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配分函数 (查看源代码)

2021年10月26日 (二) 16:25的版本

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In [[physics]], a '''partition function''' describes the [[statistics|statistical]] properties of a system in [[thermodynamic equilibrium]].{{Citation needed|reason=definition of partition function requires referencing|date=December 2016}}  Partition functions are [[function (mathematics)|functions]] of the thermodynamic [[state function|state variables]],  such as the [[temperature]] and [[volume]]. Most of the aggregate [[thermodynamics|thermodynamic]] variables of the system, such as the [[energy|total energy]], [[Thermodynamic free energy|free energy]], [[entropy]], and [[pressure]], can be expressed in terms of the partition function or its [[derivative]]s. The partition function is dimensionless, it is a pure number.
 
In [[physics]], a '''partition function''' describes the [[statistics|statistical]] properties of a system in [[thermodynamic equilibrium]].{{Citation needed|reason=definition of partition function requires referencing|date=December 2016}}  Partition functions are [[function (mathematics)|functions]] of the thermodynamic [[state function|state variables]],  such as the [[temperature]] and [[volume]]. Most of the aggregate [[thermodynamics|thermodynamic]] variables of the system, such as the [[energy|total energy]], [[Thermodynamic free energy|free energy]], [[entropy]], and [[pressure]], can be expressed in terms of the partition function or its [[derivative]]s. The partition function is dimensionless, it is a pure number.
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In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium.  Partition functions are functions of the thermodynamic state variables,  such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number.
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在物理学中，一个配分函数描述了一个[[热力学平衡系统]]的统计特性。配分函数是热力学状态变量的函数，比如温度和体积。体系中的大多数[[热力学]]变量，如总能量、自由能、[[熵]]和压力，都可以用配分函数或其衍生物来表示。配分函数是无量纲的，它是一个纯数。
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在物理学中，一个配分函数描述了一个热力学平衡系统的统计特性。配分函数是热力学状态变量的函数，比如温度和体积。体系中的大多数热力学变量，如总能量、自由能、熵和压力，都可以用配分函数或其衍生物来表示。配分函数是无量纲的，它是一个纯数。
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Each partition function is constructed to represent a particular [[statistical ensemble]] (which, in turn, corresponds to a particular [[Thermodynamic free energy|free energy]]). The most common statistical ensembles have named partition functions. The '''canonical partition function''' applies to a [[canonical ensemble]], in which the system is allowed to exchange [[heat]] with the [[Environment (systems)|environment]] at fixed temperature, volume, and [[number of particles]]. The '''grand canonical partition function''' applies to a [[grand canonical ensemble]], in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and [[chemical potential]]. Other types of partition functions can be defined for different circumstances; see [[partition function (mathematics)]] for generalizations. The partition function has many physical meanings, as discussed in [[#Meaning and significance|Meaning and significance]].
 
Each partition function is constructed to represent a particular [[statistical ensemble]] (which, in turn, corresponds to a particular [[Thermodynamic free energy|free energy]]). The most common statistical ensembles have named partition functions. The '''canonical partition function''' applies to a [[canonical ensemble]], in which the system is allowed to exchange [[heat]] with the [[Environment (systems)|environment]] at fixed temperature, volume, and [[number of particles]]. The '''grand canonical partition function''' applies to a [[grand canonical ensemble]], in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and [[chemical potential]]. Other types of partition functions can be defined for different circumstances; see [[partition function (mathematics)]] for generalizations. The partition function has many physical meanings, as discussed in [[#Meaning and significance|Meaning and significance]].
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Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance.
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每个配分函数都代表一个特定的系综(反过来，对应一个特定的自由能)。最常见的统计集合称为配分函数。典型的配分函数适用于正则系综，系统允许在固定的温度、体积和粒子数量下与环境进行热交换。巨正则配分函数适用于巨正则系综，系统可以在固定的温度、体积和化学势下同时与环境交换热量和粒子。其他类型的配分函数可以根据不同的情况来定义; 有关一般化，请参阅数学配分函数。配分函数有许多物理意义，正如在意义和重要性中讨论的那样。
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每个配分函数都代表一个特定的系综(这里指的是一个特定的自由能)。常见的系综都有配分函数。典型的配分函数适用于[[正则系综]]，系统允许在固定的温度、体积和粒子数量下与环境进行热交换。巨正则配分函数适用于[[巨正则系综]]，系统可以在固定的温度、体积和化学势下同时与环境交换热量和粒子。其他类型的配分函数可以根据不同的情况来定义。配分函数有许多物理意义。
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== Canonical partition function ==
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== Canonical partition function 标准配分函数 ==
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=== Definition ===
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=== Definition定义 ===
    
Initially, let us assume that a thermodynamically large system is in [[thermal contact]] with the environment, with a temperature ''T'', and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of systems comprises an ensemble called a [[canonical ensemble]]. The appropriate [[mathematical expression]] for the canonical partition function depends on the [[degrees of freedom]] of the system, whether the context is [[classical mechanics]] or [[quantum mechanics]], and whether the spectrum of states is [[Discrete mathematics|discrete]] or [[Probability distribution#Continuous probability distribution|continuous]].{{Citation needed|reason=definition of partition function requires referencing|date=December 2016}}
 
Initially, let us assume that a thermodynamically large system is in [[thermal contact]] with the environment, with a temperature ''T'', and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of systems comprises an ensemble called a [[canonical ensemble]]. The appropriate [[mathematical expression]] for the canonical partition function depends on the [[degrees of freedom]] of the system, whether the context is [[classical mechanics]] or [[quantum mechanics]], and whether the spectrum of states is [[Discrete mathematics|discrete]] or [[Probability distribution#Continuous probability distribution|continuous]].{{Citation needed|reason=definition of partition function requires referencing|date=December 2016}}
   −
Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of systems comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous.
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假设一个热力学大系统与大环境进行温度为 T的热接触，系统的体积和组成粒子的数量都是固定的。众多这类系统组合成的集合体叫做正则系综。它代表了与恒温热库接触而处于热平衡的系统所有可能状态的集合。由于系统可以与热库交换能量，系统可能的微观状态可以具有不同的能量。正则配分函数的数学表达式取决于系统的自由度，不同的能量表示取决于不同的假设是经典力学还是量子力学、以及状态谱是离散的还是连续的。
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首先，让我们假设一个热力学大系统与环境热接触，温度为 t，系统的体积和组成粒子的数量都是固定的。这类系统的集合包括一个叫做正则系综的集合体。正则配分函数的数学表达式取决于系统的自由度，上下文是经典力学还是量子力学，以及状态谱是离散的还是连续的。
            
;Classical discrete system
 
;Classical discrete system
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  −
Classical discrete system
      
经典离散系统
 
经典离散系统
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  −
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as
      
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as
 
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as
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: <math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math>
 
: <math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math>
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  <math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math>
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  <math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math><br />
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[数学] z = sum { i } mathrm { e } ^ {-beta e _ i } ，[数学]
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  −
 
  −
 
  −
where
      
where
 
where
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: <math> i </math> is the index for the [[Microstate (statistical mechanics)|microstates]] of the system;
 
: <math> i </math> is the index for the [[Microstate (statistical mechanics)|microstates]] of the system;
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  <math> i </math> is the index for the microstates of the system;
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  <math> i </math> 是系统微观状态的指标；
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  −
是系统微观状态的指标
      
: <math> \mathrm{e} </math> is [[e (mathematical constant)|Euler's number]];
 
: <math> \mathrm{e} </math> is [[e (mathematical constant)|Euler's number]];
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  <math> \mathrm{e} </math> is Euler's number;
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  <math> \mathrm{e} </math> 是欧拉的数字;
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  −
数学是欧拉的数字;
      
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
 
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
   −
  <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
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  <math> \beta </math> 是热力学beta，定义为 <math> \tfrac{1}{k_\text{B} T} </math>;
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  −
是热力学beta，定义为 < math > tfrac
      
: <math> E_i </math> is the total energy of the system in the respective [[Microstate (statistical mechanics)|microstate]].
 
: <math> E_i </math> is the total energy of the system in the respective [[Microstate (statistical mechanics)|microstate]].
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  <math> E_i </math> is the total energy of the system in the respective microstate.
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  <math> E_i </math> 是系统在各自微观状态下的总能量。
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是系统在各自微观状态下的总能量。
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The [[Exponential function|exponential]] factor <math> \mathrm{e}^{-\beta E_i} </math> is otherwise known as the [[Boltzmann factor]].
 
The [[Exponential function|exponential]] factor <math> \mathrm{e}^{-\beta E_i} </math> is otherwise known as the [[Boltzmann factor]].
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The exponential factor <math> \mathrm{e}^{-\beta E_i} </math> is otherwise known as the Boltzmann factor.
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指数因子 <math> \mathrm{e}^{-\beta E_i} </math> 也被称为玻尔兹曼因子。
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指数因子 < math > mathrm { e } ^ {-beta e _ i } </math > 也被称为玻尔兹曼因子。
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|}
      
|}
 
|}
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In [[classical mechanics]], the [[Position (vector)|position]] and [[Momentum vector|momentum]] variables of a particle can vary continuously, so the set of microstates is actually [[uncountable set|uncountable]]. In ''classical'' statistical mechanics, it is rather inaccurate to express the partition function as a [[Sum (mathematics)|sum]] of discrete terms. In this case we must describe the partition function using an [[integral]] rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as
 
In [[classical mechanics]], the [[Position (vector)|position]] and [[Momentum vector|momentum]] variables of a particle can vary continuously, so the set of microstates is actually [[uncountable set|uncountable]]. In ''classical'' statistical mechanics, it is rather inaccurate to express the partition function as a [[Sum (mathematics)|sum]] of discrete terms. In this case we must describe the partition function using an [[integral]] rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as
   −
In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as
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在[[经典力学]]中，一个粒子的位置和动量变量可以连续变化，所以微观状态的集合实际上是无法计算的。在[[古典统计力学]]中，将配分函数表示为离散项的和是相当不准确的。在这种情况下，我们必须用积分而不是和来描述配分函数。对于一个经典的连续正则系综，典型的配分函数被定义为
   −
在经典力学中，一个粒子的位置和动量变量可以连续变化，所以微观状态的集合实际上是无法计算的。在古典统计力学中，将配分函数表示为离散项的和是相当不准确的。在这种情况下，我们必须用积分而不是和来描述配分函数。对于一个经典的连续正则系综，典型的配分函数被定义为
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: <math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math>
 
: <math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math>
   −
  <math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math>
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  <math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math><br />
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  −
< math > z = frac {1}{ h ^ 3} int mathrm { e } ^ {-beta h (q，p)} ，mathrm { d } ^ 3 q，mathrm { d } ^ 3 p，</math >  
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  −
 
      
where
 
where
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  −
where
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在哪里
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: <math> h </math> is the [[Planck constant]];
 
: <math> h </math> is the [[Planck constant]];
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  <math> h </math> is the Planck constant;
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  <math> h </math> 是普朗克常数;
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  −
是普朗克常数;
      
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
 
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
   −
  <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
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  <math> \beta </math> 是热力学beta，定义为 <math> \tfrac{1}{k_\text{B} T} </math>;
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  −
是热力学beta，定义为 < math > tfrac
      
: <math> H(q, p) </math> is the [[Hamiltonian mechanics|Hamiltonian]] of the system;
 
: <math> H(q, p) </math> is the [[Hamiltonian mechanics|Hamiltonian]] of the system;
   −
  <math> H(q, p) </math> is the Hamiltonian of the system;
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  <math> H(q, p) </math> 是系统的哈密顿函数;
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  −
H (q，p) </math > 是系统的哈密顿函数;
      
: <math> q </math> is the [[Canonical coordinates|canonical position]];
 
: <math> q </math> is the [[Canonical coordinates|canonical position]];
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  <math> q </math> is the canonical position;
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  <math> q </math> 是典型的位置
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  −
Q </math > 是典型的位置
      
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
 
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
   −
  <math> p </math> is the canonical momentum.
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  <math> p </math> 是典型的动量。
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是典型的动量。
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To make it into a dimensionless quantity, we must divide it by ''h'', which is some quantity with units of [[action (physics)|action]] (usually taken to be [[Planck's constant]]).
 
To make it into a dimensionless quantity, we must divide it by ''h'', which is some quantity with units of [[action (physics)|action]] (usually taken to be [[Planck's constant]]).
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To make it into a dimensionless quantity, we must divide it by h, which is some quantity with units of action (usually taken to be Planck's constant).
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为了使它成为一个无量纲量，我们必须将它除以带有作用单位的量 h(通常被认为是普朗克常数)。
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为了使它成为一个无量纲量，我们必须将它除以 h，这是一个带有作用单位的量(通常被认为是普朗克常数)。
            
;Classical continuous system (multiple identical particles)
 
;Classical continuous system (multiple identical particles)
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  −
Classical continuous system (multiple identical particles)
      
经典的连续系统(多全同粒子)
 
经典的连续系统(多全同粒子)
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  −
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is
      
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is
 
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is
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  <math> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N </math>
 
  <math> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N </math>
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[数学] z = frac {1}{ n! h ^ {3N } int，exp left (- beta sum { i = 1} ^ n h (textbf q _ i，textbf p _ i) right) ; mathrm { d } ^ 3 q _ 1 cdots mathrm { d } ^ 3 q _ n，mathrm { d } ^ 3 p _ 1 cdots mathrm { d } ^ 3 p _ n </math >
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  −
      
where
 
where
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where
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在哪里
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: <math> h </math> is the [[Planck constant]];
 
: <math> h </math> is the [[Planck constant]];
   −
  <math> h </math> is the Planck constant;
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  <math> h </math> 是普朗克常数;
−
 
  −
是普朗克常数;
      
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
 
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
   −
  <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
+
  <math> \beta </math> 是热力学beta，定义为  <math> \tfrac{1}{k_\text{B} T} </math>;
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  −
是热力学beta，定义为 < math > tfrac
      
: <math> i </math> is the index for the particles of the system;
 
: <math> i </math> is the index for the particles of the system;
   −
  <math> i </math> is the index for the particles of the system;
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  <math> i </math> 是系统粒子的指数
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  −
是系统粒子的指数
      
: <math> H </math> is the [[Hamiltonian mechanics|Hamiltonian]] of a respective particle;
 
: <math> H </math> is the [[Hamiltonian mechanics|Hamiltonian]] of a respective particle;
   −
  <math> H </math> is the Hamiltonian of a respective particle;
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  <math> H </math> 是一个粒子的哈密顿量;
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  −
是一个粒子的哈密顿量;
      
: <math> q_i </math> is the [[Canonical coordinates|canonical position]] of the respective particle;
 
: <math> q_i </math> is the [[Canonical coordinates|canonical position]] of the respective particle;
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  <math> q_i </math> is the canonical position of the respective particle;
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  <math> q_i </math> 是各个粒子的标准位置;
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  −
是各个粒子的标准位置;
      
: <math> p_i </math> is the [[Canonical coordinates|canonical momentum]] of the respective particle;
 
: <math> p_i </math> is the [[Canonical coordinates|canonical momentum]] of the respective particle;
   −
  <math> p_i </math> is the canonical momentum of the respective particle;
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  <math> p_i </math> 是各个粒子的正则动量;
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  −
是各个粒子的正则动量;
      
: <math> \mathrm{d}^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
 
: <math> \mathrm{d}^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
   −
  <math> \mathrm{d}^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
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  <math> \mathrm{d}^3 </math> 是一个简写符号，用来表示 <math> q_i </math> 和 <math> p_i </math> 是三维空间中的向量。<br />
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  −
是一个简写符号，用来表示 < math > q _ i </math > 和 < math > p _ i </math > 是三维空间中的向量。
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  −
 
      
The reason for the [[factorial]] factor ''N''! is discussed [[#Partition functions of subsystems|below]]. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not [[dimensionless]]. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by ''h''<sup>3''N''</sup> (where ''h'' is usually taken to be Planck's constant).
 
The reason for the [[factorial]] factor ''N''! is discussed [[#Partition functions of subsystems|below]]. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not [[dimensionless]]. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by ''h''<sup>3''N''</sup> (where ''h'' is usually taken to be Planck's constant).
   −
The reason for the factorial factor N! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h<sup>3N</sup> (where h is usually taken to be Planck's constant).
+
为什么使用阶乘因子 n 的原因会在下面讨论。在分母中引入了额外的常数因子是因为离子的离散形式。上面显示的连续形式不是无量纲的。正如前面的章节所说，为了使它成为一个无量纲量，我们必须用''h''<sup>3''N''</sup> (h 通常被认为是普朗克常数)来除以它。
   −
阶乘因子 n 的原因！下面将讨论。在分母中引入了额外的常数因子，因为与离散形式不同，上面显示的连续形式不是无量纲的。正如前面的章节所说，为了使它成为一个无量纲量，我们必须用 h < sup > 3N </sup > (h 通常被认为是普朗克常数)来除以它。
            
;Quantum mechanical discrete system
 
;Quantum mechanical discrete system
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  −
Quantum mechanical discrete system
      
量子力学离散系统
 
量子力学离散系统
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For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the [[trace (linear algebra)|trace]] of the Boltzmann factor:
 
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the [[trace (linear algebra)|trace]] of the Boltzmann factor:
−
  −
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:
      
对于量子力学和离散的正则系综，典型的配分函数被定义为玻尔兹曼因子的轨迹:
 
对于量子力学和离散的正则系综，典型的配分函数被定义为玻尔兹曼因子的轨迹:
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  <math> Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), </math>
 
  <math> Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), </math>
−
  −
[ math > z = operatorname { tr }(mathrm { e } ^ {-beta hat { h }) ，</math >
  −
  −
  −
  −
where:
      
where:
 
where:
   −
在哪里:
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: <math> \operatorname{tr} ( \circ ) </math> is the [[trace (linear algebra)|trace]] of a matrix;
 
: <math> \operatorname{tr} ( \circ ) </math> is the [[trace (linear algebra)|trace]] of a matrix;
   −
  <math> \operatorname{tr} ( \circ ) </math> is the trace of a matrix;
+
  <math> \operatorname{tr} ( \circ ) </math> 是矩阵的轨迹;
−
 
  −
是矩阵的轨迹;
      
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
 
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
   −
  <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
+
  <math> \beta </math> 是热力学beta，定义为  <math> \tfrac{1}{k_\text{B} T} </math>;
−
 
  −
是热力学beta，定义为 < math > tfrac
      
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]].
 
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]].
   −
  <math> \hat{H} </math> is the Hamiltonian operator.
+
  <math> \hat{H} </math> 是哈密尔顿算符。
−
 
  −
是哈密尔顿算符。
  −
 
  −
 
      
The [[dimension]] of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of [[energy eigenstates]] of the system.
 
The [[dimension]] of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of [[energy eigenstates]] of the system.
   −
The dimension of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of energy eigenstates of the system.
+
系统的能量本征态个数 <math> \mathrm{e}^{-\beta \hat{H}} </math> 是系统的能量本征态个数。
   −
系统的能量本征态个数是系统的能量本征态个数。
            
;Quantum mechanical continuous system
 
;Quantum mechanical continuous system
−
  −
Quantum mechanical continuous system
      
量子力学连续系统
 
量子力学连续系统
      −
  −
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as
      
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as
 
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as
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  <math> Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \, \mathrm{d} q \, \mathrm{d} p, </math>
 
  <math> Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \, \mathrm{d} q \, \mathrm{d} p, </math>
−
  −
1}{ h } int langle q，p | mathrm { e } ^ {-beta hat { h } | q，p rangle，mathrm { d } q，mathrm { d } p，</math >
  −
  −
      
where:
 
where:
   −
where:
  −
  −
在哪里:
        第891行： 第788行：  
: <math> h </math> is the [[Planck constant]];
 
: <math> h </math> is the [[Planck constant]];
   −
  <math> h </math> is the Planck constant;
+
  <math> h </math> 是普朗克常数;
−
 
  −
是普朗克常数;
      
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
 
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
   −
  <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
+
  <math> \beta </math> 是热力学beta，定义为 <math> \tfrac{1}{k_\text{B} T} </math>;
−
 
  −
是热力学beta，定义为 < math > tfrac
      
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]];
 
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]];
   −
  <math> \hat{H} </math> is the Hamiltonian operator;
+
  <math> \hat{H} </math> 是哈密尔顿算符；
−
 
  −
哈密尔顿算符是哈密尔顿算符
      
: <math> q </math> is the [[Canonical coordinates|canonical position]];
 
: <math> q </math> is the [[Canonical coordinates|canonical position]];
   −
  <math> q </math> is the canonical position;
+
  <math> q </math>是典型的位置；
−
 
  −
Q </math > 是典型的位置
      
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
 
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
   −
  <math> p </math> is the canonical momentum.
+
  <math> p </math> 是典型的动量。
   −
是典型的动量。
           −
In systems with multiple [[quantum states]] ''s'' sharing the same energy ''E<sub>s</sub>'', it is said that the [[energy levels]] of the system are [[Degenerate energy levels|degenerate]]. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by ''j'') as follows:
+
In systems with multiple [[quantum states]] ''s'' sharing the same energy ''E<sub>s</sub>'', it is said that the [[energy levels]] of the system are [[Degenerate energy levels|degenerate]]. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by ''j'') as follows.
   −
In systems with multiple quantum states s sharing the same energy E<sub>s</sub>, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows:
+
在具有多个量子态''s''共享相同能量的系统中，系统的能级''E<sub>s</sub>''是简并的。在简并能级的情况下，我们可以用能级（ ''j'' ）的贡献来表示配分函数，如下:
   −
在具有多个量子态共享相同能量的系统中，系统的能级是简并的。在简并能级的情况下，我们可以用能级的贡献来表示配分函数，如下:
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  <math> Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j},</math>
 
  <math> Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j},</math>
   −
[数学，数学]
        第939行： 第825行：  
where ''g<sub>j</sub>'' is the degeneracy factor, or number of quantum states ''s'' that have the same energy level defined by ''E<sub>j</sub>'' = ''E<sub>s</sub>''.
 
where ''g<sub>j</sub>'' is the degeneracy factor, or number of quantum states ''s'' that have the same energy level defined by ''E<sub>j</sub>'' = ''E<sub>s</sub>''.
   −
where g<sub>j</sub> is the degeneracy factor, or number of quantum states s that have the same energy level defined by E<sub>j</sub> = E<sub>s</sub>.
+
其中''g<sub>j</sub>''是简并因子，或者是由 ''E<sub>j</sub>'' = ''E<sub>s</sub>'' 定义的具有相同能级的量子态 ''s'' 的数目。
   −
其中 g < sub > j </sub > 是简并因子，或者 e < sub > j </sub > = e < sub > s </sub > 定义的具有相同能级的量子态的数目。
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The above treatment applies to ''quantum'' [[statistical mechanics]], where a physical system inside a [[Particle in a box|finite-sized box]] will typically have a discrete set of energy eigenstates, which we can use as the states ''s'' above. In quantum mechanics, the partition function can be more formally written as a trace over the [[mathematical formulation of quantum mechanics|state space]] (which is independent of the choice of [[basis (linear algebra)|basis]]):
 
The above treatment applies to ''quantum'' [[statistical mechanics]], where a physical system inside a [[Particle in a box|finite-sized box]] will typically have a discrete set of energy eigenstates, which we can use as the states ''s'' above. In quantum mechanics, the partition function can be more formally written as a trace over the [[mathematical formulation of quantum mechanics|state space]] (which is independent of the choice of [[basis (linear algebra)|basis]]):
   −
The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above. In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis):
+
上述的处理方法适用于量子统计力学，在有限大小的盒子里的物理系统通常会有一组离散的能量本征态，我们可以用它作为上面的状态。在量子力学中，配分函数可以更正式地写成状态空间上的迹(这与基的选择无关) :
   −
上述的处理方法适用于量子统计力学，在有限大小的盒子里的物理系统通常会有一组离散的能量本征态，我们可以用它作为上面的状态。在量子力学中，配分函数可以更正式地写成状态空间上的跟踪(这与基的选择无关) :
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  <math>Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ),</math>
 
  <math>Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ),</math>
−
  −
[ math > z = operatorname { tr }(mathrm { e } ^ {-beta hat { h }) ，</math >
  −
  −
      
where ''Ĥ'' is the [[Hamiltonian (quantum mechanics)|quantum Hamiltonian operator]]. The exponential of an operator can be defined using the [[Characterizations of the exponential function|exponential power series]].
 
where ''Ĥ'' is the [[Hamiltonian (quantum mechanics)|quantum Hamiltonian operator]]. The exponential of an operator can be defined using the [[Characterizations of the exponential function|exponential power series]].
   −
where Ĥ is the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series.
+
Ĥ是量子哈密顿算符。算子的指数可以用指数幂级数来定义。
   −
量子哈密顿算符在哪里。算子的指数可以用指数幂级数来定义。
           −
The classical form of ''Z'' is recovered when the trace is expressed in terms of [[coherent state]]s<ref>{{cite book |first1=John R. |last1=Klauder |first2=Bo-Sture |last2=Skagerstam
+
The classical form of ''Z'' is recovered when the trace is expressed in terms of [[coherent state]]s<ref name=":0">{{cite book |first1=John R. |last1=Klauder |first2=Bo-Sture |last2=Skagerstam
    
The classical form of Z is recovered when the trace is expressed in terms of coherent states
 
The classical form of Z is recovered when the trace is expressed in terms of coherent states
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are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:
 
are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:
   −
被认为是微不足道的。在形式上，使用胸罩符号，在每个自由度的跟踪下插入一个标识:
+
经典形式的Z 可以用迹的相干态<ref name=":0" />来表示通常被视为微不足道。正式地，使用 bra-ket 符号，在每个自由度的迹线下插入恒等式：
    
and when quantum-mechanical [[uncertainty principle|uncertainties]] in the position and momentum of a particle  
 
and when quantum-mechanical [[uncertainty principle|uncertainties]] in the position and momentum of a particle  
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</math>
 
</math>
−
  −
数学
      
  \boldsymbol{1} = \int |x, p\rangle \langle x,p| \frac{dx \,dp}{h},
 
  \boldsymbol{1} = \int |x, p\rangle \langle x,p| \frac{dx \,dp}{h},
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where x, p is a normalised Gaussian wavepacket centered at
 
where x, p is a normalised Gaussian wavepacket centered at
   −
其中 x，p 是以? 为中心的正态高斯波包
+
其中 x，p 是一个正态高斯波包被
    
</math>
 
</math>
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position x and momentum p. Thus
 
position x and momentum p. Thus
   −
位置 x 和动量 p
+
位置 x 和动量 p包围
    
where {{!}}''x'', ''p''{{rangle}} is a [[Normalizing constant|normalised]] [[Wave packet#Gaussian wave packets in quantum mechanics|Gaussian wavepacket]] centered at
 
where {{!}}''x'', ''p''{{rangle}} is a [[Normalizing constant|normalised]] [[Wave packet#Gaussian wave packets in quantum mechanics|Gaussian wavepacket]] centered at
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</math>
 
</math>
−
  −
数学
      
   = \int \langle x,p| \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \frac{dx \,dp}{h}.
 
   = \int \langle x,p| \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \frac{dx \,dp}{h}.
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A coherent state is an approximate eigenstate of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, hence also of the Hamiltonian Ĥ, with errors of the size of the uncertainties. If Δx and Δp can be regarded as zero, the action of Ĥ reduces to multiplication by the classical Hamiltonian, and Z reduces to the classical configuration integral.
 
A coherent state is an approximate eigenstate of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, hence also of the Hamiltonian Ĥ, with errors of the size of the uncertainties. If Δx and Δp can be regarded as zero, the action of Ĥ reduces to multiplication by the classical Hamiltonian, and Z reduces to the classical configuration integral.
   −
相干态是两个算符的近似本征态，因此也是哈密顿量的近似本征态，误差大小与不确定性有关。如果 δx 和 δp 可以看作为零，则经典哈密顿量的作用减为乘法，z 的作用减为经典构型积分。
+
相干态是两个算符<math> \hat{x} </math> 和 <math> \hat{p} </math>的近似本征态，因此也是哈密顿量 ''Ĥ'',的近似本征态，误差大小与不确定性有关。如果Δ''x'' 和 Δ''p''可以看作为零，则经典哈密顿量 ''Ĥ'' 的作用减为乘法， ''Z''的作用减为经典构型积分。
   −
</math>
  −
  −
A coherent state is an approximate eigenstate of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, hence also of the Hamiltonian ''Ĥ'', with errors of the size of the uncertainties. If Δ''x'' and Δ''p'' can be regarded as zero, the action of ''Ĥ'' reduces to multiplication by the classical Hamiltonian, and ''Z'' reduces to the classical configuration integral.
           −
=== Connection to probability theory ===
+
=== Connection to probability theory 与概率论的结合 ===
    
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
 
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
   −
为了简单起见，我们将在本节中使用配分函数的离散形式。我们的结果同样适用于连续型。
+
为了简单起见，我们将在本节中使用配分函数的离散形式。结果同样适用于连续形式。
    +
Consider a system ''S'' embedded into a [[heat bath]] ''B''. Let the total [[energy]] of both systems be ''E''. Let ''p<sub>i</sub>'' denote the [[probability]] that the system ''S'' is in a particular [[Microstate (statistical mechanics)|microstate]], ''i'', with energy ''E<sub>i</sub>''. According to the [[Statistical mechanics#Fundamental postulate|fundamental postulate of statistical mechanics]] (which states that all attainable microstates of a system are equally probable), the probability ''p<sub>i</sub>'' will be proportional to the number of microstates of the total [[Closed system (thermodynamics)|closed system]] (''S'', ''B'') in which ''S'' is in microstate ''i'' with energy ''E<sub>i</sub>''. Equivalently, ''p<sub>i</sub>'' will be proportional to the number of microstates of the heat bath ''B'' with energy ''E'' − ''E<sub>i</sub>'':
   −
 
+
考虑一个系统 S 嵌入到一个热浴缸 B 中。设两个系统的总能量均为E。p<sub>i</sub> 表示系统 S 处于特定微观状态的概率 I，有能量E<sub>i。</sub>根据统计力学的基本假设(即系统中所有可达到的微观状态概率相等) ，p<sub>i</sub> 的概率将与总封闭系统 (S, B) 中 S 处于能量为  E<sub>i</sub> 的微观状态的数量成正比。等价地，p<sub>i</sub> 将与热浴 B 中能量 E − E<sub>i</sub> 的微观状态数成正比:
−
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
  −
 
  −
Consider a system S embedded into a heat bath B. Let the total energy of both systems be E. Let p<sub>i</sub> denote the probability that the system S is in a particular microstate, i, with energy E<sub>i</sub>. According to the fundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probability p<sub>i</sub> will be proportional to the number of microstates of the total closed system (S, B) in which S is in microstate i with energy E<sub>i</sub>. Equivalently, p<sub>i</sub> will be proportional to the number of microstates of the heat bath B with energy E − E<sub>i</sub>:
  −
 
  −
考虑一个系统 s 嵌入到一个热浴缸 b 中。设两个系统的总能量均为 e，p < sub > i </sub > 表示系统 s 处于特定微观状态的概率，i，能量 e < sub > i </sub > 。根据统计力学的基本假设(即系统中所有可达到的微观状态概率相等) ，p < sub > i </sub > 的概率将与总封闭系统(s，b)中 s 处于能量为 e < sub > i </sub > 的微观状态的数量成正比。等价地，p < sub > i </sub > 将与热浴 b 中能量 e-e < sub > i </sub > 的微观状态数成正比:
  −
 
  −
 
  −
 
  −
Consider a system ''S'' embedded into a [[heat bath]] ''B''. Let the total [[energy]] of both systems be ''E''. Let ''p<sub>i</sub>'' denote the [[probability]] that the system ''S'' is in a particular [[Microstate (statistical mechanics)|microstate]], ''i'', with energy ''E<sub>i</sub>''. According to the [[Statistical mechanics#Fundamental postulate|fundamental postulate of statistical mechanics]] (which states that all attainable microstates of a system are equally probable), the probability ''p<sub>i</sub>'' will be proportional to the number of microstates of the total [[Closed system (thermodynamics)|closed system]] (''S'', ''B'') in which ''S'' is in microstate ''i'' with energy ''E<sub>i</sub>''. Equivalently, ''p<sub>i</sub>'' will be proportional to the number of microstates of the heat bath ''B'' with energy ''E'' − ''E<sub>i</sub>'':
      
<math>
 
<math>
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</math>
 
</math>
−
  −
数学
      
  p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.
 
  p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.
−
  −
</math>
      
Assuming that the heat bath's internal energy is much larger than the energy of S (E ≫ E<sub>i</sub>), we can Taylor-expand <math>\Omega_B</math> to first order in E<sub>i</sub> and use the thermodynamic relation <math>\partial S_B/\partial E = 1/T</math>, where here <math>S_B</math>, <math>T</math> are the entropy and temperature of the bath respectively:
 
Assuming that the heat bath's internal energy is much larger than the energy of S (E ≫ E<sub>i</sub>), we can Taylor-expand <math>\Omega_B</math> to first order in E<sub>i</sub> and use the thermodynamic relation <math>\partial S_B/\partial E = 1/T</math>, where here <math>S_B</math>, <math>T</math> are the entropy and temperature of the bath respectively:
   −
假设热水池的内能远大于热水池的内能(e e < sub > i </sub >) ，我们可以在 e < sub > i </sub > 中泰勒展开欧米加 b </math > 到一级，并利用热力学关系式 < math > 部分 s _ b/部分 e = 1/T </math > ，这里 s _ b </math > ，< math > t </math > 分别是热水池的熵和温度:
+
假设热水池的内能远大于热水池的内能''S'' (''E'' ≫ ''E<sub>i</sub>'') ，我们可以对E<sub>i</sub> 进行一阶泰勒展开 <math>\Omega_B</math> ，并利用热力学关系式  <math>\partial S_B/\partial E = 1/T</math>，这里<math>S_B</math>, <math>T</math> 分别是热水池的熵和温度:
−
 
  −
 
  −
 
  −
Assuming that the heat bath's internal energy is much larger than the energy of ''S'' (''E'' ≫ ''E<sub>i</sub>''), we can [[Taylor expansion|Taylor-expand]] <math>\Omega_B</math> to first order in ''E<sub>i</sub>'' and use the thermodynamic relation <math>\partial S_B/\partial E = 1/T</math>, where here <math>S_B</math>, <math>T</math> are the entropy and temperature of the bath respectively:
      
<math>
 
<math>
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</math>
 
</math>
−
  −
数学
      
\end{align}
 
\end{align}
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Thus
 
Thus
   −
因此
+
因此得到
−
 
  −
 
      
<math>
 
<math>
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</math>
 
</math>
−
  −
数学
      
  p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}.
 
  p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}.
   −
</math>
+
Since the total probability to find the system in ''some'' microstate (the sum of all ''p<sub>i</sub>'') must be equal to&nbsp;1, we know that the constant of proportionality must be the [[Normalizing constant|normalization constant]],  and so, we can define the partition function to be this constant:
−
 
  −
Since the total probability to find the system in some microstate (the sum of all p<sub>i</sub>) must be equal to&nbsp;1, we know that the constant of proportionality must be the normalization constant,  and so, we can define the partition function to be this constant:
  −
 
  −
由于发现系统处于某种微观状态的总概率(所有 p < sub > i </sub > 的和)必须等于1，我们知道比例常数必须是归一化常数，因此，我们可以将配分函数定义为这个常数:
  −
 
  −
 
     −
Since the total probability to find the system in ''some'' microstate (the sum of all ''p<sub>i</sub>'') must be equal to&nbsp;1, we know that the constant of proportionality must be the [[Normalizing constant|normalization constant]],  and so, we can define the partition function to be this constant:
+
由于发现系统处于某种微观状态的总概率(所有 p<sub>i</sub>的和)必须等于1，我们知道比例常数必须是归一化常数，因此，我们可以将配分函数定义为这个常数:
    
<math>
 
<math>
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</math>
 
</math>
   −
数学
+
Z =  \sum_i e^{-\beta E_i} = \frac{\Omega_{(S,B)}(E)}{\Omega_B(E)}.<br />
   −
Z = \sum_i e^{-\beta E_i} = \frac{\Omega_{(S,B)}(E)}{\Omega_B(E)}.
+
=== Calculating the thermodynamic total energy 计算热力学总能 ===
   −
</math>
     −
 
+
In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the [[expected value]], or [[ensemble average]] for the energy, which is the sum of the microstate energies weighted by their probabilities:
−
 
  −
=== Calculating the thermodynamic total energy ===
  −
 
  −
In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value, or ensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities:
      
为了证明配分函数的有用性，让我们计算总能量的热力学值。这仅仅是能量的期望值，或者说总体均值，它是微状态能量的总和，加上它们的概率:
 
为了证明配分函数的有用性，让我们计算总能量的热力学值。这仅仅是能量的期望值，或者说总体均值，它是微状态能量的总和，加上它们的概率:
−
  −
  −
  −
In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the [[expected value]], or [[ensemble average]] for the energy, which is the sum of the microstate energies weighted by their probabilities:
      
  <math>\langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s
 
  <math>\langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s
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</math>
 
</math>
−
  −
数学
      
Z(\beta, E_1, E_2, \cdots) = - \frac{\partial \ln Z}{\partial \beta}
 
Z(\beta, E_1, E_2, \cdots) = - \frac{\partial \ln Z}{\partial \beta}
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或者，等价地说,
 
或者，等价地说,
−
  −
      
or, equivalently,
 
or, equivalently,
    
  <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math>
 
  <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math>
−
  −
[数学]长角 e rangle = k _ b t ^ 2 frac { partial ln z }{ partial t }
  −
  −
      
: <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math>
 
: <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math>
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Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
 
Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
   −
顺便说一句，我们应该注意到，如果微态能量依赖于参数 λ 的方式
+
如果微态能量依赖于参数 λ 的方式
−
 
  −
 
  −
 
  −
Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
  −
 
   
  <math>E_s = E_s^{(0)} + \lambda A_s \qquad \mbox{for all}\; s </math>
 
  <math>E_s = E_s^{(0)} + \lambda A_s \qquad \mbox{for all}\; s </math>
   第1,281行： 第1,103行：  
then the expected value of A is
 
then the expected value of A is
   −
那么 a 的期望值就是
+
那么 A 的期望值就是
−
 
  −
 
  −
 
  −
then the expected value of ''A'' is
  −
 
   
  <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta}
 
  <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta}
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\frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math>
 
\frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math>
−
  −
{ partial }{ partial lambda } ln z (beta，lambda) . </math >
  −
   
: <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta}
 
: <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta}
    
\frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math>
 
\frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math>
−
  −
This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set λ to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory.
  −
  −
这为我们提供了一种计算许多微观量的期望值的方法。我们将这个量人为地加到微态能量上(或者用量子力学的语言，加到哈密顿量上) ，计算出新的配分函数和期望值，然后在最终的表达式中将 λ 设置为零。这类似于量子场论路径积分表述中使用的源场方法。
  −
         
This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set ''λ'' to zero in the final expression. This is analogous to the [[source field]] method used in the [[path integral formulation]] of [[quantum field theory]].{{citation needed|date=December 2015}}
 
This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set ''λ'' to zero in the final expression. This is analogous to the [[source field]] method used in the [[path integral formulation]] of [[quantum field theory]].{{citation needed|date=December 2015}}
   −
 
+
这为我们提供了一种计算许多微观量的期望值的方法。我们将这个量人为地加到微态能量上(或者用量子力学的语言，加到哈密顿量上) ，计算出新的配分函数和期望值，然后在最终的表达式中将 λ 设置为零。这类似于量子场论路径积分表述中使用的源场方法。
−
 
+
=== Relation to thermodynamic variables 与热力学变量的关联 ===
−
=== Relation to thermodynamic variables ===
      
In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.
 
In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.
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在这一节中，我们将陈述配分函数和系统的各种热力学参数之间的关系。这些结果可用前面的方法和各种热力学关系式推导出来。
 
在这一节中，我们将陈述配分函数和系统的各种热力学参数之间的关系。这些结果可用前面的方法和各种热力学关系式推导出来。
   −
  −
  −
In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.
      
As we have already seen, the thermodynamic energy is
 
As we have already seen, the thermodynamic energy is
    
正如我们已经看到的，热力学能
 
正如我们已经看到的，热力学能
−
  −
  −
  −
As we have already seen, the thermodynamic energy is
  −
   
  <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
 
  <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
−
  −
[数学][数学]
  −
  −
      
: <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
 
: <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
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Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ζ<sub>1</sub>, ζ<sub>2</sub>, ..., ζ<sub>N</sub>, then the partition function of the entire system is the product of the individual partition functions:
 
Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ζ<sub>1</sub>, ζ<sub>2</sub>, ..., ζ<sub>N</sub>, then the partition function of the entire system is the product of the individual partition functions:
   −
假设一个系统被细分为 n 个相互作用能可忽略的子系统，也就是说，我们可以假定这些粒子基本上是不相互作用的。如果子系统的配分函数是 ζ < sub > 1 </sub > ，ζ < sub > 2 </sub > ，... ，ζ < sub > n </sub > ，那么整个系统的配分函数就是单个配分函数的乘积:
+
假设一个系统被细分为 n 个相互作用能可忽略的子系统，也就是说，我们可以假定这些粒子基本上是不相互作用的。如果子系统的配分函数是 ζ < sub > 1 ，ζ < sub > 2 ，... ，ζ < sub > n ，那么整个系统的配分函数就是单个配分函数的乘积:
 +
 
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If the sub-systems have the same physical properties, then their partition functions are equal, ζ<sub>1</sub> = ζ<sub>2</sub> = ... = ζ, in which case
 
If the sub-systems have the same physical properties, then their partition functions are equal, ζ<sub>1</sub> = ζ<sub>2</sub> = ... = ζ, in which case
   −
如果子系统具有相同的物理性质，那么它们的配分函数是相等的，ζ < sub > 1 </sub > = ζ < sub > 2 </sub > = ... = ζ，在这种情况下
+
如果子系统具有相同的物理性质，那么它们的配分函数是相等的，ζ < sub > 1 = ζ < sub > 2 = ... = ζ，在这种情况下
 +
 
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It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature T and the microstate energies E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
 
It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature T and the microstate energies E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
   −
为什么我们上面已经定义过的配分函数是一个重要的数量，这可能并不明显。首先，考虑一下里面有什么。配分函数是温度 t 和微态能量 e < sub > 1 </sub > ，e < sub > 2 </sub > ，e < sub > 3 </sub > 等的函数。微观能量是由其他热力学变量决定的，例如粒子的数量和体积，以及组成粒子的质量等微观量。这种对微观变量的依赖是统计力学的中心点。通过建立一个系统的微观组分模型，我们可以计算出系统的微观能量，从而计算出配分函数，这样我们就可以计算出系统的所有其他热力学性质。
+
为什么我们上面已经定义过的配分函数是一个重要的数量，这可能并不明显。首先，考虑一下里面有什么。配分函数是温度 t 和微态能量 e < sub > 1 ，e < sub > 2 ，e < sub > 3 等的函数。微观能量是由其他热力学变量决定的，例如粒子的数量和体积，以及组成粒子的质量等微观量。这种对微观变量的依赖是统计力学的中心点。通过建立一个系统的微观组分模型，我们可以计算出系统的微观能量，从而计算出配分函数，这样我们就可以计算出系统的所有其他热力学性质。
 +
 
     
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