洛斯密特悖论

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or 脚本错误：没有“lang”这个模块。,^[1] 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，也被称为可逆性悖论，不可逆性悖论，或者，是不可能从时间对称动力学推导出不可逆性的反对意见。这使得(几乎)所有已知的低层次基本物理过程的时间反转对称性与任何试图从它们推断描述宏观系统行为的热力学第二定律不一致。这两个原则在物理学中都是被广泛接受的原则，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

原产地

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 t₀ to time t₁ to time t₂ 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 t₁, 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 t₀ to time t₁ to time t₂ 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 t₁, 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 的批评是由玻尔兹曼的 h 定理挑起的，该定理利用动力学理论来解释理想气体在非平衡状态下熵的增加，当气体的分子被允许碰撞时。1876年，Loschmidt 指出，如果一个系统的运动从时间 t 小于0 / sub 到时间 t 小于1 / sub 到时间 t 小于2 / sub，导致 h (熵增加)随时间稳定减少，那么在 t 小于1 / sub 时，系统还有另一个允许的运动状态，通过反转所有的速度得到，h 必须增加。这揭示了玻尔兹曼的一个关键假设，分子混沌，或者，Stosszahlansatz，即所有粒子速度是完全不相关的，并不遵循牛顿动力学。一个人可以断言可能的相关性是无趣的，因此决定忽略它们; 但是如果一个人这样做了，他就改变了概念系统，通过这个行为注入了一个时间不对称的元素。

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.

可逆的运动定律不能解释为什么我们的世界在此刻处于一个相对较低的熵状态(相对于宇宙热死的平衡熵) ，并且在过去处于一个更低的熵状态。

Before Loschmidt

在 Loschmidt 之前

In 1874, two years before the Loschmidt paper, William Thomson defended the second law against the time reversal objection.^[2]

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 · 汤姆森为第二定律辩护，反对时间反转的异议。

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.

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箭头。这个术语只是指对时间不对称性的观察; 它并不意味着对这种不对称性提出解释。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，如何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，一个看起来完全像是向后播放的第一个过程的胶片的反转版本，将同样符合相同的基本定律，甚至同样可能，如果一个人从该系统的所有可能状态的相空间中随机挑选系统的初始状态。

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箭头都被认为是热力学箭头的特殊情况，但也有少数被认为是不相关的，比如宇宙学的时间箭头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，而它们尊重相关的称为 CPT 对称性的事实。在宇宙学箭头的例子中，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，其中热力学箭头在这个阶段会逆转)。在粒子物理学中违反时间对称性的情况下，它们发生的情况是罕见的，只知道涉及少数几种介子粒子。此外，由于 CPT 时间方向的对称性反转等价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.

动力系统的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。中心论点是基于这样的主张，即研究宏观系统动力学的正确方法是研究与微观运动方程相对应的转移算子。因此，有人认为，转让经营者不是单一的(即，转让经营者不是单一的。是不可逆的) ，但有严格小于一的本征值，这些本征值对应于衰减物理状态。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.

用于研究耗散系统的抽象数学工具一般包括混合、游荡集和遍历理论的定义。

Fluctuation theorem

涨落定理

模板:Unreferenced section

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.^[4] The result is obtained with the exact time reversible dynamical equations of motion and the Axiom of Causality. 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 Axiom of Causality. 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 Loschmidt 悖论的一个方法是由 Denis Evans 和 Debra Searles 启发性地推导出的涨落定理，它给出了一个数值估计，一个远离平衡的系统在一定时间内的耗散函数(通常是一个类似熵的性质)具有一定值的概率。这个结果是由精确的时间可逆动力学运动方程和因果关系公理得到的。涨落定理是利用动力学是时间可逆的这一事实获得的。这个定理的定量预测已经在澳大利亚国立大学的实验室实验中得到证实。使用光镊仪器。这个定理适用于瞬态系统，它最初可能处于平衡状态，然后逐渐消失(就像 Sevick 等人的第一个实验那样)或者其他任意的初始状态，包括向平衡的放松。对于任何时候都处于非平衡稳态的系统，也有一个渐近结果。

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.

在《涨落定理有一个关键点，与 Loschmidt 构建这个悖论的方式不同。Loschmidt 考虑了观测单一轨迹的概率，这类似于在相空间中观测单一点的概率问题。在这两种情况下，概率总是为零。为了能够有效地解决这个问题，你必须考虑在一个相空间的小区域中的一组点的概率密度，或者一组轨迹。涨落定理分析考虑了最初在相空间无限小区域内的所有轨迹的概率密度。这直接导致了在正向或反向轨道组中找到一条轨道的可能性，这取决于初始概率分布以及随着系统演化所做的耗散。正是这种方法上的关键差异，使得涨落定理能够正确地解决这个悖论。

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.

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，时间的箭头完全是由远离大爆炸的方向决定的，一个假设的宇宙具有最大熵大爆炸不会有时间的箭头。宇宙膨胀理论试图解释为什么早期的宇宙熵如此之低。

References

参考资料

↑ 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

A toy system with time-reversible discrete dynamics showing entropy increase

! -- 范畴: 理论物理 --

! -- 类别: 统计力学 --

! -- 分类: 热力学 --

Category:Philosophy of thermal and statistical physics

类别: 热力学和统计物理学哲学

Category:Non-equilibrium thermodynamics

类别: 非平衡态热力学

Category:Physical paradoxes

类别: 物理悖论

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

[1] 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.

[2] Thomson, W. (Lord Kelvin) (1874/1875). The kinetic theory of the dissipation of energy, Nature, Vol. IX, 1874-04-09, 441–444.

[3] Dean J. Driebe, Fully Chaotic Maps and Broken Time Symmetry, (1999) Kluwer Academic
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[4] D. J. Evans and D. J. Searles, Adv. Phys. 51, 1529 (2002).

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洛斯密特悖论

目录

Origin

Origin

Before Loschmidt

Before Loschmidt

Arrow of time

Arrow of time

Dynamical systems

Dynamical systems

Fluctuation theorem

Fluctuation theorem

The Big Bang

The Big Bang

See also

See also

References

References

External links

External links

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