistic when played forwards, but unrealistic when played backwards.

Consider the situation in which a large container is filled with two separated liquids, for example a dye on one side and water on the other. With no barrier between the two liquids, the random jostling of their molecules will result in them becoming more mixed as time passes. However, if the dye and water are mixed then one does not expect them to separate out again when left to themselves. A movie of the mixing would seem realistic when played forwards, but unrealistic when played backwards.

考虑这样一种情况: 一个大容器装有两种分开的液体，例如一边是染料，另一边是水。由于两种液体之间没有屏障，随着时间的推移，它们的分子之间的随机碰撞将导致它们变得更加混合。然而，如果染料和水是混合的，那么当把它们留给它们自己时，就不会期望它们再次分离出来。一部混合的电影在向前播放的时候看起来很真实，但是向后播放的时候就不真实了。

If the large container is observed early on in the mixing process, it might be found only partially mixed. It would be reasonable to conclude that, without outside intervention, the liquid reached this state because it was more ordered in the past, when there was greater separation, and will be more disordered, or mixed, in the future.

If the large container is observed early on in the mixing process, it might be found only partially mixed. It would be reasonable to conclude that, without outside intervention, the liquid reached this state because it was more ordered in the past, when there was greater separation, and will be more disordered, or mixed, in the future.

如果观察大容器的早期混合过程，则可能发现其内部只是部分混合。由此可以合理地得出这样的结论: 在没有外界干预的情况下，液体达到这种状态是因为在过去分离程度较大时，液体的有序程度较高，而在未来，液体的无序程度或混合程度更高。

Now imagine that the experiment is repeated, this time with only a few molecules, perhaps ten, in a very small container. One can easily imagine that by watching the random jostling of the molecules it might occur — by chance alone — that the molecules became neatly segregated, with all dye molecules on one side and all water molecules on the other. That this can be expected to occur from time to time can be concluded from the fluctuation theorem; thus it is not impossible for the molecules to segregate themselves. However, for a large numbers of molecules it is so unlikely that one would have to wait, on average, many times longer than the age of the universe for it to occur. Thus a movie that showed a large number of molecules segregating themselves as described above would appear unrealistic and one would be inclined to say that the movie was being played in reverse. See Boltzmann's Second Law as a law of disorder.

Now imagine that the experiment is repeated, this time with only a few molecules, perhaps ten, in a very small container. One can easily imagine that by watching the random jostling of the molecules it might occur — by chance alone — that the molecules became neatly segregated, with all dye molecules on one side and all water molecules on the other. That this can be expected to occur from time to time can be concluded from the fluctuation theorem; thus it is not impossible for the molecules to segregate themselves. However, for a large numbers of molecules it is so unlikely that one would have to wait, on average, many times longer than the age of the universe for it to occur. Thus a movie that showed a large number of molecules segregating themselves as described above would appear unrealistic and one would be inclined to say that the movie was being played in reverse. See Boltzmann's Second Law as a law of disorder.

现在想象重复这个实验，这一次在一个非常小的容器里只有几个分子，也许是十个分子。容易想象，通过观察分子的随机碰撞，可能会发现这种情况——分子整齐地分开，所有的染料分子在一边，所有的水分子在另一边，而这仅仅是偶然的。根据涨落定理可以得出这种情况时有发生的结论；因此，分子是可能自行分离的。然而，对于大量的分子来说，这种现象不太可能发生，以至于人们平均需要等待比宇宙年龄长很多倍的时间才能发生这种情况。因此，上述大量分子相互分离不切实际，人们可能会倾向于说这部电影是在倒放。把波尔兹曼第二定律看作无序定律。

## Mathematics of the arrow

The mathematics behind the *arrow of time*, entropy, and basis of the second law of thermodynamics derive from the following set-up, as detailed by Carnot (1824), Clapeyron (1832), and Clausius (1854):

The mathematics behind the arrow of time, entropy, and basis of the second law of thermodynamics derive from the following set-up, as detailed by Carnot (1824), Clapeyron (1832), and Clausius (1854):

时间箭头、熵和热力学第二定律的基础数学背景来源于以下的设置，详见卡诺(1824)、克拉佩龙(1832)和克劳修斯(1854) :

Here, as common experience demonstrates, when a hot body *T _{1}*, such as a furnace, is put into physical contact, such as being connected via a body of fluid (working body), with a cold body

*T*, such as a stream of cold water, energy will invariably flow from hot to cold in the form of heat

_{2}*Q*, and given

**time**the system will reach equilibrium. Entropy, defined as Q/T, was conceived by Rudolf Clausius as a function to measure the molecular irreversibility of this process, i.e. the dissipative work the atoms and molecules do on each other during the transformation.

Here, as common experience demonstrates, when a hot body T_{1}, such as a furnace, is put into physical contact, such as being connected via a body of fluid (working body), with a cold body T_{2}, such as a stream of cold water, energy will invariably flow from hot to cold in the form of heat Q, and given time the system will reach equilibrium. Entropy, defined as Q/T, was conceived by Rudolf Clausius as a function to measure the molecular irreversibility of this process, i.e. the dissipative work the atoms and molecules do on each other during the transformation.

这里，如一般经验所示，当热物体T1（例如炉子）通过流体(工作物体)与冷物体T2（例如冷水）物理接触时，能量总是以热 Q 的形式从热的物体流向冷的物体，并且系统将在给定时间内达到平衡。熵定义为 q / t，由 Rudolf Clausius 提出，作为一个函数来衡量这个过程的分子不可逆性，即原子和分子在转变过程中对彼此做的耗散功。

In this diagram, one can calculate the entropy change Δ*S* for the passage of the quantity of heat *Q* from the temperature *T _{1}*, through the "working body" of fluid (see heat engine), which was typically a body of steam, to the temperature

*T*. Moreover, one could assume, for the sake of argument, that the working body contains only two molecules of water.

_{2}In this diagram, one can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T_{1}, through the "working body" of fluid (see heat engine), which was typically a body of steam, to the temperature T_{2}. Moreover, one could assume, for the sake of argument, that the working body contains only two molecules of water.

在此图中，我们可以计算热量 Q 通过通常是蒸汽的流体的“工作体”(见热机)从温度T1到温度 T2 的熵变 ΔS。此外，为了讨论的目的，可以假设工作物体只含有两个水分子。

Next, if we make the assignment, as originally done by Clausius:

Next, if we make the assignment, as originally done by Clausius:

接下来，如果我们像克劳修斯最初所做的那样:

- [math]\displaystyle{ S= \frac {Q}{T} }[/math]

[math]\displaystyle{ S= \frac {Q}{T} }[/math]

Then the entropy change or "equivalence-value" for this transformation is:

Then the entropy change or "equivalence-value" for this transformation is:

那么这个转化过程的熵变或“等价值”是:

- [math]\displaystyle{ \Delta S = S_{\mathit{final}} - S_{\mathit{initial}} \, }[/math]

[math]\displaystyle{ \Delta S = S_{\mathit{final}} - S_{\mathit{initial}} \, }[/math]

which equals:

which equals:

等于:

- [math]\displaystyle{ \Delta S = \left(\frac {Q}{T_2} - \frac {Q}{T_1}\right) }[/math]

[math]\displaystyle{ \Delta S = \left(\frac {Q}{T_2} - \frac {Q}{T_1}\right) }[/math]

and by factoring out Q, we have the following form, as was derived by Clausius:

and by factoring out Q, we have the following form, as was derived by Clausius:

通过分解 Q，我们得到了下面的形式，正如克劳修斯所推导的:

- [math]\displaystyle{ \Delta S = Q\left(\frac {1}{T_2} - \frac {1}{T_1}\right) }[/math]

[math]\displaystyle{ \Delta S = Q\left(\frac {1}{T_2} - \frac {1}{T_1}\right) }[/math]

Thus, for example, if Q was 50 units, *T _{1}* was initially 100 degrees, and

*T*was initially 1 degree, then the entropy change for this process would be 49.5. Hence, entropy increased for this process, the process took a certain amount of "time", and one can correlate entropy increase with the passage of time. For this system configuration, subsequently, it is an "absolute rule". This rule is based on the fact that all natural processes are irreversible by virtue of the fact that molecules of a system, for example two molecules in a tank, not only do external work (such as to push a piston), but also do internal work on each other, in proportion to the heat used to do work (see: Mechanical equivalent of heat) during the process. Entropy accounts for the fact that internal inter-molecular friction exists.

_{2}Thus, for example, if Q was 50 units, T_{1} was initially 100 degrees, and T_{2} was initially 1 degree, then the entropy change for this process would be 49.5. Hence, entropy increased for this process, the process took a certain amount of "time", and one can correlate entropy increase with the passage of time. For this system configuration, subsequently, it is an "absolute rule". This rule is based on the fact that all natural processes are irreversible by virtue of the fact that molecules of a system, for example two molecules in a tank, not only do external work (such as to push a piston), but also do internal work on each other, in proportion to the heat used to do work (see: Mechanical equivalent of heat) during the process. Entropy accounts for the fact that internal inter-molecular friction exists.

因此，举例来说，如果 Q有50个单位，T1 最初是100度，T2 最初是1度，那么这个过程的熵变为49.5，因此，这个过程的熵增加了。过程需要一定的“时间” 并且熵增与时间流逝相关，对于该系统的配置来说，这是“绝对规则”。这一规则是基于这样一个事实，即所有的自然过程都是不可逆的，因为一个系统的分子（例如一个罐子中的两个分子）不仅做外部功(如推动活塞) ，而且按过程中所用的热量(见: 热量的机械等效物)成比例地对彼此做内部功。熵解释了分子间存在内部摩擦力的事实。

## Correlations

An important difference between the past and the future is that in any system (such as a gas of particles) its initial conditions are usually such that its different parts are uncorrelated, but as the system evolves and its different parts interact with each other, they become correlated.^{[3]} For example, whenever dealing with a gas of particles, it is always assumed that its initial conditions are such that there is no correlation between the states of different particles (i.e. the speeds and locations of the different particles are completely random, up to the need to conform with the macrostate of the system). This is closely related to the Second Law of Thermodynamics.

An important difference between the past and the future is that in any system (such as a gas of particles) its initial conditions are usually such that its different parts are uncorrelated, but as the system evolves and its different parts interact with each other, they become correlated. For example, whenever dealing with a gas of particles, it is always assumed that its initial conditions are such that there is no correlation between the states of different particles (i.e. the speeds and locations of the different particles are completely random, up to the need to conform with the macrostate of the system). This is closely related to the Second Law of Thermodynamics.

过去和未来的一个重要区别是，在任何系统(如粒子气体)中，其初始条件通常如此：系统不同部分是不相关的，但随着系统的演化和不同部分相互作用，它们会变得相互关联。例如，每当处理粒子气体时，总是假设其初始条件是这样的，即不同粒子的状态之间没有相关性(即，不同粒子的速度和位置是完全随机的，直到需要符合系统的宏观状态)。这与热力学第二定律密切相关。

Take for example (experiment A) a closed box that is, at the beginning, half-filled with ideal gas. As time passes, the gas obviously expands to fill the whole box, so that the final state is a box full of gas. This is an irreversible process, since if the box is full at the beginning (experiment B), it does not become only half-full later, except for the very unlikely situation where the gas particles have very special locations and speeds. But this is precisely because we always assume that the initial conditions are such that the particles have random locations and speeds. This is not correct for the final conditions of the system, because the particles have interacted between themselves, so that their locations and speeds have become dependent on each other, i.e. correlated. This can be understood if we look at experiment A backwards in time, which we'll call experiment C: now we begin with a box full of gas, but the particles do not have random locations and speeds; rather, their locations and speeds are so particular, that after some time they all move to one half of the box, which is the final state of the system (this is the initial state of experiment A, because now we're looking at the same experiment backwards!). The interactions between particles now do not create correlations between the particles, but in fact turn them into (at least seemingly) random, "canceling" the pre-existing correlations. The only difference between experiment C (which defies the Second Law of Thermodynamics) and experiment B (which obeys the Second Law of Thermodynamics) is that in the former the particles are uncorrelated at the end, while in the latter the particles are uncorrelated at the beginning.^{[citation needed]}

Take for example (experiment A) a closed box that is, at the beginning, half-filled with ideal gas. As time passes, the gas obviously expands to fill the whole box, so that the final state is a box full of gas. This is an irreversible process, since if the box is full at the beginning (experiment B), it does not become only half-full later, except for the very unlikely situation where the gas particles have very special locations and speeds. But this is precisely because we always assume that the initial conditions are such that the particles have random locations and speeds. This is not correct for the final conditions of the system, because the particles have interacted between themselves, so that their locations and speeds have become dependent on each other, i.e. correlated. This can be understood if we look at experiment A backwards in time, which we'll call experiment C: now we begin with a box full of gas, but the particles do not have random locations and speeds; rather, their locations and speeds are so particular, that after some time they all move to one half of the box, which is the final state of the system (this is the initial state of experiment A, because now we're looking at the same experiment backwards!). The interactions between particles now do not create correlations between the particles, but in fact turn them into (at least seemingly) random, "canceling" the pre-existing correlations. The only difference between experiment C (which defies the Second Law of Thermodynamics) and experiment B (which obeys the Second Law of Thermodynamics) is that in the former the particles are uncorrelated at the end, while in the latter the particles are uncorrelated at the beginning.

例如(实验A)，一个封闭的盒子，一开始是半满的理想气体。随着时间的推移，气体明显膨胀到装满整个盒子，所以最终的状态是一个装满气体的盒子。这是一个不可逆转的过程，因为如果盒子在一开始是满的(实验B)，它不会在之后变得只有半满，除了非常不可能的情况，即气体颗粒具有非常特殊的位置和速度。但这恰恰是因为我们总是假设初始条件中粒子具有随机的位置和速度。这对于系统的最终条件是不正确的，因为粒子之间已经进行了相互作用，以至于它们的位置和速度已经变得相互依赖，即相互关联。如果我们倒着看实验A（称之为实验C），这是可以理解的：现在我们从一个装满气体的盒子开始，但粒子没有随机的位置和速度；相反，它们的位置和速度很特殊，以至于一段时间后，它们都移动到盒子的半边，达到系统的最终状态(这是实验A的初始状态，因为现在我们正在反向管看相同的实验！)。粒子之间的相互作用现在不会使粒子之间产生关联，而实际上是将它们变成(至少看起来是)随机的，“取消”先前存在的关联。实验C(违反热力学第二定律)和实验B(遵守热力学第二定律)之间的唯一区别是，在前者中，粒子在结束时不相关，而在后者中，粒子在开始时不相关。

In fact, if all the microscopic physical processes are reversible (see discussion below), then the Second Law of Thermodynamics can be proven for any isolated system of particles with initial conditions in which the particles states are uncorrelated. To do this, one must acknowledge the difference between the measured entropy of a system—which depends only on its macrostate (its volume, temperature etc.)—and its information entropy,^{[4]} which is the amount of information (number of computer bits) needed to describe the exact microstate of the system. The measured entropy is independent of correlations between particles in the system, because they do not affect its macrostate, but the information entropy **does** depend on them, because correlations lower the randomness of the system and thus lowers the amount of information needed to describe it.^{[5]} Therefore, in the absence of such correlations the two entropies are identical, but otherwise the information entropy is smaller than the measured entropy, and the difference can be used as a measure of the amount of correlations.

In fact, if all the microscopic physical processes are reversible (see discussion below), then the Second Law of Thermodynamics can be proven for any isolated system of particles with initial conditions in which the particles states are uncorrelated. To do this, one must acknowledge the difference between the measured entropy of a system—which depends only on its macrostate (its volume, temperature etc.)—and its information entropy, which is the amount of information (number of computer bits) needed to describe the exact microstate of the system. The measured entropy is independent of correlations between particles in the system, because they do not affect its macrostate, but the information entropy does depend on them, because correlations lower the randomness of the system and thus lowers the amount of information needed to describe it. Therefore, in the absence of such correlations the two entropies are identical, but otherwise the information entropy is smaller than the measured entropy, and the difference can be used as a measure of the amount of correlations.

事实上，如果所有的微观物理过程都是可逆的(见下面的讨论) ，那么对于任何一个孤立的粒子系统，只要其初始条件中的粒子状态是不相关的，那么热力学第二定律就可以被证明。要做到这一点，我们必须认识到一个系统测量熵和信息熵（描述系统精确微状态所需的信息量，即计算机位数）的差别ーー这个差别仅仅取决于它的宏观状态(体积、温度等)。测量熵与系统中粒子之间的相关性无关，因为它们不影响系统的宏观状态，但熵确实依赖于粒子间的相关性，因为相关性降低了系统的随机性，从而降低了描述系统所需的信息量。因此，在没有这种相关性的情况下，两种熵是相同的，否则信息熵比测量熵小，这一差值可以用来衡量相关性程度。

Now, by Liouville's theorem, time-reversal of all microscopic processes implies that the amount of information needed to describe the exact microstate of an isolated system (its information-theoretic joint entropy) is constant in time. This joint entropy is equal to the marginal entropy (entropy assuming no correlations) plus the entropy of correlation (mutual entropy, or its negative mutual information). If we assume no correlations between the particles initially, then this joint entropy is just the marginal entropy, which is just the initial thermodynamic entropy of the system, divided by Boltzmann's constant. However, if these are indeed the initial conditions (and this is a crucial assumption), then such correlations form with time. In other words, there is a decreasing mutual entropy (or increasing mutual information), and for a time that is not too long—the correlations (mutual information) between particles only increase with time. Therefore, the thermodynamic entropy, which is proportional to the marginal entropy, must also increase with time ^{[6]} (note that "not too long" in this context is relative to the time needed, in a classical version of the system, for it to pass through all its possible microstates—a time that can be roughly estimated as [math]\displaystyle{ \tau e^S }[/math], where [math]\displaystyle{ \tau }[/math] is the time between particle collisions and S is the system's entropy. In any practical case this time is huge compared to everything else). Note that the correlation between particles is not a fully objective quantity. One cannot measure the mutual entropy, one can only measure its change, assuming one can measure a microstate. Thermodynamics is restricted to the case where microstates cannot be distinguished, which means that only the marginal entropy, proportional to the thermodynamic entropy, can be measured, and, in a practical sense, always increases.

Now, by Liouville's theorem, time-reversal of all microscopic processes implies that the amount of information needed to describe the exact microstate of an isolated system (its information-theoretic joint entropy) is constant in time. This joint entropy is equal to the marginal entropy (entropy assuming no correlations) plus the entropy of correlation (mutual entropy, or its negative mutual information). If we assume no correlations between the particles initially, then this joint entropy is just the marginal entropy, which is just the initial thermodynamic entropy of the system, divided by Boltzmann's constant. However, if these are indeed the initial conditions (and this is a crucial assumption), then such correlations form with time. In other words, there is a decreasing mutual entropy (or increasing mutual information), and for a time that is not too long—the correlations (mutual information) between particles only increase with time. Therefore, the thermodynamic entropy, which is proportional to the marginal entropy, must also increase with time (note that "not too long" in this context is relative to the time needed, in a classical version of the system, for it to pass through all its possible microstates—a time that can be roughly estimated as [math]\displaystyle{ \tau e^S }[/math], where [math]\displaystyle{ \tau }[/math] is the time between particle collisions and S is the system's entropy. In any practical case this time is huge compared to everything else). Note that the correlation between particles is not a fully objective quantity. One cannot measure the mutual entropy, one can only measure its change, assuming one can measure a microstate. Thermodynamics is restricted to the case where microstates cannot be distinguished, which means that only the marginal entropy, proportional to the thermodynamic entropy, can be measured, and, in a practical sense, always increases.

现在，根据刘维尔定理，所有微观过程的时间反转意味着描述孤立系统精确微观状态所需要的信息量(信息论中的联合熵)在时间上是恒定的。这个联合熵等于边际熵(假设没有相关性的熵)加上相关熵(互熵，或其负互信息)。如果我们假设最初粒子之间没有相关性，那么联合熵就是边际熵，也就是系统的初始热力学熵除以玻耳兹曼常数。然而，如果这些确实是初始条件(这是一个关键的假设) ，那么这种相关性随时间推移而形成。换句话说，互熵减少(互信息增加) ，并且在不太长的时间内，粒子之间的关联(互信息)只会随时间推移而增加。因此，与边际熵成正比的热力学熵也必须随着时间而增加(注意，在这种情况下，上文中的“不太长”是相对于经典系统所需的时间，以便它经过所有可能的微观状态ーー这段时间可以粗略地估计为 ，其中 是粒子碰撞之间的时间，S是系统的熵。在任何实际的情况下，这段时间比其他任何事情都重要)。注意，粒子之间的相关性并不是一个完全客观的量。不能测量互熵，只能在假设可以测量微观状态的情况下测量它的变化。热力学被限制在微观状态无法区分的情况下，这意味着只有与热力学熵成正比的边际熵才能测量，而且，在实际情况中，热力学熵总是在增加。

## The arrow of time in various phenomena

All phenomena that behave differently in one time direction can ultimately be linked to the Second Law of Thermodynamics^{[citation needed]}. This includes the fact that ice cubes melt in hot coffee rather than assembling themselves out of the coffee, that a block sliding on a rough surface slows down rather than speeding up, and that we can remember the past rather than the future. This last phenomenon, called the "psychological arrow of time", has deep connections with Maxwell's demon and the physics of information; In fact, it is easy to understand its link to the Second Law of Thermodynamics if one views memory as correlation between brain cells (or computer bits) and the outer world^{[citation needed]}.

All phenomena that behave differently in one time direction can ultimately be linked to the Second Law of Thermodynamics. This includes the fact that ice cubes melt in hot coffee rather than assembling themselves out of the coffee, that a block sliding on a rough surface slows down rather than speeding up, and that we can remember the past rather than the future. This last phenomenon, called the "psychological arrow of time", has deep connections with Maxwell's demon and the physics of information; In fact, it is easy to understand its link to the Second Law of Thermodynamics if one views memory as correlation between brain cells (or computer bits) and the outer world.

所有在一个时间方向上表现得不同的现象最终都可以与热力学第二定律联系起来。这包括：冰块在热咖啡中融化而不是从咖啡中自行组合出来；在粗糙表面上滑动的冰块会减慢而不是加速；我们能记住过去而不是未来。事最后一种现象被称为”心理时间之箭“，它与麦克斯韦妖和信息物理学有深刻联系；事实上，如果把记忆看作是脑细胞(或计算机位元)和外部世界之间的联系，就很容易理解它与热力学第二定律的联系。

## Current research

Current research focuses mainly on describing the thermodynamic arrow of time mathematically, either in classical or quantum systems, and on understanding its origin from the point of view of cosmological boundary conditions.

Current research focuses mainly on describing the thermodynamic arrow of time mathematically, either in classical or quantum systems, and on understanding its origin from the point of view of cosmological boundary conditions.

目前的研究主要集中在用数学方法描述经典系统和量子系统中的时间热力学箭头，以及从宇宙学边界条件的角度理解时间箭头的来源。

### Dynamical systems

Some current research in dynamical systems indicates a possible "explanation" for the arrow of time.^{[citation needed]} There are several ways to describe the time evolution of a dynamical system. In the classical framework, one considers a differential equation, where one of the parameters is explicitly time. By the very nature of differential equations, the solutions to such systems are inherently time-reversible. However, many of the interesting cases are either ergodic or mixing, and it is strongly suspected that mixing and ergodicity somehow underlie the fundamental mechanism of the arrow of time.

Some current research in dynamical systems indicates a possible "explanation" for the arrow of time. There are several ways to describe the time evolution of a dynamical system. In the classical framework, one considers a differential equation, where one of the parameters is explicitly time. By the very nature of differential equations, the solutions to such systems are inherently time-reversible. However, many of the interesting cases are either ergodic or mixing, and it is strongly suspected that mixing and ergodicity somehow underlie the fundamental mechanism of the arrow of time.

目前对动力系统的一些研究提出了一种对时间之箭的可能的“解释”。有几种方法可以描述动力系统的时间演变。在经典的框架中，考虑一个微分方程，其中一个参数必然是时间。由于微分方程的本质，这类系统的解本质上是时间可逆的。然而，许多有趣的情况要么是遍历的，要么是混合的，而且强烈怀疑混合和遍历性以某种方式构成了时间箭头的基本机制。

Mixing and ergodic systems do not have exact solutions, and thus proving time irreversibility in a mathematical sense is (模板:As of) impossible. Some progress can be made by studying discrete-time models or difference equations. Many discrete-time models, such as the iterated functions considered in popular fractal-drawing programs, are explicitly not time-reversible, as any given point "in the present" may have several different "pasts" associated with it: indeed, the set of all pasts is known as the Julia set. Since such systems have a built-in irreversibility, it is inappropriate to use them to explain why time is not reversible.

Mixing and ergodic systems do not have exact solutions, and thus proving time irreversibility in a mathematical sense is () impossible. Some progress can be made by studying discrete-time models or difference equations. Many discrete-time models, such as the iterated functions considered in popular fractal-drawing programs, are explicitly not time-reversible, as any given point "in the present" may have several different "pasts" associated with it: indeed, the set of all pasts is known as the Julia set. Since such systems have a built-in irreversibility, it is inappropriate to use them to explain why time is not reversible.

混合系统和遍历系统没有精确解，因此从数学意义上证明时间的不可逆性是不可能的。研究离散时间模型或差分方程可以取得一些进展。许多离散时间模型，例如在流行的分形绘图程序中考虑的迭代函数，显然是不可逆的，因为可能有几个不同的“过去”与任何给定点“在当前”相关联: 实际上，所有过去的集合都被称为 Julia 集。因为这种系统具有内在的不可逆性，所以不宜用它们来解释时间为什么不可逆。

There are other systems that are chaotic, and are also explicitly time-reversible: among these is the baker's map, which is also exactly solvable. An interesting avenue of study is to examine solutions to such systems not by iterating the dynamical system over time, but instead, to study the corresponding Frobenius-Perron operator or transfer operator for the system. For some of these systems, it can be explicitly, mathematically shown that the transfer operators are not trace-class. This means that these operators do not have a unique eigenvalue spectrum that is independent of the choice of basis. In the case of the baker's map, it can be shown that several unique and inequivalent diagonalizations or bases exist, each with a different set of eigenvalues. It is this phenomenon that can be offered as an "explanation" for the arrow of time. That is, although the iterated, discrete-time system is explicitly time-symmetric, the transfer operator is not. Furthermore, the transfer operator can be diagonalized in one of two inequivalent ways: one that describes the forward-time evolution of the system, and one that describes the backwards-time evolution.

There are other systems that are chaotic, and are also explicitly time-reversible: among these is the baker's map, which is also exactly solvable. An interesting avenue of study is to examine solutions to such systems not by iterating the dynamical system over time, but instead, to study the corresponding Frobenius-Perron operator or transfer operator for the system. For some of these systems, it can be explicitly, mathematically shown that the transfer operators are not trace-class. This means that these operators do not have a unique eigenvalue spectrum that is independent of the choice of basis. In the case of the baker's map, it can be shown that several unique and inequivalent diagonalizations or bases exist, each with a different set of eigenvalues. It is this phenomenon that can be offered as an "explanation" for the arrow of time. That is, although the iterated, discrete-time system is explicitly time-symmetric, the transfer operator is not. Furthermore, the transfer operator can be diagonalized in one of two inequivalent ways: one that describes the forward-time evolution of the system, and one that describes the backwards-time evolution.

还有一些系统是混沌的，也是明确的时间可逆的: 其中有面包师的地图，它也是完全可解的。一个有趣的研究方法是不通过在时间上迭代动力系统来检验这类系统的解，而是与研究系统相应的 Frobenius-Perron 算子或转移算子。对于其中一些系统，可以明确地从数学上证明传输运算符不是跟踪类的。这意味着这些算子没有独立于基的选择的唯一特征值谱。在面包师地图的例子中，可以证明存在几个唯一的和不等价的对角化或基，每个都有一组不同的特征值。正是这种现象可以作为时间之箭的“解释”。也就是说，尽管迭代的离散时间系统是显式时间对称的，但传递算子不是。此外，传递算子可以用两种不等价的方式之一对角化: 一种描述系统的前向演化，另一种描述系统的后向演化。

As of 2006, this type of time-symmetry breaking has been demonstrated for only a very small number of exactly-solvable, discrete-time systems. The transfer operator for more complex systems has not been consistently formulated, and its precise definition is mired in a variety of subtle difficulties. In particular, it has not been shown that it has a broken symmetry for the simplest exactly-solvable continuous-time ergodic systems, such as Hadamard's billiards, or the Anosov flow on the tangent space of PSL(2,R).

As of 2006, this type of time-symmetry breaking has been demonstrated for only a very small number of exactly-solvable, discrete-time systems. The transfer operator for more complex systems has not been consistently formulated, and its precise definition is mired in a variety of subtle difficulties. In particular, it has not been shown that it has a broken symmetry for the simplest exactly-solvable continuous-time ergodic systems, such as Hadamard's billiards, or the Anosov flow on the tangent space of PSL(2,R).

截至2006年，这种类型的时间对称性破缺只在很少的精确可解的离散时间系统中得到证明。对于更复杂的系统，转移算子还没有统一的表述，其精确定义也陷入了各种微妙的困难中。特别是，最简单精确可解的连续时间遍历系统，如 Hadamard 台球系统，或 PSL (2，r、R)切线空间上的 Anosov 流，还没有证明它具有破对称性。

### Quantum mechanics

Research on irreversibility in quantum mechanics takes several different directions. One avenue is the study of rigged Hilbert spaces, and in particular, how discrete and continuous eigenvalue spectra intermingle^{[citation needed]}. For example, the rational numbers are completely intermingled with the real numbers, and yet have a unique, distinct set of properties. It is hoped that the study of Hilbert spaces with a similar inter-mingling will provide insight into the arrow of time.

Research on irreversibility in quantum mechanics takes several different directions. One avenue is the study of rigged Hilbert spaces, and in particular, how discrete and continuous eigenvalue spectra intermingle. For example, the rational numbers are completely intermingled with the real numbers, and yet have a unique, distinct set of properties. It is hoped that the study of Hilbert spaces with a similar inter-mingling will provide insight into the arrow of time.

对量子力学不可逆性的研究有几个不同方向。其中一个途径是研究受操纵的希尔伯特空间，特别是离散和连续特征值谱如何混合。例如，有理数与实数完全混合在一起，但它们具有一组独特的、不同的性质。希望对具有类似混合的希尔伯特空间的研究也能提供时间之箭的洞见。

Another distinct approach is through the study of quantum chaos by which attempts are made to quantize systems as classically chaotic, ergodic or mixing.^{[citation needed]} The results obtained are not dissimilar from those that come from the transfer operator method. For example, the quantization of the Boltzmann gas, that is, a gas of hard (elastic) point particles in a rectangular box reveals that the eigenfunctions are space-filling fractals that occupy the entire box, and that the energy eigenvalues are very closely spaced and have an "almost continuous" spectrum (for a finite number of particles in a box, the spectrum must be, of necessity, discrete). If the initial conditions are such that all of the particles are confined to one side of the box, the system very quickly evolves into one where the particles fill the entire box. Even when all of the particles are initially on one side of the box, their wave functions do, in fact, permeate the entire box: they constructively interfere on one side, and destructively interfere on the other. Irreversibility is then argued by noting that it is "nearly impossible" for the wave functions to be "accidentally" arranged in some unlikely state: such arrangements are a set of zero measure. Because the eigenfunctions are fractals, much of the language and machinery of entropy and statistical mechanics can be imported to discuss and argue the quantum case.^{[citation needed]}

Another distinct approach is through the study of quantum chaos by which attempts are made to quantize systems as classically chaotic, ergodic or mixing. The results obtained are not dissimilar from those that come from the transfer operator method. For example, the quantization of the Boltzmann gas, that is, a gas of hard (elastic) point particles in a rectangular box reveals that the eigenfunctions are space-filling fractals that occupy the entire box, and that the energy eigenvalues are very closely spaced and have an "almost continuous" spectrum (for a finite number of particles in a box, the spectrum must be, of necessity, discrete). If the initial conditions are such that all of the particles are confined to one side of the box, the system very quickly evolves into one where the particles fill the entire box. Even when all of the particles are initially on one side of the box, their wave functions do, in fact, permeate the entire box: they constructively interfere on one side, and destructively interfere on the other. Irreversibility is then argued by noting that it is "nearly impossible" for the wave functions to be "accidentally" arranged in some unlikely state: such arrangements are a set of zero measure. Because the eigenfunctions are fractals, much of the language and machinery of entropy and statistical mechanics can be imported to discuss and argue the quantum case.

另一种不同的方法是通过量子混沌的研究，试图将系统量子化为经典的混沌、遍历或混合。结果与转移算子法得出的没有什么不同。例如，玻耳兹曼气体的量子化，即矩形盒中的硬(弹性)点粒子气体的量子化揭示了本征函数是占据整个盒子的填充空间的分形，能量本征值的间隔非常紧密，并且有一个“几乎连续”的谱(对于盒子中有限数目的粒子，谱必然是离散的)。如果初始条件是所有粒子都被限制在盒子的一边，系统很快演变为一个粒子填满整个盒子的系统。实际上，即使所有的粒子最初都在盒子的一边，它们的波函数也会渗透到整个盒子里: 它们在一边建设性地干涉，在另一边破坏性地干涉。不可逆性则是通过指出波函数”几乎不可能”被”意外地”安排在某种不可能的状态: 这种排列是一组零测度。因为本征函数是分形的，所以可以引入许多关于熵和统计力学的语言和机制来讨论和论证量子情况。

### Cosmology

!-链接由玻尔兹曼大脑 # 玻尔兹曼大脑悖论-

Some processes that involve high energy particles and are governed by the weak force (such as K-meson decay) defy the symmetry between time directions. However, all known physical processes *do* preserve a more complicated symmetry (CPT symmetry), and are therefore unrelated to the second law of thermodynamics, or to the day-to-day experience of the arrow of time. A notable exception is the wave function collapse in quantum mechanics, which is an irreversible process. It has been conjectured that the collapse of the wave function may be the reason for the Second Law of Thermodynamics. However it is more accepted today that the opposite is correct, namely that the (possibly merely apparent) wave function collapse is a consequence of quantum decoherence, a process that is ultimately an outcome of the Second Law of Thermodynamics.

Some processes that involve high energy particles and are governed by the weak force (such as K-meson decay) defy the symmetry between time directions. However, all known physical processes *do* preserve a more complicated symmetry (CPT symmetry), and are therefore unrelated to the second law of thermodynamics, or to the day-to-day experience of the arrow of time. A notable exception is the wave function collapse in quantum mechanics, which is an irreversible process. It has been conjectured that the collapse of the wave function may be the reason for the Second Law of Thermodynamics. However it is more accepted today that the opposite is correct, namely that the (possibly merely apparent) wave function collapse is a consequence of quantum decoherence, a process that is ultimately an outcome of the Second Law of Thermodynamics.

一些涉及高能粒子并受弱力支配的过程(如 k 介子衰变)违背了时间方向之间的对称性。然而，所有已知的物理过程都保留了一种更加复杂的对称性(CPT 对称) ，因此与热力学第二定律或时间箭头的日常经验无关。一个值得注意的例外是量子力学中波函数的坍缩是一个不可逆过程。据猜测，波函数的坍缩可能是热力学第二定律成立的原因。然而，今天被更接受的是，相反的观点是正确的，即(可能只是表面上)波函数坍缩是量子退相干的结果，这个过程最终是热力学第二定律的结果。

The universe was in a uniform, high density state at its very early stages, shortly after the Big Bang. The hot gas in the early universe was near thermodynamic equilibrium (giving rise to the horizon problem) and hence in a state of maximum entropy, given its volume. Expansion of a gas increases its entropy, however, and expansion of the universe has therefore enabled an ongoing increase in entropy. Viewed from later eras, the early universe can thus be considered to be highly ordered. The uniformity of this early near-equilibrium state has been explained by the theory of cosmic inflation.

The universe was in a uniform, high density state at its very early stages, shortly after the Big Bang. The hot gas in the early universe was near thermodynamic equilibrium (giving rise to the horizon problem) and hence in a state of maximum entropy, given its volume. Expansion of a gas increases its entropy, however, and expansion of the universe has therefore enabled an ongoing increase in entropy. Viewed from later eras, the early universe can thus be considered to be highly ordered. The uniformity of this early near-equilibrium state has been explained by the theory of cosmic inflation.

在大爆炸后不久的早期阶段，宇宙处于一中均匀的高密度状态。早期宇宙中的热气体接近热力学平衡(引起视界问题) ，因此考虑体积时，宇宙处于最大熵的状态。气体的膨胀会增加它的熵，然而，宇宙的膨胀使得熵不断增加。从后来的时代来看，早期的宇宙是高度有序的。这种早期近平衡态的均匀性已经被宇宙膨胀理论解释了。

According to this theory the universe (or, rather, its accessible part, a radius of 46 billion light years around Earth) evolved from a tiny, totally uniform volume (a portion of a much bigger universe), which expanded greatly; hence it was highly ordered. Fluctuations were then created by quantum processes related to its expansion, in a manner supposed to be such that these fluctuations are uncorrelated for any practical use. This is supposed to give the desired initial conditions needed for the Second Law of Thermodynamics.

According to this theory the universe (or, rather, its accessible part, a radius of 46 billion light years around Earth) evolved from a tiny, totally uniform volume (a portion of a much bigger universe), which expanded greatly; hence it was highly ordered. Fluctuations were then created by quantum processes related to its expansion, in a manner supposed to be such that these fluctuations are uncorrelated for any practical use. This is supposed to give the desired initial conditions needed for the Second Law of Thermodynamics.

根据这一理论，宇宙(或者更确切地说，其可接近的部分，围绕地球460亿光年的半径)是从一个极小的、完全一致的体积(更大的宇宙的一部分)演化而来的，这个体积膨胀得很大，因此它是高度有序的。波动是由与宇宙膨胀相关的量子过程产生的，在某种程度上，这些波动在任何实际应用中都是不相关的。这就给出了给出热力学第二定律所需要的理想初始条件。

The universe is apparently an open universe, so that its expansion will never terminate, but it is an interesting thought experiment to imagine what would have happened had the universe been closed. In such a case, its expansion would stop at a certain time in the distant future, and then begin to shrink. Moreover, a closed universe is finite.

The universe is apparently an open universe, so that its expansion will never terminate, but it is an interesting thought experiment to imagine what would have happened had the universe been closed. In such a case, its expansion would stop at a certain time in the distant future, and then begin to shrink. Moreover, a closed universe is finite.

宇宙显然是一个开放的宇宙，所以它的膨胀永远不会终止，但想象如果宇宙是封闭的会发生什么，是一个有趣的思想实验。在这种情况下，它的膨胀会在遥远的未来的某个时间停止，然后开始收缩。此外，一个封闭的宇宙是有限的。

It is unclear what would happen to the Second Law of Thermodynamics in such a case. One could imagine at least three different scenarios (in fact, only the third one is plausible, since the first two require a smooth cosmic evolution, contrary to what is observed):

It is unclear what would happen to the Second Law of Thermodynamics in such a case. One could imagine at least three different scenarios (in fact, only the third one is plausible, since the first two require a smooth cosmic evolution, contrary to what is observed):

目前还不清楚在这种情况下，热力学第二定律会发生什么变化。我们可以设想至少有三种不同的情况(事实上，只有第三种情况是可信的，因为前两种情况需要宇宙平稳演化，这与我们观察到的情况相反) :

- A highly controversial view is that in such a case the arrow of time will reverse.
^{[7]}The quantum fluctuations—which in the meantime have evolved into galaxies and stars—will be in superposition in such a way that the whole process described above is reversed—i.e., the fluctuations are erased by destructive interference and total uniformity is achieved once again. Thus the universe ends in a Big Crunch, which is similar to its beginning in the Big Bang. Because the two are totally symmetric, and the final state is very highly ordered, entropy must decrease close to the end of the universe, so that the Second Law of Thermodynamics reverses when the universe shrinks. This can be understood as follows: in the very early universe, interactions between fluctuations created entanglement (quantum correlations) between particles spread all over the universe; during the expansion, these particles became so distant that these correlations became negligible (see quantum decoherence). At the time the expansion halts and the universe starts to shrink, such correlated particles arrive once again at contact (after circling around the universe), and the entropy starts to decrease—because highly correlated initial conditions may lead to a decrease in entropy. Another way of putting it, is that as distant particles arrive, more and more order is revealed because these particles are highly correlated with particles that arrived earlier. - 一种极具争议的观点认为，在这种情况下，时间之箭会逆转。量子波动将以一种完全颠倒的方式叠加在一起——在此期间已经演化成星系和恒星。相消干涉消除了波动，再次实现了完全均一。因此，宇宙在大坍缩中结束，这与它在大爆炸开始时的情况类似。因为这两者是完全对称的，而且终态是高度有序的，熵必须在接近宇宙尽头时减小，所以当宇宙缩小时，热力学第二定律会发生逆转。这可以理解为:在非常早期的宇宙中，波动之间的相互作用在遍布宇宙的粒子之间产生了纠缠(量子关联);在膨胀过程中，这些粒子变得十分遥远，以至于这些相关性变得微不足道(见量子退相干)。当膨胀停止，宇宙开始收缩时，这些相关的粒子(在环绕宇宙后)再次接触，熵开始减少——因为高度相关的初始条件可能导致熵的减少。换句话说，随着遥远粒子的到达，越来越多的顺序被揭示出来，因为这些粒子与更早到达的粒子高度相关。

- It could be that this is the crucial point where the wavefunction collapse is important: if the collapse is real, then the quantum fluctuations will not be in superposition any longer; rather they had collapsed to a particular state (a particular arrangement of galaxies and stars), thus creating a Big Crunch, which is very different from the Big Bang. Such a scenario may be viewed as adding boundary conditions (say, at the distant future) that dictate the wavefunction collapse.
^{[8]} - 波函数坍缩的关键点十分重要:如果坍缩是真实的，那么量子波动将不再是叠加的;相反，它们已经坍缩到一个特定的状态(星系和恒星的特定排列)，从而产生了大坍缩，这与大爆炸非常不同。这样的情形可以看作是增加了边界条件(例如，在遥远的将来)来决定波函数的崩溃。

- The broad consensus among the scientific community today is that smooth initial conditions lead to a highly non-smooth final state, and that this is in fact the source of the thermodynamic arrow of time.
^{[9]}Highly non-smooth gravitational systems tend to collapse to black holes, so the wavefunction of the whole universe evolves from a superposition of small fluctuations to a superposition of states with many black holes in each. It may even be that it is impossible for the universe to have both a smooth beginning and a smooth ending. Note that in this scenario the energy density of the universe in the final stages of its shrinkage is much larger than in the corresponding initial stages of its expansion (there is no destructive interference, unlike in the first scenario described above), and consists of mostly black holes rather than free particles.

当今科学界普遍的共识是，光滑的初始状态会导致高度不光滑的最终状态，而这实际上就是热力学时间箭头的来源。高度不光滑的引力系统倾向于坍陷为黑洞，所以整个宇宙的波函数从小波动的叠加演化为每个状态中有许多黑洞的叠加。甚至可能是，宇宙不可能既有一个光滑的开始，又有一个光滑的结束。注意,在这些场景中,宇宙能量密度在最后阶段的收缩量远远大于相应初始阶段的扩张量(没有相消干涉,不像在第一个场景中所述),以及大多由黑洞而不是自由粒子组成。

In the first scenario, the cosmological arrow of time is the reason for both the thermodynamic arrow of time and the quantum arrow of time. Both will slowly disappear as the universe will come to a halt, and will later be reversed.

In the first scenario, the cosmological arrow of time is the reason for both the thermodynamic arrow of time and the quantum arrow of time. Both will slowly disappear as the universe will come to a halt, and will later be reversed.

在第一种情况下，宇宙时间之箭是热力学时间之箭和量子时间之箭存在的原因。两者都会随着宇宙的停止而慢慢消失，并将在随后发生逆转。

In the second and third scenarios, it is the difference between the initial state and the final state of the universe that is responsible for the thermodynamic arrow of time. This is independent of the cosmological arrow of time. In the second scenario, the quantum arrow of time may be seen as the deep reason for this.

In the second and third scenarios, it is the difference between the initial state and the final state of the universe that is responsible for the thermodynamic arrow of time. This is independent of the cosmological arrow of time. In the second scenario, the quantum arrow of time may be seen as the deep reason for this.

在第二种和第三种情况下，宇宙的初始状态和最终状态之间的差异决定了时间的热力学箭头。这与宇宙时间之箭无关。在第二种情况下，时间的量子箭头可能被视为这种现象的深层原因。

## See also

## References

- ↑ Price, Huw (2004). "The Thermodynamic Arrow: Puzzles and Pseudo-puzzles". arXiv:physics/0402040.
- ↑ Penrose, R.
*The Road to Reality*pp. 686-734 - ↑
*Physical Origins of Time Asymmetry*, p. 109. - ↑
*Physical Origins of Time Asymmetry*, p. 35. - ↑
*Physical Origins of Time Asymmetry*, pp. 35-38. - ↑ "Some Misconceptions about Entropy". Archived from the original on 2012-02-04. Retrieved 2011-02-13.
- ↑ Hawking, S. W. (1985). "Arrow of time in cosmology".
*Physical Review D*.**32**(10): 2489–2495. Bibcode:1985PhRvD..32.2489H. doi:10.1103/PhysRevD.32.2489. PMID 9956019. - ↑ Gruss, Eyal Y.; Aharonov, Yakir (2005). "Two-time interpretation of quantum mechanics g". arXiv:quant-ph/0507269.
- ↑ Lebowitz, Joel (2008). "Time's arrow and Boltzmann's entropy".
*Scholarpedia*.**3**(4): 3448. Bibcode:2008SchpJ...3.3448L. doi:10.4249/scholarpedia.3448.

## Further reading

- Halliwell, J.J. (1994).
*Physical Origins of Time Asymmetry*. Cambridge. ISBN 0-521-56837-4. (technical).

- Mackey

最后一个麦基, Michael C.

首先是迈克尔 · c。 (1992

1992年). *Time's Arrow: The Origins of Thermodynamic Behavior*

时间之箭: 热力学行为的起源*. Berlin Heidelberg New York: Springer*

出版商斯普林格. ISBN 3-540-94093-6. OCLC [//www.worldcat.org/oclc/28585247

28585247 28585247 28585247]. "... it is shown that for there to be a global evolution of the entropy to its maximal value ... it is necessary and sufficient that the system have a property known as exactness. ... these criteria suggest that all currently formulated physical laws may not be at the foundation of the thermodynamic behavior we observe every day of our lives. (page xi)

有证据表明，要使熵值达到最大值，系统具有精确性是必要的，也是充分的。... 这些标准表明，所有目前公式化的物理定律可能不是我们日常生活中观察到的热力学行为的基础。(第十一页)"

Dover has reprinted the monograph in 2003 (

). For a short paper listing "the essential points of that argument, correcting presentation points that were confusing ... and emphasizing conclusions more forcefully than previously" see Mackey, Michael C. (2001). "Microscopic Dynamics and the Second Law of Thermodynamics". In Mugnai, C.; Ranfagni, A.; Schulman, L.S.. *Time's Arrow, Quantum Measurement and Superluminal Behavior*. Rome: Consiglio Nazionale Delle Ricerche. pp. 49–65. ISBN 88-8080-024-8. http://www.cnd.mcgill.ca/bios/mackey/pdf_pub/newfinalnaples.pdf.

}}

Dover has reprinted the monograph in 2003 (). For a short paper listing "the essential points of that argument, correcting presentation points that were confusing ... and emphasizing conclusions more forcefully than previously" see

} br / Dover 在2003年重印了该专著。一篇简短的论文列出“论点的要点，纠正混乱的陈述观点... ... 并比以前更有力地强调结论”见

## External links

Category:Thermodynamic entropy

类别: 熵

Category:Asymmetry

分类: 不对称

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