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小第1行：
第1行： − 此词条暂由彩云小译翻译，翻译字数共1721，未经人工整理和审校，带来阅读不便，请见谅。+ 第247行：
第247行： − + 第311行：
第311行： − 由于这个方程对任何变化都适用+ 第463行：
第463行： − 左(frac { λ _ 2}{ k _ text { b }} e _ i right)。+ 第515行：
第515行： − & =-lambda _ 2 u + k _ text { b } ln (z).+ 第617行：
第617行： − 2. s & = frac { u }{ t } + k text { b } ln z.+ 第653行：
第653行： − 在经典力学中，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，典型的配分函数定义为+ 第701行：
第701行： − P 是典型的动量。+ 第773行：
第773行： − Q _ i </math > 是各个粒子的标准位置;+ 第976行：
第976行： + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 第995行：
第1,685行： − position ''x'' and momentum ''p''. Thus+
Moved page from wikipedia:en:Partition function (statistical mechanics) (history)
此词条暂由彩云小译翻译，翻译字数共3425，未经人工整理和审校，带来阅读不便，请见谅。
{{About|statistical mechanics|other uses|partition function (disambiguation)}}
{{About|statistical mechanics|other uses|partition function (disambiguation)}}
Varying and extremizing <math> \mathcal{L} </math> with respect to <math> \rho_i </math> leads to
Varying and extremizing <math> \mathcal{L} </math> with respect to <math> \rho_i </math> leads to
变化和极值化 < math > > mathcal { l } </math > 关于 < math > rho _ i </math >
变化和极值化 < math > </math > 关于 < math > rho _ i </math > 导致
Since this equation should hold for any variation <math> \delta ( \rho_i ) </math>, it implies that
Since this equation should hold for any variation <math> \delta ( \rho_i ) </math>, it implies that
由于这个方程适用于任何变化 < math > delta (rho _ i) </math > ，它暗示
\rho_i = \frac{1}{Z} \exp \left( \frac{\lambda_2}{k_\text{B}} E_i \right) .
\rho_i = \frac{1}{Z} \exp \left( \frac{\lambda_2}{k_\text{B}} E_i \right) .
左边(frac { λ _ 2}{ k _ text { b }} e _ i right)。
</math>
</math>
&= - \lambda_2 U + k_\text{B} \ln(Z) .
&= - \lambda_2 U + k_\text{B} \ln(Z) .
2 u + k _ text { b } ln (z).
\end{align}
\end{align}
S & = \frac{U}{T} + k_\text{B} \ln Z .
S & = \frac{U}{T} + k_\text{B} \ln Z .
s & = frac { u }{ t } + k _ text { b } ln z.
\end{align}
\end{align}
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
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
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
<math> p </math> is the canonical momentum.
<math> p </math> is the canonical momentum.
是典型的动量。
<math> q_i </math> is the canonical position of the respective particle;
<math> q_i </math> is the canonical position of the respective particle;
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;
|title=Coherent States: Applications in Physics and Mathematical Physics
|title=Coherent States: Applications in Physics and Mathematical Physics
and when quantum-mechanical uncertainties in the position and momentum of a particle
当粒子位置和动量的量子力学不确定性
|publisher=World Scientific |date=1985 |pages=71–73 |isbn=978-9971-966-52-2 }}</ref>
|publisher=World Scientific |date=1985 |pages=71–73 |isbn=978-9971-966-52-2 }}</ref>
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:
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
<math>
《数学》
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:
\boldsymbol{1} = \int |x, p\rangle \langle x,p| \frac{dx \,dp}{h},
1} = int | x，p rangle langle x，p | frac { dx，dp }{ h } ,
:<math>
:<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},
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
</math>
</math>
position x and momentum p. Thus
位置 x 和动量 p
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
<math>
《数学》
position ''x'' and momentum ''p''. Thus
Z = \int \operatorname{tr} \left( \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \langle x, p| \right) \frac{dx \,dp}{h}
Z = int operatorname { tr } left (mathrm { e } ^ {-beta hat { h } | x，p rangle langle x，p | right) frac { dx，dp }{ h }
:<math>
= \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 }.
Z = \int \operatorname{tr} \left( \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \langle x, p| \right) \frac{dx \,dp}{h}
</math>
数学
= \int \langle x,p| \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \frac{dx \,dp}{h}.
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。
</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 ===
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。
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>
《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>
</math>
数学
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:
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:
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>
《数学》
\begin{align}
开始{ align }
:<math>
k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_{(S,B)}(E) \\[5pt]
K ln p _ i & = k ln Omega _ b (e-e _ i)-k ln Omega _ (s，b)}(e)[5 pt ]
\begin{align}
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
大约-frac { partial big (k ln Omega _ b (e) big)}{ partial e } e _ i + k ln Omega _ b (e)-k ln Omega _ {(s，b)}(e))
k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_{(S,B)}(E) \\[5pt]
\\[5pt]
[5 pt ]
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
&\approx -\frac{\partial S_B}{\partial E} E_i + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)} \\[5pt]
大约-frac { partial s _ b }{ partial e } e _ i + k ln frac { Omega _ b (e)}{ Omega _ {(s，b)}(e)}[5 pt ]
\\[5pt]
&\approx -\frac{E_i}{T} + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)}
约-frac { e _ i }{ t } + k ln frac { Omega _ b (e)}{ Omega _ {(s，b)}(e)}
&\approx -\frac{\partial S_B}{\partial E} E_i + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)} \\[5pt]
\end{align}
结束{ align }
&\approx -\frac{E_i}{T} + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)}
</math>
数学
\end{align}
</math>
Thus
因此
<math>
《数学》
Thus
p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}.
P _ i propto e ^ {-e _ i/(kT)} = e ^ {-beta e _ i }.
:<math>
</math>
数学
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 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 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:
<math>
《数学》
Z = \sum_i e^{-\beta E_i} = \frac{\Omega_{(S,B)}(E)}{\Omega_B(E)}.
Z = sum _ i e ^ {-beta e _ i } = frac { Omega _ {(s，b)}(e)}{ Omega _ b (e)}.
:<math>
</math>
数学
Z = \sum_i e^{-\beta E_i} = \frac{\Omega_{(S,B)}(E)}{\Omega_B(E)}.
</math>
=== 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:
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
[数学]长角 e rangle = sum _ s e _ s p _ s = frac {1}{ z } sum _ s e _ s
e^{- \beta E_s} = - \frac{1}{Z} \frac{\partial}{\partial \beta}
E ^ {-beta e _ s } =-frac {1}{ z } frac { partial beta }
: <math>\langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s
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 }
e^{- \beta E_s} = - \frac{1}{Z} \frac{\partial}{\partial \beta}
</math>
数学
Z(\beta, E_1, E_2, \cdots) = - \frac{\partial \ln Z}{\partial \beta}
</math>
or, equivalently,
或者，等价地说,
or, equivalently,
<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>
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
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>
[所有} ; s </math >
: <math>E_s = E_s^{(0)} + \lambda A_s \qquad \mbox{for all}\; s </math>
then the expected value of A is
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}
1. a rangle = sum _ s a _ s p _ s =-frac {1}{ beta }
\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}
\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}}
=== 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.
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
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>
[数学][数学]
: <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
The variance in the energy (or "energy fluctuation") is
能量(或“能量波动”)的方差是
The [[variance]] in the energy (or "energy fluctuation") is
<math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
2rangle equiv langle (e-langle)2 rangle equiv langle
E\rangle)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2}.</math>
2 rangle = frac { partial ^ 2 ln z }{ partial beta ^ 2} . </math >
: <math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
E\rangle)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2}.</math>
The heat capacity is
热容为
The [[heat capacity]] is
<math>C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle.</math>
[ math > c _ v = frac { partial langle e rangle }{ partial t } = frac {1}{ k _ b t ^ 2} langle (Delta e) ^ 2 rangle
: <math>C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle.</math>
In general, consider the extensive variable X and intensive variable Y where X and Y form a pair of conjugate variables. In ensembles where Y is fixed (and X is allowed to fluctuate), then the average value of X will be:
一般来说，考虑扩展变量 x 和密集变量 y，其中 x 和 y 形成一对共轭变量。在 y 固定(x 允许波动)的系综中，x 的平均值是:
In general, consider the [[extensive variable]] X and [[intensive variable]] Y where X and Y form a pair of [[conjugate variables]]. In ensembles where Y is fixed (and X is allowed to fluctuate), then the average value of X will be:
<math>\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math>
[数学][数学]
: <math>\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math>
The sign will depend on the specific definitions of the variables X and Y. An example would be X = volume and Y = pressure. Additionally, the variance in X will be
符号将取决于变量 x 和 y 的具体定义。一个例子是 x = 体积和 y = 压强。另外，x 中的方差是
The sign will depend on the specific definitions of the variables X and Y. An example would be X = volume and Y = pressure. Additionally, the variance in X will be
<math>\langle (\Delta X)^2 \rangle \equiv \langle (X - \langle
2rangle equiv langle (x-langle
X\rangle)^2 \rangle = \frac{\partial \langle X \rangle}{\partial \beta Y} = \frac{\partial^2 \ln Z}{\partial (\beta Y)^2}.</math>
2 rangle = frac { partial angle x rangle }{ partial beta y } = frac { partial ^ 2 ln z }{ partial (beta y) ^ 2} . </math >
: <math>\langle (\Delta X)^2 \rangle \equiv \langle (X - \langle
X\rangle)^2 \rangle = \frac{\partial \langle X \rangle}{\partial \beta Y} = \frac{\partial^2 \ln Z}{\partial (\beta Y)^2}.</math>
In the special case of entropy, entropy is given by
在熵的特殊情况下，熵是由
In the special case of [[entropy]], entropy is given by
<math>S \equiv -k_B\sum_s P_s\ln P_s= k_B (\ln Z + \beta \langle E\rangle)=\frac{\partial}{\partial T}(k_B T \ln Z) =-\frac{\partial A}{\partial T}</math>
= frac { partial }{ t }(k _ b t ln z) =-frac { partial a }{ partial t } </math >
: <math>S \equiv -k_B\sum_s P_s\ln P_s= k_B (\ln Z + \beta \langle E\rangle)=\frac{\partial}{\partial T}(k_B T \ln Z) =-\frac{\partial A}{\partial T}</math>
where A is the Helmholtz free energy defined as A = U − TS, where U = E is the total energy and S is the entropy, so that
其中 a 是定义为 a = u-TS 的亥姆霍兹自由能，其中 u = e 是总能量，s 是熵，所以
where ''A'' is the [[Helmholtz free energy]] defined as ''A'' = ''U'' − ''TS'', where ''U'' = {{langle}}''E''{{rangle}} is the total energy and ''S'' is the [[entropy]], so that
<math>A = \langle E\rangle -TS= - k_B T \ln Z.</math>
A = langle e rangle-TS =-k _ b t ln z
: <math>A = \langle E\rangle -TS= - k_B T \ln Z.</math>
=== Partition functions of subsystems ===
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:
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:
<math>Z =\prod_{j=1}^{N} \zeta_j.</math>
[数学] z = prod { j = 1} ^ { n } zeta _ j
: <math>Z =\prod_{j=1}^{N} \zeta_j.</math>
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
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
<math>Z = \zeta^N.</math>
Z = zeta ^ n
: <math>Z = \zeta^N.</math>
However, there is a well-known exception to this rule. If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a N! (N factorial):
然而，这条规则有一个众所周知的例外。如果这些子系统实际上是全同粒子的，从量子力学的意义上说，即使在原则上也无法区分它们，那么总配分函数必须除以 n！(n 阶乘) :
However, there is a well-known exception to this rule. If the sub-systems are actually [[identical particles]], in the [[quantum mechanics|quantum mechanical]] sense that they are impossible to distinguish even in principle, the total partition function must be divided by a ''N''! (''N'' [[factorial]]):
<math>Z = \frac{\zeta^N}{N!}.</math>
如果你想要的话，你可以在这里找到！} . </math >
: <math>Z = \frac{\zeta^N}{N!}.</math>
This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox.
这是为了确保我们不会“过多计算”微型状态的数量。虽然这看起来似乎是一个奇怪的要求，但实际上有必要为这样的系统保留一个热力学极限。这就是所谓的吉布斯悖论。
This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the [[Gibbs paradox]].
=== Meaning and significance ===
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。
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.
The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability P<sub>s</sub> that the system occupies microstate s is
配分函数可能与热力学性质有关，因为它具有非常重要的统计意义。系统处于微观状态的概率为
The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability ''P<sub>s</sub>'' that the system occupies microstate ''s'' is
<math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math>
< math > p _ s = frac {1}{ z } mathrm { e } ^ {-beta e _ s }.数学
: <math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math>
Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does not depend on s), ensuring that the probabilities sum up to one:
因此，如上所示，配分函数常数扮演了一个正常化常数的角色(注意它不依赖于 s) ，确保概率总和为1:
Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does ''not'' depend on ''s''), ensuring that the probabilities sum up to one:
<math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
1}{ z } sum s mathrm { e } ^ {-beta e _ s } = frac {1}{ z } z
= 1. </math>
= 1.数学
: <math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
= 1. </math>
This is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter Z stands for the German word Zustandssumme, "sum over states". The usefulness of the partition function stems from the fact that it can be used to relate macroscopic thermodynamic quantities to the microscopic details of a system through the derivatives of its partition function. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies.
这就是把 z 称为“配分函数”的原因: 它编码概率如何在不同的微观状态之间分配，基于它们各自的能量。字母 z 代表德语单词 Zustandssumme，“ sum over states”。配分函数的有用性源于这样一个事实，即它可以通过一个系统的配分函数导数，将宏观的热力学量与系统的微观细节联系起来。找到配分函数也等同于执行从能域到 β 域的态密度函数的拉普拉斯变换，而拉普拉斯逆变换配分函数重新要求能量的态密度函数。
This is the reason for calling ''Z'' the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter ''Z'' stands for the [[German language|German]] word ''Zustandssumme'', "sum over states". The usefulness of the partition function stems from the fact that it can be used to relate macroscopic [[thermodynamic state|thermodynamic quantities]] to the microscopic details of a system through the derivatives of its partition function. Finding the partition function is also equivalent to performing a [[Laplace transform]] of the density of states function from the energy domain to the β domain, and the [[inverse Laplace transform]] of the partition function reclaims the state density function of energies.
==Grand canonical partition function==
{{Main|Grand canonical ensemble}}
We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. The reservoir has a constant temperature T, and a chemical potential μ.
我们可以定义一个巨典型的配分函数，它描述了一个恒定体积系统的统计数据，这个系统可以同时与一个巨正则系综库交换热量和粒子。储层具有恒定的温度 t 和化学势 μ。
We can define a '''grand canonical partition function''' for a [[grand canonical ensemble]], which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. The reservoir has a constant temperature ''T'', and a [[chemical potential]] ''μ''.
The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over microstates
巨典型配分函数，表示为 < math > mathcal { z } </math > ，是微状态上的和
<math> \mathcal{Z}(\mu, V, T) = \sum_{i} \exp\left(\frac{N_i\mu - E_i}{k_B T} \right). </math>
(mu，v，t) = sum _ { i } exp left (frac { n _ i mu-e _ i }{ k _ b _ t } right) .数学
The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over [[microstate (statistical mechanics)|microstates]]
Here, each microstate is labelled by <math>i</math>, and has total particle number <math>N_i</math> and total energy <math>E_i</math>. This partition function is closely related to the grand potential, <math>\Phi_{\rm G}</math>, by the relation
在这里，每个微观状态都用 < math > i </math > 标记，并且有总粒子数 < math > n _ i </math > 和总能量 < math > e _ i </math > 。这种配分函数与巨大的潜力密切相关，通过这种关系
:<math> \mathcal{Z}(\mu, V, T) = \sum_{i} \exp\left(\frac{N_i\mu - E_i}{k_B T} \right). </math>
<math> -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. </math>
数学上的-k _ b _ t = Phi _ { rm g } = langle e rangle-TS-mu rangle n rangle。数学
Here, each microstate is labelled by <math>i</math>, and has total particle number <math>N_i</math> and total energy <math>E_i</math>. This partition function is closely related to the [[grand potential]], <math>\Phi_{\rm G}</math>, by the relation
This can be contrasted to the canonical partition function above, which is related instead to the Helmholtz free energy.
这可以与上面提到的权威配分函数相对照，后者与亥姆霍兹自由能相关。
:<math> -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. </math>
This can be contrasted to the canonical partition function above, which is related instead to the [[Helmholtz free energy]].
It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, since here we consider not only variations in energy but also in particle number. Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state <math>i</math>:
值得注意的是，巨正则系综中的微观态数量可能远远大于正则系综中的微观态数量，因为这里我们不仅考虑了能量的变化，还考虑了粒子数量的变化。同样，巨典型配分函数的效用在于它与系统处于状态的概率有关:
<math> p_i = \frac{1}{\mathcal Z} \exp\left(\frac{N_i\mu - E_i}{k_B T}\right).</math>
{ mathcal z } exp left (frac { n _ i mu-e _ i }{ k _ b t } right) . </math >
It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, since here we consider not only variations in energy but also in particle number. Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state <math>i</math>:
:<math> p_i = \frac{1}{\mathcal Z} \exp\left(\frac{N_i\mu - E_i}{k_B T}\right).</math>
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas (Fermi–Dirac statistics for fermions, Bose–Einstein statistics for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.
巨正则系综理论的一个重要应用是精确地导出没有相互作用的多体量子气体的统计数据(费米-狄拉克统计费米子，玻色子玻色子玻色-爱因斯坦统计) ，然而，它的应用范围要比这广泛得多。巨正则系综也可以用来描述经典系统，甚至相互作用的量子气体。
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas ([[Fermi–Dirac statistics]] for fermions, [[Bose–Einstein statistics]] for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.
The grand partition function is sometimes written (equivalently) in terms of alternate variables as
大配分函数有时候是用交替变量来表示的
<math> \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), </math>
< math > mathcal { z }(z，v，t) = sum _ { n _ i } z (n _ i，v，t) ，</math >
The grand partition function is sometimes written (equivalently) in terms of alternate variables as<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref>
where <math>z \equiv \exp(\mu/kT)</math> is known as the absolute activity (or fugacity) and <math>Z(N_i, V, T)</math> is the canonical partition function.
其中，z equiv exp (mu/kT) </math > 被称为绝对活度(或逸度) ，而 < math > z (n _ i，v，t) </math > 则是典范配分函数。
:<math> \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), </math>
where <math>z \equiv \exp(\mu/kT)</math> is known as the absolute [[activity (chemistry)|activity]] (or [[fugacity]]) and <math>Z(N_i, V, T)</math> is the canonical partition function.
==See also==
* [[Partition function (mathematics)]]
* [[Partition function (quantum field theory)]]
* [[Virial theorem]]
* [[Widom insertion method]]
==References==
<references />
* Huang, Kerson, "Statistical Mechanics", John Wiley & Sons, New York, 1967.
* A. Isihara, "Statistical Physics", Academic Press, New York, 1971.
* Kelly, James J, [http://www.physics.umd.edu/courses/Phys603/kelly/Notes/IdealQuantumGases.pdf (Lecture notes)]
* L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996.
* Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].
{{Statistical mechanics topics}}
Category:Concepts in physics
Category:Concepts in physics
分类: 物理概念
分类: 物理概念
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