# 正则系综

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In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1902, an **ensemble** (also **statistical ensemble**) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system.^{[1]}

In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1902, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system.

在数学物理学中，特别是 j. Willard Gibbs 在1902年介绍到统计力学和热力学中的，一个集合体(也叫系综)是一个理想化的系统，由一个系统的大量虚拟副本(有时是无限多的)组成，一次考虑所有的副本，每个副本代表一个真实系统可能处于的状态。换句话说，系综是体制状态的概率分布。

A **thermodynamic ensemble** is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.^{[2]}^{[3]}

A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.

热力学系综是处于统计平衡状态(定义如下)的系综的一种特殊变体，用于从经典定律或量子力学定律推导热力学系统的性质。

## Physical considerations

The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.

The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.

这个整体将实验者在相同的宏观条件下一次又一次重复实验，但无法控制微观细节，可能期望观察到一系列不同的结果这一概念正式化。

The notional size of ensembles in thermodynamics, statistical mechanics and quantum statistical mechanics can be very large, including every possible microscopic state the system could be in, consistent with its observed macroscopic properties. For many important physical cases, it is possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate partition function.

The notional size of ensembles in thermodynamics, statistical mechanics and quantum statistical mechanics can be very large, including every possible microscopic state the system could be in, consistent with its observed macroscopic properties. For many important physical cases, it is possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate partition function.

在热力学、统计力学和量子统计力学中，系综的概念尺寸可以非常大，包括系统可能处于的每一种微观状态，与其观察到的宏观性质一致。对于许多重要的物理情况，可以直接计算整个热力学系综的平均值，以获得许多利息的热力学量的显式公式，通常是根据适当的配分函数。

The concept of an equilibrium or stationary ensemble is crucial to many applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it contains all past and future phases of the system. Such a statistical ensemble, one that does not change over time, is called *stationary* and can be said to be in *statistical equilibrium*.^{[1]}

The concept of an equilibrium or stationary ensemble is crucial to many applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it contains all past and future phases of the system. Such a statistical ensemble, one that does not change over time, is called stationary and can be said to be in statistical equilibrium.

平衡或稳态系综的概念对于统计系综的许多应用是至关重要的。虽然一个机械系统肯定会随着时间的推移而演化，但是整体并不一定要演化。事实上，如果整体包含了系统的所有过去和未来阶段，它就不会发展。这样的系综，一个不随时间变化的，被称为静止的，可以说是在统计平衡。

### Terminology

- The word "ensemble" is also used for a smaller set of possibilities sampled from the full set of possible states. For example, a collection of walkers in a Markov chain Monte Carlo iteration is called an ensemble in some of the literature.

- The term "ensemble" is often used in physics and the physics-influenced literature. In probability theory, the term probability space is more prevalent.

## Principal ensembles of statistical thermodynamics

Visual representation of five statistical ensembles.

五种统计集合的直观表示。

The study of thermodynamics is concerned with systems that appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics. Three important thermodynamic ensembles were defined by Gibbs:^{[1]}

The study of thermodynamics is concerned with systems that appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics. Three important thermodynamic ensembles were defined by Gibbs:

热力学研究所涉及的系统在人类看来是”静态的”(尽管它们的内部部分在运动) ，并且可以简单地用一组宏观上可观察到的变量来描述。这些系统可以用依赖于几个可观测参数的统计集合来描述，这些参数处于统计均衡状态。吉布斯指出，不同的宏观约束导致不同类型的集合，具有特定的统计特征。吉布斯定义了三个重要的热力学系综:

- Microcanonical ensemble or
*NVE*ensemble—a statistical ensemble where the total energy of the system and the number of particles in the system are each fixed to particular values; each of the members of the ensemble are required to have the same total energy and particle number. The system must remain totally isolated (unable to exchange energy or particles with its environment) in order to stay in statistical equilibrium.^{[1]}

- Canonical ensemble or
*NVT*ensemble—a statistical ensemble where the energy is not known exactly but the number of particles is fixed. In place of the energy, the temperature is specified. The canonical ensemble is appropriate for describing a closed system which is in, or has been in, weak thermal contact with a heat bath. In order to be in statistical equilibrium, the system must remain totally closed (unable to exchange particles with its environment) and may come into weak thermal contact with other systems that are described by ensembles with the same temperature.^{[1]}

- Grand canonical ensemble or
*μVT*ensemble—a statistical ensemble where neither the energy nor particle number are fixed. In their place, the temperature and chemical potential are specified. The grand canonical ensemble is appropriate for describing an open system: one which is in, or has been in, weak contact with a reservoir (thermal contact, chemical contact, radiative contact, electrical contact, etc.). The ensemble remains in statistical equilibrium if the system comes into weak contact with other systems that are described by ensembles with the same temperature and chemical potential.^{[1]}

The calculations that can be made using each of these ensembles are explored further in their respective articles.

The calculations that can be made using each of these ensembles are explored further in their respective articles.

使用这些集合中的每一个都可以进行的计算将在它们各自的文章中进一步探讨。

Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived.

Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived.

其他热力学系综也可以定义，对应不同的物理要求，其类似的公式往往可以类似地导出。

## Representations of statistical ensembles in statistical mechanics

The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a probability distribution over the microstates. In quantum mechanics, this notion, due to von Neumann, is a way of assigning a probability distribution over the results of each complete set of commuting observables.

The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a probability distribution over the microstates. In quantum mechanics, this notion, due to von Neumann, is a way of assigning a probability distribution over the results of each complete set of commuting observables.

系综的精确数学表达式根据所考虑的力学类型(量子力学或经典力学)有不同的形式。在古典情况下，这种合奏就是微观态上的概率分布。在量子力学，这个概念，由于冯诺依曼，是一种方式分配一个概率分布对每个完整的可交换观测量的结果。

In classical mechanics, the ensemble is instead written as a probability distribution in phase space; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.

In classical mechanics, the ensemble is instead written as a probability distribution in phase space; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.

在经典力学中，系综被改写为相空间中的概率分布; 微观状态是将相空间划分为相同大小的单元的结果，尽管这些单元的大小可以任意选择。

### Requirements for representations

Putting aside for the moment the question of how statistical ensembles are generated operationally, we should be able to perform the following two operations on ensembles *A*, *B* of the same system:

Putting aside for the moment the question of how statistical ensembles are generated operationally, we should be able to perform the following two operations on ensembles A, B of the same system:

暂且不论如何在操作上生成统计集合的问题，我们应该能够对同一系统的集合 a、 b 进行以下两个操作:

- Test whether
*A*,*B*are statistically equivalent.

- If
*p*is a real number such that 0 <*p*< 1, then produce a new ensemble by probabilistic sampling from*A*with probability*p*and from*B*with probability*1 – p*.

Under certain conditions, therefore, equivalence classes of statistical ensembles have the structure of a convex set.

Under certain conditions, therefore, equivalence classes of statistical ensembles have the structure of a convex set.

因此，在一定条件下，统计集合的等价类具有凸集的结构。

### Quantum mechanical

A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by [math]\displaystyle{ \hat{\rho} }[/math]. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable *X* in quantum mechanics can be written as an operator, *X̂*. The expectation value of this operator on the statistical ensemble [math]\displaystyle{ \rho }[/math] is given by the following trace:

A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by [math]\displaystyle{ \hat{\rho} }[/math]. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable in quantum mechanics can be written as an operator, . The expectation value of this operator on the statistical ensemble [math]\displaystyle{ \rho }[/math] is given by the following trace:

一个系综在量子力学(也称为混合状态)最常用密度矩阵表示，用 math / rho 表示。密度矩阵提供了一个完全通用的工具，可以统一地包括量子不确定性(即使系统的状态完全已知)和经典不确定性(由于缺乏知识)。量子力学中的任何物理可见都可以写成算符。这个算子在期望值上的系综由下面的轨迹给出:

- [math]\displaystyle{ \langle X \rangle = \operatorname{Tr}(\hat X \rho). }[/math]

[math]\displaystyle{ \langle X \rangle = \operatorname{Tr}(\hat X \rho). }[/math]

Math langle x rangle operatorname { Tr }(hat x rho) . / math

This can be used to evaluate averages (operator *X̂*), variances (using operator *X̂*^{ 2}), covariances (using operator *X̂Ŷ*), etc. The density matrix must always have a trace of 1: [math]\displaystyle{ \operatorname{Tr}{\hat{\rho}}=1 }[/math] (this essentially is the condition that the probabilities must add up to one).

This can be used to evaluate averages (operator ), variances (using operator ), covariances (using operator ), etc. The density matrix must always have a trace of 1: [math]\displaystyle{ \operatorname{Tr}{\hat{\rho}}=1 }[/math] (this essentially is the condition that the probabilities must add up to one).

这可以用来计算平均值(算子) ，方差(使用算子) ，协方差(使用算子)等。密度矩阵必须始终有一个1: math operatorname { Tr }1 / math 的迹(这本质上是概率加起来必须等于1的条件)。

In general, the ensemble evolves over time according to the von Neumann equation.

In general, the ensemble evolves over time according to the von Neumann equation.

一般来说，根据冯 · 诺依曼方程，这个乐团会随着时间而演化。

Equilibrium ensembles (those that do not evolve over time, [math]\displaystyle{ d\hat{\rho}/dt=0 }[/math]) can be written solely as a function of conserved variables. For example, the microcanonical ensemble and canonical ensemble are strictly functions of the total energy, which is measured by the total energy operator *Ĥ* (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator *N̂*. Such equilibrium ensembles are a diagonal matrix in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In bra–ket notation, the density matrix is

Equilibrium ensembles (those that do not evolve over time, [math]\displaystyle{ d\hat{\rho}/dt=0 }[/math]) can be written solely as a function of conserved variables. For example, the microcanonical ensemble and canonical ensemble are strictly functions of the total energy, which is measured by the total energy operator (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator . Such equilibrium ensembles are a diagonal matrix in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In bra–ket notation, the density matrix is

平衡系综(那些不随时间演化的，math d { rho } / dt 0 / math)可以单独写成守恒变量的函数。例如，微正则系综和正则系综是总能量的严格函数，总能量由总能量算符(哈密顿量)测量。此外，巨正则系综是粒子数的函数，由粒子总数算符测量。这样的平衡系综是正交基中的对角矩阵，它同时对角化每个守恒变量。用文胸符号表示，密度矩阵为

- [math]\displaystyle{ \hat \rho = \sum_i P_i |\psi_i\rangle \langle \psi_i | }[/math]

[math]\displaystyle{ \hat \rho = \sum_i P_i |\psi_i\rangle \langle \psi_i | }[/math]

数学 rho sum i i psi i rangle langle psi i | / math

where the |*ψ*_{i}⟩, indexed by *i*, are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)

where the ψ_{i}⟩}}, indexed by , are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)

其中子 i / sub something } ，索引由，是一个完整和正交基的元素。(请注意，在其他基地，密度矩阵不一定是对角线。)

### Classical mechanical

classical systems in phase space (top). Each system consists of one massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]

相空间(顶部)中的经典系统。每个系统由一个一维势阱(红曲线，下图)中的一个大质量粒子组成。[参考译文]最初的紧凑集合随着时间的推移而变得漩涡状

In classical mechanics, an ensemble is represented by a probability density function defined over the system's phase space.^{[1]} While an individual system evolves according to Hamilton's equations, the density function (the ensemble) evolves over time according to Liouville's equation.

In classical mechanics, an ensemble is represented by a probability density function defined over the system's phase space. While an individual system evolves according to Hamilton's equations, the density function (the ensemble) evolves over time according to Liouville's equation.

在经典力学中，系综由定义在系统相空间上的概率密度函数表示。当一个单独的系统根据哈密顿方程进化时，密度函数(集合)则根据刘维尔方程随时间进化。

In a mechanical system with a defined number of parts, the phase space has *n* generalized coordinates called *q*_{1}, ... *q*_{n}, and *n* associated canonical momenta called *p*_{1}, ... *p*_{n}. The ensemble is then represented by a joint probability density function *ρ*(*p*_{1}, ... *p*_{n}, *q*_{1}, ... *q*_{n}).

In a mechanical system with a defined number of parts, the phase space has generalized coordinates called , and associated canonical momenta called . The ensemble is then represented by a joint probability density function .

在一个力学系统中，有一定数量的部分，相空间叫做广义坐标，相关的正则动量叫做动量。然后，合奏团由一个联合概率密度函数代表。

If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers *N*_{1} (first kind of particle), *N*_{2} (second kind of particle), and so on up to *N*_{s} (the last kind of particle; *s* is how many different kinds of particles there are). The ensemble is then represented by a joint probability density function *ρ*(*N*_{1}, ... *N*_{s}, *p*_{1}, ... *p*_{n}, *q*_{1}, ... *q*_{n}). The number of coordinates *n* varies with the numbers of particles.

If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers (first kind of particle), (second kind of particle), and so on up to (the last kind of particle; is how many different kinds of particles there are). The ensemble is then represented by a joint probability density function . The number of coordinates varies with the numbers of particles.

如果系统中的部分数量允许在系综中不同系统之间变化(就像在一个大系综中，粒子数量是一个随机量) ，那么它就是一个扩展相空间上的概率分布，扩展相空间包括更多的变量，如粒子数量(第一类粒子) ，第二类粒子) ，等等，直到(最后一类粒子; 是有多少不同类型的粒子)。然后，合奏团由一个联合概率密度函数代表。坐标的数目随粒子的数目而变化。

Any mechanical quantity *X* can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by *ρ*:

Any mechanical quantity can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by :

任何机械量都可以写成系统相位的函数。任何这样的量的期望值是由这个量在整个相空间上的积分加权得到的:

- [math]\displaystyle{ \langle X \rangle = \sum_{N_1 = 0}^{\infty} \ldots \sum_{N_s = 0}^{\infty} \int \ldots \int \rho X \, dp_1 \ldots dq_n. }[/math]

[math]\displaystyle{ \langle X \rangle = \sum_{N_1 = 0}^{\infty} \ldots \sum_{N_s = 0}^{\infty} \int \ldots \int \rho X \, dp_1 \ldots dq_n. }[/math]

数学角 x = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 10 = 1

The condition of probability normalization applies, requiring

The condition of probability normalization applies, requiring

概率归一化的条件适用，要求

- [math]\displaystyle{ \sum_{N_1 = 0}^{\infty} \ldots \sum_{N_s = 0}^{\infty} \int \ldots \int \rho \, dp_1 \ldots dq_n = 1. }[/math]

[math]\displaystyle{ \sum_{N_1 = 0}^{\infty} \ldots \sum_{N_s = 0}^{\infty} \int \ldots \int \rho \, dp_1 \ldots dq_n = 1. }[/math]

数学{ n10} ^ { infty } ldots sum { ns0} ^ { infty } ldots int rho，dp 1 ldots dqn1. / math

Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability *density* in phase space to a probability *distribution* over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume.^{[note 1]} In particular, the probability density function in phase space, *ρ*, is related to the probability distribution over microstates, *P* by a factor

Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability density in phase space to a probability distribution over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume. In particular, the probability density function in phase space, , is related to the probability distribution over microstates, by a factor

相空间是一个连续的空间，在任何一个小区域内都包含无限多个不同的物理状态。为了将相空间中的概率密度与微态上的概率分布相联系，有必要以某种方式将相空间分割成代表系统不同状态的块。事实证明，正确的方法只会导致相同大小的正则相空间块，所以经典力学中的微状态是正则坐标相空间中具有特定体积的扩展区域。特别是，位相空间中的概率密度函数与微态上的概率分布有一定的关系

- [math]\displaystyle{ \rho = \frac{1}{h^n C} P, }[/math]

[math]\displaystyle{ \rho = \frac{1}{h^n C} P, }[/math]

数学，数学，数学

where

where

在哪里

*h*is an arbitrary but predetermined constant with the units of energy×time, setting the extent of the microstate and providing correct dimensions to*ρ*.^{[note 2]}

*C*is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.

Since *h* can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of *h* influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of *h* when comparing different systems.

Since can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of when comparing different systems.

由于可以任意选择，一个微观状态的概念大小也是任意的。尽管如此，价值的影响量的偏移量，如熵和化学势，所以它是重要的是一致的价值时，比较不同的体系。

#### Correcting overcounting in phase space

Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:

Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:

通常，相空间在多个不同的位置包含相同物理状态的重复项。这是物理状态编码为数学坐标的结果; 最简单的坐标系选择通常允许以多种方式对状态进行编码。这方面的一个例子是全同粒子的气体，其状态是由粒子各自的位置和动量表示的: 当两个粒子交换时，相空间中的结果点是不同的，但它相当于系统的一个相同的物理状态。认识到相空间只是一个数学结构，并且在相空间上积分时不要天真地过多计算实际的物理状态，这在统计力学理论中是很重要的。数量过多会导致严重的问题:

- Dependence of derived quantities (such as entropy and chemical potential) on the choice of coordinate system, since one coordinate system might show more or less overcounting than another.
^{[note 3]}

- Erroneous conclusions that are inconsistent with physical experience, as in the mixing paradox.
^{[1]}

- Foundational issues in defining the chemical potential and the grand canonical ensemble.
^{[1]}

It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.

It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.

一般来说，很难找到一个对每种物理状态进行唯一编码的坐标系。因此，通常需要使用带有每个状态的多个副本的坐标系，然后识别并删除多计数。

A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' *x* coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor *C* introduced above would be set to *C* = 1, and the integral would be restricted to the selected subregion of phase space.)

A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor introduced above would be set to 1}}, and the integral would be restricted to the selected subregion of phase space.)

消除多计的一个粗略方法是人工定义一个只包含每个物理状态一次的相空间分区，然后排除相空间的所有其他部分。例如，在气体中，可以只包括粒子坐标按升序排列的那些相。虽然这样可以解决这个问题，但是由于相空间的边界形状不同寻常，因此计算相空间上的积分将是非常繁琐的。(在这种情况下，上面介绍的因子将设置为1} ，并且积分将局限于选定的相空间分区。)

A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor *C* introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates,^{[note 4]} so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, *C* does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.

A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates, so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.

纠正过计数的一种较简单的方法是对所有相空间进行积分，但要减小各相的权重，以便准确地补偿过计数。这是通过上面介绍的因子来实现的，该因子是一个整数，表示物理状态在相空间中可以用多少种方式表示。它的值不会随着连续正则坐标的变化而变化，因此可以简单地通过对整个正则坐标范围进行积分，然后将结果除以重复计数因子来纠正重复计数。然而，随着离散变量的变化，例如粒子数量的变化很大，因此在对粒子数量进行求和之前必须进行求和。

As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using^{[1]}

As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using

如上所述，这种过度计算的典型例子是包含各种粒子的流体系统，其中同种类的任何两个粒子是不可区分和可交换的。当状态用粒子的单个位置和动量来表示时，那么与全同粒子交换有关的过度计算就可以通过使用

- [math]\displaystyle{ C = N_1! N_2! \ldots N_s!. }[/math]

[math]\displaystyle{ C = N_1! N_2! \ldots N_s!. }[/math]

数学 c n 1！N 2！数学! 数学

This is known as "correct Boltzmann counting".

This is known as "correct Boltzmann counting".

这就是所谓的“正确玻尔兹曼计数”。

## Ensembles in statistics

The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the canonical ensemble or Gibbs measure serves to maximize the entropy of a system, subject to a set of constraints: this is the principle of maximum entropy. This principle has now been widely applied to problems in linguistics, robotics, and the like.

The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the canonical ensemble or Gibbs measure serves to maximize the entropy of a system, subject to a set of constraints: this is the principle of maximum entropy. This principle has now been widely applied to problems in linguistics, robotics, and the like.

在物理学中使用的统计系综的公式现在已经被其他领域广泛采用，部分原因是已经认识到正则系综或吉布斯测量服务于最大化一个系统的熵，受到一系列约束: 这是最大熵原理。这一原则现已广泛应用于语言学、机器人学等领域的问题。

In addition, statistical ensembles in physics are often built on a principle of locality: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example, lattice models, such as the Ising model, model ferromagnetic materials by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the Markov property in the broad sense; nearest neighbors are now Markov blankets. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to Markov random fields, which again find broad applicability; for example in Hopfield networks.

In addition, statistical ensembles in physics are often built on a principle of locality: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example, lattice models, such as the Ising model, model ferromagnetic materials by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the Markov property in the broad sense; nearest neighbors are now Markov blankets. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to Markov random fields, which again find broad applicability; for example in Hopfield networks.

此外，物理学中的统计集合通常是建立在定域性原理的基础上的: 所有的相互作用只存在于相邻的原子或分子之间。因此，例如，格子模型，如伊辛模型，通过自旋间的最近邻相互作用来模拟铁磁材料。定域性原理的统计公式现在被认为是广义的马尔可夫性的一种形式; 最近的邻居现在是马尔可夫的毯子。因此，具有最近邻相互作用的系综的一般概念导致马尔可夫随机场，这再次找到广泛的适用性，例如在 Hopfield 网络。

## Operational interpretation

In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble *itself* (not the consequent results) is a precisely defined object mathematically. For instance,

In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble itself (not the consequent results) is a precisely defined object mathematically. For instance,

在迄今为止给出的讨论中，尽管很严谨，但我们理所当然地认为集合的概念是先验有效的，就像通常在物理环境中所做的那样。但是还没有证明的是整体本身(而不是结果)在数学上是一个精确定义的对象。比如说,

- It is not clear where this
*very large set of systems*exists (for example, is it a*gas*of particles inside a container?)

- It is not clear how to physically generate an ensemble.

In this section, we attempt to partially answer this question.

In this section, we attempt to partially answer this question.

在本节中，我们试图部分回答这个问题。

Suppose we have a *preparation procedure* for a system in a physics

Suppose we have a preparation procedure for a system in a physics

假设我们有一个物理系统的准备程序

lab: For example, the procedure might involve a physical apparatus and

lab: For example, the procedure might involve a physical apparatus and

例如，这个过程可能会涉及到一个物理装置和

some protocols for manipulating the apparatus. As a result of this preparation procedure, some system

some protocols for manipulating the apparatus. As a result of this preparation procedure, some system

一些操纵仪器的程序。由于这一准备程序的结果，一些制度

is produced and maintained in isolation for some small period of time.

is produced and maintained in isolation for some small period of time.

在一小段时间内独立生产和维护。

By repeating this laboratory preparation procedure we obtain a

By repeating this laboratory preparation procedure we obtain a

通过重复这个实验室准备程序，我们得到了一个

sequence of systems *X*_{1}, *X*_{2},

sequence of systems X_{1}, X_{2},

系统序列 x 子1 / 子，x 子2 / 子,

....,*X*_{k}, which in our mathematical idealization, we assume is an infinite sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble.

....,X_{k}, which in our mathematical idealization, we assume is an infinite sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble.

在我们的数学理想化中，我们假设系统是一个无穷序列。这些系统是相似的，因为它们都是以相同的方式生产的。这个无限序列是一个集合。

In a laboratory setting, each one of these prepped systems might be used as input

In a laboratory setting, each one of these prepped systems might be used as input

在实验室环境中，这些预先准备好的系统中的每一个都可以用作输入

for *one* subsequent *testing procedure*. Again, the testing procedure

for one subsequent testing procedure. Again, the testing procedure

进行后续测试。同样，测试过程

involves a physical apparatus and some protocols; as a result of the

involves a physical apparatus and some protocols; as a result of the

涉及到一个物理装置和一些协议

testing procedure we obtain a *yes* or *no* answer.

testing procedure we obtain a yes or no answer.

测试程序，我们得到是或不是的答案。

Given a testing procedure *E* applied to each prepared system, we obtain a sequence of values

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

给出了应用于每个准备系统的测试程序 e，我们得到了一系列的值

Meas (*E*, *X*_{1}), Meas (*E*, *X*_{2}),

Meas (E, X_{1}), Meas (E, X_{2}),

Meas (e，x 子1 / sub) ，Meas (e，x 子2 / sub) ,

...., Meas (*E*, *X*_{k}). Each one of these values is a 0 (or no) or a 1 (yes).

...., Meas (E, X_{k}). Each one of these values is a 0 (or no) or a 1 (yes).

... ，Meas (e，x 子 k / sub)。这些值中的每一个都是0(或否)或1(是)。

Assume the following time average exists:

Assume the following time average exists:

假设存在以下时间平均值:

- [math]\displaystyle{ \sigma(E) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \operatorname{Meas}(E, X_k) }[/math]

[math]\displaystyle{ \sigma(E) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \operatorname{Meas}(E, X_k) }[/math]

Math sigma (e) lim { n right tarrow infty } frac {1}{ n }{ k1} ^ n operatorname { Meas }(e，xk) / math

For quantum mechanical systems, an important assumption made in the

For quantum mechanical systems, an important assumption made in the

对于量子力学系统，一个重要的假设

quantum logic approach to quantum mechanics is the identification of *yes-no* questions to the

quantum logic approach to quantum mechanics is the identification of yes-no questions to the

量子逻辑方法的量子力学是识别是否的问题

lattice of closed subspaces of a Hilbert space. With some additional

lattice of closed subspaces of a Hilbert space. With some additional

Hilbert 空间的闭子空间格。加上一些额外的

technical assumptions one can then infer that states are given by

technical assumptions one can then infer that states are given by

我们可以推断，状态是由

density operators *S* so that:

density operators S so that:

密度算符 s，以便:

- [math]\displaystyle{ \sigma(E) = \operatorname{Tr}(E S). }[/math]

[math]\displaystyle{ \sigma(E) = \operatorname{Tr}(E S). }[/math]

数学 sigma (e)运营商名称{ Tr }(e s)。数学

We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.

We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.

我们看到这反映了一般量子态的定义: 量子态是从可观测量到其期望值的映射。

## See also

## Notes

- ↑ This equal-volume partitioning is a consequence of Liouville's theorem, i. e., the principle of conservation of extension in canonical phase space for Hamiltonian mechanics. This can also be demonstrated starting with the conception of an ensemble as a multitude of systems. See Gibbs'
*Elementary Principles*, Chapter I. - ↑ (Historical note) Gibbs' original ensemble effectively set
*h*= 1 [energy unit]×[time unit], leading to unit-dependence in the values of some thermodynamic quantities like entropy and chemical potential. Since the advent of quantum mechanics,*h*is often taken to be equal to Planck's constant in order to obtain a semiclassical correspondence with quantum mechanics. - ↑ In some cases the overcounting error is benign. An example is the choice of coordinate system used for representing orientations of three-dimensional objects. A simple encoding is the 3-sphere (e. g., unit quaternions) which is a double cover—each physical orientation can be encoded in two ways. If this encoding is used without correcting the overcounting, then the entropy will be higher by
*k*log 2 per rotatable object and the chemical potential lower by*kT*log 2. This does not actually lead to any observable error since it only causes unobservable offsets. - ↑ Technically, there are some phases where the permutation of particles does not even yield a distinct specific phase: for example, two similar particles can share the exact same trajectory, internal state, etc.. However, in classical mechanics these phases only make up an infinitesimal fraction of the phase space (they have measure zero) and so they do not contribute to any volume integral in phase space.

## References

## External links

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}^{1.7}^{1.8}^{1.9}Gibbs, Josiah Willard (1902).*Elementary Principles in Statistical Mechanics*. New York: Charles Scribner's Sons. - ↑ Kittel, Charles; Herbert Kroemer (1980).
*Thermal Physics, Second Edition*. San Francisco: W.H. Freeman and Company. pp. 31 ff. ISBN 0-7167-1088-9. - ↑ Landau, L.D.; Lifshitz, E.M. (1980).
*Statistical Physics*. Pergamon Press. pp. 9 ff. ISBN 0-08-023038-5.

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