# 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.

熵是物理科学中唯一需要特定时间方向的量(除了粒子物理学中某些罕见的相互作用，见下文) ，有时被称为时间之箭。热力学第二定律认为，随着时间的推移，孤立系统的熵可以增加，但不会减少。因此，从一个角度来看，熵测量是一种区分过去和未来的方法。然而，在不封闭的热力学系统中，熵会随着时间而减少: 许多系统，包括生命系统，以牺牲环境的增加为代价减少局部熵，导致熵的净增加。这类系统和现象的例子包括热力学中所用的典型晶体的形成、冰箱的工作原理和活的有机体。

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.

就像温度一样，尽管是一个抽象的概念，每个人对熵的影响都有一种直观的感觉。例如，通常很容易区分正在播放的视频和正在播放的前后。一段视频可能描述了一场融化附近冰块的木柴大火，反过来播放的话，就会显示出一滩水把一团烟雾变成了未燃烧的木头，并在此过程中把自己冻结了。令人惊讶的是，在这两种情况下，绝大多数物理定律都没有被这些过程打破，一个明显的例外是热力学第二定律。当一个物理定律同样适用于时间被反转的情况时，它被称为 t 对称性，在这种情况下，熵允许人们判断上面描述的视频是正向播放还是反向播放，因为我们直观地认为，只有当正向播放时，场景的熵才会增加。由于热力学第二定律的存在，熵阻止了宏观过程呈现 t 对称性。

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.

在微观尺度上进行研究时，上述判断是不能作出的。观察一个被空气冲击的单个烟雾粒子，我们不清楚视频是正向播放还是反向播放，事实上，这是不可能的，因为适用的定律显示了 t 对称性，因为它向左或向右漂移，它看起来没有什么不同。只有当你在宏观研究气体时，熵的影响才会变得明显。平均来说，你可以预期点燃的火柴周围的烟雾颗粒会相互漂移远离，扩散到整个可用空间。这将是一个天文学上不可能的事件，所有的粒子聚集在一起，但你不能评论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违反宇称守恒，但这种情况很少发生。根据 CPT 定理，这意味着它们也应该是时间不可逆的，因此建立了一个时间箭头。然而，这既与时间的热力学箭头无关，也与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.

不管时间的方向如何，热力学第二定律允许熵保持不变。如果熵在时间的任何一个方向上都是常数，那么就没有偏好的方向。然而，只有当系统处于最高可能的无序状态时，熵才可能是一个常数，例如一种气体，过去是，将来也是，均匀地散布在它的容器中。热力学时间箭头的存在意味着系统只在一个时间方向上高度有序，这个时间方向就是“过去”。因此，这个定律是关于边界条件的，而不是运动方程。

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.

热力学第二定律本质上是统计学的，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，一摩尔气体中的所有6个10加23个原子自发迁移到容器的一半是不可能的，只是不太可能ーー不太可能，以至于从来没有观察到对第二定律的宏观违反。对称性是物理定律在时间反转变换下的对称性。虽然在受限制的环境中，人们可能会发现这种对称性，但是可观测宇宙本身在时间反转下并不表现出对称性，主要是由于热力学第二定律。

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.

热力学箭头通常与宇宙学的时间箭头联系在一起，因为它最终与早期宇宙的边界条件有关。根据大爆炸理论，宇宙最初是非常热的，能量分布均匀。对于一个引力很重要的系统，比如宇宙，这是一个低熵状态(与所有物质坍缩成黑洞的高熵状态相比，系统最终可能进化到这种状态)。随着宇宙的发展，它的温度下降，这使得未来可用于工作的能量比过去要少。此外，能量密度中的扰动也在增加(最终形成星系和恒星)。因此，宇宙本身有一个定义明确的热力学时间箭头。但是这并没有解决为什么宇宙的初始状态是低熵的问题。如果宇宙膨胀由于引力而停止或逆转，那么宇宙的温度将再次升高，但是由于扰动的持续增长和最终黑洞的形成，宇宙的熵也将继续增加，直到大坍缩的后期，那时的熵将比现在低。

## 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.

如果在混合过程的早期观察到大容器，则可能发现它只是部分混合。我们可以合理地得出这样的结论: 在没有外界干预的情况下，液体之所以达到这种状态，是因为在过去，当分离程度较大时，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较高。

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.

现在想象一下这个实验重复进行，这次只有几个分子，也许十个，在一个非常小的容器里。人们可以很容易地想象，通过观察分子的随机碰撞，它可能发生---- 仅仅是偶然---- 分子整齐地分离开来，所有的染料分子在一边，所有的水分子在另一边。这种情况可以不时地发生，这可以从涨落定理中得出结论; 因此分子彼此分离并不是不可能的。然而，对于大量的分子来说，它是如此的不可能，以至于人们不得不等待，平均而言，要比宇宙的年龄长很多倍的时间才能发生。因此，如果一部电影像上面描述的那样展示了大量的分子自我分离，这看起来是不现实的，人们会倾向于说这部电影是以相反的方式播放的。把玻尔兹曼第二定律看作是无序定律。

## 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背后的数学来源于以下的设置，详见卡诺(1824)、克拉佩龙(1832)和克劳修斯(1854) :

225px / center

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，正如常见的经验所证明的，当一个热的物体 t 子1 / 子，例如一个炉子，进入物理接触，例如通过一个流体(工作物体)与一个冷的物体 t 子2 / 子，例如一股冷水，能量总是以热 q 的形式从热流向冷流，并且给定时间系统将达到平衡。熵被定义为 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。原子和分子在转变过程中相互作用的耗散功。

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.

在这个图中，我们可以计算热量 q 从温度 t1 / sub 通过流体的“工作体”(见热机)到温度 t2 / sub 的熵变 s。此外，为了讨论的目的，我们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，如果我们像最初由克劳修斯所做的那样做作业:

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

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

数学 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:

那么这个变换的熵变或“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]

数学 | Delta s { final }-s { initial } ，/ math

which equals:

which equals:

这等于:

- [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]

数学 Delta s 左( frac { t2}- frac { t1}右) / 数学

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]

数学 Delta s q 左( frac {1}{ t 2}- frac {1}{ t 1}右) / 数学

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 是50个单位，t1 / sub 最初是100度，t2 / sub 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，这个过程需要一定的“时间” ，并且熵增加与时间的流逝相关。因此，对于这种系统配置，它是一个“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，例如一个罐中的两个分子，不仅做外部功(如推动活塞) ，而且根据过程中所用的热量(见: 热量的机械等效物)相互作内部功。熵解释了分子间存在内部摩擦的事实。

## 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(例如粒子气体)中，其初始条件通常是其不同部分是不相关的，但随着系统的演化及其不同部分之间的相互作用，它们变得相关。例如，在处理粒子气体时，总是假定其初始条件是不同粒子的状态之间没有相关性(即不同粒子的状态之间没有相关性)。不同粒子的速度和位置是完全随机的，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，一开始装了一半的理想气体。随着时间的推移，气体显然会膨胀，充满整个盒子，因此最终状态是一个装满气体的盒子。这是一个不可逆性，因为如果盒子在开始时是满的(实验 b) ，它不会在以后变成只有一半满的，除了非常不可能的情况下，气体粒子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是这样的，即粒子具有随机的位置和速度。这对于系统的最终条件来说是不正确的，因为粒子之间相互作用，所以它们的位置和速度变得相互依赖。相关的。如果我们倒过来看实验 a，我们称之为实验 c: 现在我们从一个装满气体的箱子开始，但是粒子没有随机的位置和速度; 相反，它们的位置和速度是如此特别，过了一段时间它们都移动到箱子的一半，这是系统的最终状态(这是实验 a 的初始状态，因为现在我们正在倒过来看同一个实验!) .粒子之间的相互作用现在不会在粒子之间产生关联，但实际上会使它们变成(至少看起来是)随机的，“抵消”先前存在的关联。实验 c (无视热力学第二定律)和实验 b (服从热力学第二定律)的唯一区别在于，前者的粒子在最后是不相关的，而后者的粒子在开始时是不相关的。

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的测量熵之间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)。)ー及其熵，即描述系统精确微状态所需的信息量(计算机位数)。测量到的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，但熵确实依赖于它们，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，两个熵是相同的，但是除此之外，熵比测量的熵要小，这种差异可以用来衡量相关性的数量。

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(其信息论联合熵)在时间上是不变的。这个联合熵等于边际熵(假设没有相关性)加上相关熵(互熵，或其负互信息)。如果我们最初假设粒子之间没有相关性，那么这个联合熵就是边际熵，也就是系统的初始熵，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) ，那么这种相关性与时间形成。换句话说，存在一个减少的互熵(或增加的互信息) ，并且在不太长的时间内，粒子之间的关联(互信息)只随时间增加。因此，与边际熵成正比的熵也必须随着时间而增加(注意，在这种情况下，“不要太久”是相对于经典系统版本所需的时间，以便它通过所有可能的微观状态ーー这个时间可以粗略地估计为数学 τ e ^ s / math，其中 math tau / math 是粒子碰撞和 s 之间的时间，s 是系统的熵。在任何实际的情况下，这个时间是巨大的比其他任何事情)。注意，粒子之间的相关性并不是一个完全客观的量。我们不能测量互熵，我们只能测量它的变化，假设我们可以测量一个微观状态。热力学仅限于微观状态无法区分的情况，这意味着只有与熵成正比的边际熵才能被测量，而且，在实际意义上，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行为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己组合出来，在粗糙的表面上滑动的冰块减慢而不是加速，以及我们能记住过去而不是未来。事实上，如果人们把记忆看作是脑细胞(或计算机比特)和外部世界之间的联系，那么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其来源。

### 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表明，时间之箭可能存在一种“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，我们考虑一个微分方程，其中一个参数是明确的时间。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不可逆是不可能的。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，因为任何给定点“在当前”可能有几个不同的“过去”与之相关联: 实际上，所有过去的集合被称为 Julia 集。由于这种系统具有内在的不可逆性，因此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，也是显式的时间可逆的: 其中一个是面包师的映射，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，而是研究系统的相应的 Frobenius-Perron 算子或转移算子。对于这些系统中的一些，可以明确地从数学上证明传输运算符不是跟踪类。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，转移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已被证明只有很少数量的精确可解的，离散时间系统。较复杂系统的转移算子没有得到一致的规定，其精确定义也陷入了各种微妙的困难之中。特别是，对于最简单的精确可解连续时间遍历系统，如 Hadamard 台球系统，或 PSL (2，r)切空间上的 Anosov 流动，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.

关于量子力学的不可逆性的研究有几个不同的方向。其中一个途径是研究拼凑的希尔伯特空间，特别是离散和连续谱如何混合。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性质。希望对希尔伯特空间的研究也能提供时间之箭的线索。

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.

另一种截然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。得到的结果与转移算子法的结果没有什么不同。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，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(对于一个盒子中有限数目的粒子，谱必然是离散的)。如果初始条件是这样的，所有的粒子都局限在箱子的一边，系统很快演化为一个粒子填充整个箱子。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。不可逆性的论点是，波函数”几乎不可能”被”偶然地”安排在某种不可能的状态: 这种安排是一组零测度。因为本征函数是分形的，所以熵和统计力学的大部分语言和机制可以用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 em do / em 保留了一个更加复杂的对称性(CPT 对称) ，因此与热力学第二定律无关，或者与日常经验的时间箭头无关。一个值得注意的例外是波函数崩溃在量子力学，这是一个不可逆性。有人猜测，波函数的崩塌可能是热力学第二定律的原因。然而，今天人们更接受的是，相反的观点是正确的，即波函数崩溃(可能只是表面上的)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.

宇宙在宇宙大爆炸后不久的早期阶段，处于一个统一的、高密度的状态。早期宇宙中的热气体接近热力学平衡(引起视界问题) ，因此在其体积上处于熵最大的状态。然而，气体的膨胀增加了它的熵，因此宇宙的膨胀使得熵不断增加。从后来的时代来看，早期的宇宙可以被认为是高度有序的。这种早期近平衡态的均匀性已经用宇宙膨胀理论来解释。

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(一个更大的宇宙的一部分)演化而来的，这个体积极大地膨胀，因此它是高度有序的。涨落随后由与其膨胀相关的量子过程产生，以一种假定这些涨落在任何实际应用中都是不相关的方式。这是为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，但是想象一下如果宇宙被关闭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，然后开始萎缩。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。我们可以想象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.

- 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]}

- 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.

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的原因。两者都会随着宇宙的停止而慢慢消失，随后又会逆转。

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.

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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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