# 洛斯密特悖论

本词条由11初步翻译，由Fernando审校。

{{简述}反对意见，认为不可能从时间对称动力学中推断出一个不可逆的过程}}

**Loschmidt's paradox**, also known as the **reversibility paradox**, **irreversibility paradox** or *脚本错误：没有“lang”这个模块。*, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict, hence the paradox.

Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or , is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict, hence the paradox.

** 洛斯密特悖论Loschmidt's paradox **，也被称为可逆性悖论，不可逆性悖论^{[1]}，或者说是一种反对意见，它认为不可能从时间对称的动力学中推导出一个不可逆的过程。这使得（几乎）所有已知的低级基本物理过程的时间反演对称性与任何依据描述宏观系统行为的热力学第二定律进行推断的努力相矛盾。这两个原则都是物理学中公认的原则，有着可靠的观测和理论支持，但它们似乎相互冲突，因此产生了悖论。

## Origin

起源

Josef Loschmidt's criticism was provoked by the H-theorem of Boltzmann, which employed kinetic theory to explain the increase of entropy in an ideal gas from a non-equilibrium state, when the molecules of the gas are allowed to collide. In 1876, Loschmidt pointed out that if there is a motion of a system from time t_{0} to time t_{1} to time t_{2} that leads to a steady decrease of *H* (increase of entropy) with time, then there is another allowed state of motion of the system at t_{1}, found by reversing all the velocities, in which *H* must increase. This revealed that one of Boltzmann's key assumptions, molecular chaos, or, the *Stosszahlansatz*, that all particle velocities were completely uncorrelated, did not follow from Newtonian dynamics. One can assert that possible correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed the conceptual system, injecting an element of time-asymmetry by that very action.

Josef Loschmidt's criticism was provoked by the H-theorem of Boltzmann, which employed kinetic theory to explain the increase of entropy in an ideal gas from a non-equilibrium state, when the molecules of the gas are allowed to collide. In 1876, Loschmidt pointed out that if there is a motion of a system from time t_{0} to time t_{1} to time t_{2} that leads to a steady decrease of H (increase of entropy) with time, then there is another allowed state of motion of the system at t_{1}, found by reversing all the velocities, in which H must increase. This revealed that one of Boltzmann's key assumptions, molecular chaos, or, the Stosszahlansatz, that all particle velocities were completely uncorrelated, did not follow from Newtonian dynamics. One can assert that possible correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed the conceptual system, injecting an element of time-asymmetry by that very action.

约瑟夫·洛斯密特Josef Loschmidt的批评是由玻尔兹曼Boltzmann的H定理引起的，该定理运用动力学理论来解释当气体分子被允许碰撞时，来自非平衡状态的理想气体的熵的增加。1876年，Loschmidt指出，如果一个系统从时间t_{0}到时间t_{1}再到时间t_{2}有一个运动，导致H随时间的变化而稳定减少（熵增加），那么，通过逆转所有的速度发现，在t_{1}处有另一个系统允许的运动状态，这个状态下H一定会增加。这揭示了玻尔兹曼的一个关键假设--分子混沌，或者说，斯托斯扎哈兰茨（Stosszahlansatz），即所有粒子速度完全不相关，并不遵循牛顿动力学。人们会断言潜在的相关性并不吸引人，因此决定忽略它们；但如果这样做，就改变了概念体系，通过这一特别的行动注入了时间不对称的因素。

Reversible laws of motion cannot explain why we experience our world to be in such a comparatively low state of entropy at the moment (compared to the equilibrium entropy of universal heat death); and to have been at even lower entropy in the past.

Reversible laws of motion cannot explain why we experience our world to be in such a comparatively low state of entropy at the moment (compared to the equilibrium entropy of universal heat death); and to have been at even lower entropy in the past.

运动的可逆定律无法解释为什么我们的世界此刻处于一个相对较低的熵状态(与宇宙热寂的平衡熵相比) ，并且在过去处于更低的熵状态。

### Before Loschmidt

在洛斯密特之前

In 1874, two years before the Loschmidt paper, William Thomson defended the second law against the time reversal objection.

In 1874, two years before the Loschmidt paper, William Thomson defended the second law against the time reversal objection.

1874年，也就是洛斯密特论文发表的前两年，威廉 · 汤姆森William Thomson为第二定律辩护，反对时间反演的异议。^{[2]}

## Arrow of time

时间之箭

Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature. This term only refers to an observation of an asymmetry in time; it is not meant to suggest an explanation for such asymmetries. Loschmidt's paradox is equivalent to the question of how it is possible that there could be a thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws, a reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system.

Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature. This term only refers to an observation of an asymmetry in time; it is not meant to suggest an explanation for such asymmetries. Loschmidt's paradox is equivalent to the question of how it is possible that there could be a thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws, a reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system.

任何有规律地在时间的前进方向上发生但很少或从来没有在相反的方向上发生的过程，例如在一个孤立的系统中熵的增加，定义了物理学家所说的自然界中的** 时间之矢an arrow of time **。这个术语仅指对时间不对称性的观察; 它并不意味着对这种不对称性做出解释。洛斯密特悖论等价于这样一个问题: 在给定时间对称基本定律的情况下，怎么可能存在一个热力学的时间箭头? 因为时间对称性意味着，对于任何符合这些基本定律的过程，一个看起来似乎是倒放第一个过程的胶片的逆向版本也将与同样的基本定律相容，甚至如果人们从该系统所有可能的状态的相空间中随机挑选系统的初始状态，也同样有可能(相容)。

Although most of the arrows of time described by physicists are thought to be special cases of the thermodynamic arrow, there are a few that are believed to be unconnected, like the cosmological arrow of time based on the fact that the universe is expanding rather than contracting, and the fact that a few processes in particle physics actually violate time-symmetry, while they respect a related symmetry known as CPT symmetry. In the case of the cosmological arrow, most physicists believe that entropy would continue to increase even if the universe began to contract (although the physicist Thomas Gold once proposed a model in which the thermodynamic arrow would reverse in this phase). In the case of the violations of time-symmetry in particle physics, the situations in which they occur are rare and are only known to involve a few types of meson particles. Furthermore, due to CPT symmetry reversal of time direction is equivalent to renaming particles as antiparticles and *vice versa*. Therefore, this cannot explain Loschmidt's paradox.

Although most of the arrows of time described by physicists are thought to be special cases of the thermodynamic arrow, there are a few that are believed to be unconnected, like the cosmological arrow of time based on the fact that the universe is expanding rather than contracting, and the fact that a few processes in particle physics actually violate time-symmetry, while they respect a related symmetry known as CPT symmetry. In the case of the cosmological arrow, most physicists believe that entropy would continue to increase even if the universe began to contract (although the physicist Thomas Gold once proposed a model in which the thermodynamic arrow would reverse in this phase). In the case of the violations of time-symmetry in particle physics, the situations in which they occur are rare and are only known to involve a few types of meson particles. Furthermore, due to CPT symmetry reversal of time direction is equivalent to renaming particles as antiparticles and vice versa. Therefore, this cannot explain Loschmidt's paradox.

虽然物理学家描述的大多数时间之矢都被认为是热力学之矢的特殊情况，但也有少数被认为是不相关的，比如宇宙学的时间之矢是基于宇宙正在膨胀而不是收缩这一事实，以及粒子物理学中的一些过程实际上违反了时间对称性，而它们却遵守一种被称为CPT对称性的相关对称性。在宇宙学之矢的例子中，大多数物理学家认为，即使宇宙开始收缩，熵也会继续增加(尽管物理学家托马斯•戈尔德(Thomas Gold)曾提出过一个模型，模型里的热力学之箭在这个阶段会逆转)。在违反粒子物理学中的时间对称性的情况下，它们很少发生，而且只知道涉及少数几种介子粒子。此外，由于CPT对称性，时间方向的逆转相当于将粒子重命名为反粒子，反之亦然。因此，这不能解释洛斯密特悖论。

## Dynamical systems

动力系统

Current research in dynamical systems offers one possible mechanism for obtaining irreversibility from reversible systems. The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the transfer operator corresponding to the microscopic equations of motion. It is then argued that the transfer operator is not unitary (*i.e.* is not reversible) but has eigenvalues whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states. This approach is fraught with various difficulties; it works well for only a handful of exactly solvable models.^{[3]}

Current research in dynamical systems offers one possible mechanism for obtaining irreversibility from reversible systems. The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the transfer operator corresponding to the microscopic equations of motion. It is then argued that the transfer operator is not unitary (i.e. is not reversible) but has eigenvalues whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states. This approach is fraught with various difficulties; it works well for only a handful of exactly solvable models.

目前的动力学系统研究为从可逆系统中获得不可逆性提供了一种可能的机制。其中心论点是基于这样一种说法，即研究宏观系统动力学的正确方法是研究微观运动方程所对应的转移算子。因此，有人认为，转移算子不是一元的(即不可逆的)，而是具有严格小于1的特征值；这些特征值对应的是衰减的物理状态。这种方法充满了各种困难; 它只适用于少数几个完全可以解决的模型。

Abstract mathematical tools used in the study of dissipative systems include definitions of mixing, wandering sets, and ergodic theory in general.

Abstract mathematical tools used in the study of dissipative systems include definitions of mixing, wandering sets, and ergodic theory in general.

用于研究耗散系统的抽象数学工具一般包括混合、游走集和** 各态经历理论ergodic theory in general **的定义。

## Fluctuation theorem

波动定理

One approach to handling Loschmidt's paradox is the fluctuation theorem, derived heuristically by Denis Evans and Debra Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain value for the dissipation function (often an entropy like property) over a certain amount of time. The result is obtained with the exact time reversible dynamical equations of motion and the universal causation proposition. The fluctuation theorem is obtained using the fact that dynamics is time reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments at the Australian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus. This theorem is applicable for transient systems, which may initially be in equilibrium and then driven away (as was the case for the first experiment by Sevick et al.) or some other arbitrary initial state, including relaxation towards equilibrium. There is also an asymptotic result for systems which are in a nonequilibrium steady state at all times.

One approach to handling Loschmidt's paradox is the fluctuation theorem, derived heuristically by Denis Evans and Debra Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain value for the dissipation function (often an entropy like property) over a certain amount of time. The result is obtained with the exact time reversible dynamical equations of motion and the universal causation proposition. The fluctuation theorem is obtained using the fact that dynamics is time reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments at the Australian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus. This theorem is applicable for transient systems, which may initially be in equilibrium and then driven away (as was the case for the first experiment by Sevick et al.) or some other arbitrary initial state, including relaxation towards equilibrium. There is also an asymptotic result for systems which are in a nonequilibrium steady state at all times.

处理洛斯密特悖论的一种方法是波动定理，由丹尼斯·埃文斯Denis Evans和黛布拉·塞尔斯Debra Searles以启发式的方式导出，它给出了一个数值上的估计，即一个远离平衡的系统在一定时间内耗散函数(通常表示类似熵的性质)为特定值的概率。^{[4]}该结果是在精确的时间可逆动力学运动方程和普遍因果命题下得到的。由动力学是时间可逆的事实，可以推导出波动定理。在澳大利亚国立大学由伊迪丝·塞维克Edith M.Sevick等人利用光学镊子仪器进行的实验室实验中，已经证实了该定理的定量预测。该定理适用于这样的瞬态系统，它最初可能处于平衡状态，然后离开平衡（如Sevick等人的第一个实验）或处于其他一些任意的初始状态，包括向平衡状态的弛豫。对于始终处于非平衡稳定状态的系统，也有一个渐进结果。

There is a crucial point in the fluctuation theorem, that differs from how Loschmidt framed the paradox. Loschmidt considered the probability of observing a single trajectory, which is analogous to enquiring about the probability of observing a single point in phase space. In both of these cases the probability is always zero. To be able to effectively address this you must consider the probability density for a set of points in a small region of phase space, or a set of trajectories. The fluctuation theorem considers the probability density for all of the trajectories that are initially in an infinitesimally small region of phase space. This leads directly to the probability of finding a trajectory, in either the forward or the reverse trajectory sets, depending upon the initial probability distribution as well as the dissipation which is done as the system evolves. It is this crucial difference in approach that allows the fluctuation theorem to correctly solve the paradox.

There is a crucial point in the fluctuation theorem, that differs from how Loschmidt framed the paradox. Loschmidt considered the probability of observing a single trajectory, which is analogous to enquiring about the probability of observing a single point in phase space. In both of these cases the probability is always zero. To be able to effectively address this you must consider the probability density for a set of points in a small region of phase space, or a set of trajectories. The fluctuation theorem considers the probability density for all of the trajectories that are initially in an infinitesimally small region of phase space. This leads directly to the probability of finding a trajectory, in either the forward or the reverse trajectory sets, depending upon the initial probability distribution as well as the dissipation which is done as the system evolves. It is this crucial difference in approach that allows the fluctuation theorem to correctly solve the paradox.

在波动定理中，有一个关键点与Loschmidt构建悖论的方式不同。Loschmidt考虑的是观察到单一轨迹的概率，这类似于调查观察到相空间中单个点的概率。在这两种情况下，概率总是零。为了能够有效地解决这个问题，必须考虑在相空间的一个小区域中的一组点或者一组轨迹的概率密度。波动定理考虑的是最初处于相空间无限小区域的所有轨迹的概率密度。这直接导致了在正向或反向轨迹集中找到轨迹的概率，这轨迹取决于初始概率分布以及随着系统演化所做的耗散。正是方法中的这一关键差异，使得波动定理能够正确解决这一悖论。

## The Big Bang

大爆炸

Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy starting point: the Big Bang. From this point of view, the arrow of time is determined entirely by the direction that leads away from the Big Bang, and a hypothetical universe with a maximum-entropy Big Bang would have no arrow of time. The theory of cosmic inflation tries to give reason why the early universe had such a low entropy.

Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy starting point: the Big Bang. From this point of view, the arrow of time is determined entirely by the direction that leads away from the Big Bang, and a hypothetical universe with a maximum-entropy Big Bang would have no arrow of time. The theory of cosmic inflation tries to give reason why the early universe had such a low entropy.

处理洛斯密特悖论的另一种方法是把第二定律看作是一系列边界条件的表达，在这些边界条件中，我们的宇宙时间坐标的起点是低熵的: 大爆炸。从这个角度来看，时间之矢完全是由远离大爆炸的方向决定的，一个具有最大熵大爆炸的假设宇宙将没有时间之矢。宇宙膨胀理论试图解释早期宇宙为什么有如此低的熵。

## See also

请参阅

- Maximum entropy thermodynamics for one particular perspective on entropy, reversibility and the Second Law
- 最大熵热力学对熵、可逆性和第二定律的一种特殊观点

## References

参考

- ↑ Wu, Ta-You (December 1975). "Boltzmann's H theorem and the Loschmidt and the Zermelo paradoxes".
*International Journal of Theoretical Physics*.**14**(5): 289. doi:10.1007/BF01807856. - ↑ Thomson, W. (Lord Kelvin) (1874/1875). The kinetic theory of the dissipation of energy,
*Nature*, Vol. IX, 1874-04-09, 441–444. - ↑ Dean J. Driebe,
*Fully Chaotic Maps and Broken Time Symmetry*, (1999) Kluwer Academic - ↑ D. J. Evans and D. J. Searles, Adv. Phys.
**51**, 1529 (2002).

- J. Loschmidt, Sitzungsber. Kais. Akad. Wiss. Wien, Math. Naturwiss. Classe 73, 128–142 (1876)

## External links

外部链接

- Reversible laws of motion and the arrow of time by Mark Tuckerman

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This page was moved from wikipedia:en:Loschmidt's paradox. Its edit history can be viewed at 洛斯密特悖论/edithistory