遍历假设

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文件:Ergodic hypothesis w reflecting rays.jpg
The question of ergodicity in a perfectly collisionless ideal gas with specular reflections.

The question of ergodicity in a perfectly collisionless ideal gas with specular reflections.

完全无碰撞情况下的遍历性问题[[带有镜面反射的理想气体]

文件:Fruit fly trap.jpg
This device can trap fruit flies, but if it trapped atoms when placed in gas that already uniformly fills the available phase space, then both Liouville's theorem and the second law of thermodynamics would be violated.

This device can trap fruit flies, but if it trapped atoms when placed in gas that already uniformly fills the available phase space, then both Liouville's theorem and the second law of thermodynamics would be violated.

这个装置可以捕捉果蝇,但是如果把它放在已经均匀地充满可用相空间的气体中,就会捕捉到原子[[那么 Liouville 定理和热力学第二定律就会被破坏]。]

In physics and thermodynamics, the ergodic hypothesis[1] says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.

In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.

在物理学和热力学中,遍历假设说,在很长一段时间内,一个系统在相空间的某个区域所花费的时间与这个区域的体积成正比,也就是说,在很长一段时间内,所有可能的微观状态都是等概率的。


Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does not imply that the ergodic hypothesis holds for all Hamiltonian systems.

Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does not imply that the ergodic hypothesis holds for all Hamiltonian systems.

刘维尔定理指出,对于哈密顿系统,当观察者与系综一起移动时,沿着粒子路径穿过相空间的局部微观态密度是常数(即对流时间导数为零)。因此,如果微观态最初均匀分布在相空间中,那么它们将始终保持均匀分布。但刘维尔定理并不意味着遍历假设定理适用于所有的哈密顿系统。


The ergodic hypothesis is often assumed in the statistical analysis of computational physics. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. This assumption—that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system—is not always correct. (See, for example, the Fermi–Pasta–Ulam–Tsingou experiment of 1953.)

The ergodic hypothesis is often assumed in the statistical analysis of computational physics. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. This assumption—that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system—is not always correct. (See, for example, the Fermi–Pasta–Ulam–Tsingou experiment of 1953.)

遍历假设通常被认为是计算物理学的统计分析。分析师会假设一个过程参数随时间的平均值和系综的平均值是相同的。这种假设——长时间模拟一个系统和对同一个系统进行许多独立实现一样好——并不总是正确的。(例如,参见1953年的费米-通心粉-乌拉姆-青岛实验。)


Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible.

Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible.

遍历假设的假设可以证明第二类永动机的某些类型的机器是不可能的。


Phenomenology

In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking.

In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking.

在宏观系统中,一个系统能够真正探索其自身相空间的整体的时间尺度可以足够大为热力学平衡状态表现出某种形式的遍历性破坏。一个常见的例子是铁磁系统中的自发磁化,即低于居里点时,系统优先采用非零磁化,即使遍历假设意味着,由于系统探索所有时均磁化应为零的状态,净磁化不应存在。事实上,宏观系统经常违反遍历假设的字面形式是一个例子,自发对称性破缺。


However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In practice this means that on sufficiently short time scales (e.g. those of parts of seconds, minutes, or a few hours) the systems may behave as solids, i.e. with a positive shear modulus, but on extremely long scales, e.g. over millennia or eons, as liquids, or with two or more time scales and plateaux in between.[2]

However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In practice this means that on sufficiently short time scales (e.g. those of parts of seconds, minutes, or a few hours) the systems may behave as solids, i.e. with a positive shear modulus, but on extremely long scales, e.g. over millennia or eons, as liquids, or with two or more time scales and plateaux in between.

然而,复杂的无序系统,如自旋玻璃表现出一种更加复杂的遍历性破坏形式,在实践中看到的热力学平衡态的性质更加难以纯粹通过对称性论证来预测。还有传统的眼镜(例如:。窗玻璃)以复杂的方式违反遍历性。在实践中,这意味着在足够短的时间范围内(例如:。这些系统可以表现为固体,例如:。具有正剪切模量,但是在极长的尺度上,例如:。数千年或数亿年,或是液体,或是两个或两个以上的时间尺度和高原之间。


Mathematics

Ergodic theory is a branch of mathematics which deals with dynamical systems that satisfy a version of this hypothesis, phrased in the language of measure theory.

Ergodic theory is a branch of mathematics which deals with dynamical systems that satisfy a version of this hypothesis, phrased in the language of measure theory.

遍历理论是数学的一个分支,它研究的是满足这一假设的动力系统,用测量理论的语言来表述。


See also

  • Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity


References

  1. Originally due to L. Boltzmann. See part 2 of Vorlesungen über Gastheorie. Leipzig: J. A. Barth. 1898. OCLC 01712811. https://archive.org/details/vorlesungenberg02boltgoog.  ('Ergoden' on p.89 in the 1923 reprint.) It was used to prove equipartition of energy in the kinetic theory of gases
  2. The introduction of the practical aspect of ergodicity breaking by introducing a "non-ergodicity time scale" is due to Palmer, R. G. (1982). "Broken ergodicity". Advances in Physics. 31 (6): 669. Bibcode:1982AdPhy..31..669P. doi:10.1080/00018738200101438.. Also related to these time-scale phenomena are the properties of ageing and the Mode-Coupling theory of Götze, W. (2008). Dynamics of Glass Forming Liquids. Oxford Univ. Press. 

Category:Ergodic theory

范畴: 遍历理论

Category:Statistical mechanics

类别: 统计力学

Category:Philosophy of thermal and statistical physics

类别: 热力学和统计物理学哲学

Category:Concepts in physics

分类: 物理概念


This page was moved from wikipedia:en:Ergodic hypothesis. Its edit history can be viewed at 遍历假设/edithistory