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第1行： − 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)}} − − {{Use American English|date = February 2019}} − − {{Short description|Function in thermodynamics and statistical physics}} − − {{statistical mechanics}} 第614行：
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第792行： − + − + 第1,047行：
第1,039行： − + 第1,073行：
第1,065行： − − </math> − − Z(\beta, E_1, E_2, \cdots) = - \frac{\partial \ln Z}{\partial \beta} 第1,083行：
第1,071行： − − or, equivalently, 第1,094行：
第1,080行： − − [所有} ; s </math > − − 第1,136行：
第1,118行： − − − − The [[variance]] in the energy (or "energy fluctuation") is 第1,148行：
第1,126行： − − 2 rangle = frac { partial ^ 2 ln z }{ partial beta ^ 2} . </math > − 第1,158行：
第1,133行： − + − − − 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 − − − − 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: 第1,179行：
第1,141行： − <math>\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math> − − [数学][数学] + + 第1,189行：
第1,150行： − + − − − − 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 第1,202行：
第1,159行： − − 2 rangle = frac { partial angle x rangle }{ partial beta y } = frac { partial ^ 2 ln z }{ partial (beta y) ^ 2} . </math > − 第1,211行：
第1,165行： − 在熵的特殊情况下，熵是由+ − + − − − In the special case of [[entropy]], entropy is given by − − − − = frac { 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 − − − <math>A = \langle E\rangle -TS= - k_B T \ln Z.</math>+ − − − + 第1,243行：
第1,180行： − + − + − − − − − 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 − + 第1,264行：
第1,192行： − + + + − + − 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 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]]): − − − − 如果你想要的话，你可以在这里找到！} . </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。 − − + − + − − 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。+ − − − − − 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. − 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 }.数学 − − 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: − − + 第1,347行：
第1,236行： − − = 1.数学 − − 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。 − + − + − − 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。 − − − 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 − + − (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]] − 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+ 第1,396行：
第1,268行： − 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+ − − − 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]]. − 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:+ − − − { 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. − 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，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。 − − 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.
无编辑摘要
此词条暂由彩云小译翻译，翻译字数共3425，AvecSally审校
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: <math> q </math> is the [[Canonical coordinates|canonical position]];
: <math> q </math> is the [[Canonical coordinates|canonical position]];
<math> q </math> 是典型的位置
<math> q </math> 是正则位置
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
<math> p </math> 是典型的动量。
<math> p </math> 是正则动量。
: <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;
<math> q_i </math> 是各个粒子的标准位置;
<math> q_i </math> 是各个粒子的正则位置;
: <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> q </math> is the [[Canonical coordinates|canonical position]];
: <math> q </math> is the [[Canonical coordinates|canonical position]];
<math> q </math>是典型的位置；
<math> q </math>是正则位置；
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
<math> p </math> 是典型的动量。
<math> p </math> 是正则动量。
</math>
</math>
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 ===
=== Calculating the thermodynamic total energy 计算热力学总能 ===
e^{- \beta E_s} = - \frac{1}{Z} \frac{\partial}{\partial \beta}
e^{- \beta E_s} = - \frac{1}{Z} \frac{\partial}{\partial \beta}
</math>
</math>
或者，等价地说,
或者，等价地说,
<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>
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
<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>
: <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>
能量(或“能量波动”)的方差是
能量(或“能量波动”)的方差是
<math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
<math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
E\rangle)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2}.</math>
E\rangle)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2}.</math>
: <math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
: <math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
热容为
热容为
<math>C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle.</math><br />
<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>
: <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:
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:
<math>\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math><br />
: <math>\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math>
: <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
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
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
<math>\langle (\Delta X)^2 \rangle \equiv \langle (X - \langle
<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>
X\rangle)^2 \rangle = \frac{\partial \langle 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
: <math>\langle (\Delta X)^2 \rangle \equiv \langle (X - \langle
In the special case of entropy, entropy is given by
In the special case of entropy, entropy is given by
在[[熵]]的特殊情况下，熵是由
<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><br />
<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>
: <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>
: <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'' = {{langle}}''E''{{rangle}} is the total energy and ''S'' is the [[entropy]], so that
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
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
A = langle e rangle-TS =-k _ b t ln z
<math>A = \langle E\rangle -TS= - k_B T \ln Z.</math><br />
: <math>A = \langle E\rangle -TS= - k_B T \ln Z.</math>
: <math>A = \langle E\rangle -TS= - k_B T \ln Z.</math>
=== Partition functions of subsystems ===
=== Partition functions of subsystems 子系统配分函数 ===
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 ，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:
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:
<math>Z =\prod_{j=1}^{N} \zeta_j.</math><br />
: <math>Z =\prod_{j=1}^{N} \zeta_j.</math>
: <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
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 = ζ < sub > 2 = ... = 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</sub> = ζ<sub>2</sub> = ... = ζ，在这种情况下
<math>Z = \zeta^N.</math><br />
: <math>Z = \zeta^N.</math><br />
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]]):
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) :
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) :
<math>Z = \frac{\zeta^N}{N!}.</math><br />
<math>Z = \frac{\zeta^N}{N!}.</math>
: <math>Z = \frac{\zeta^N}{N!}.</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]].
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]]。
=== Meaning and significance 影响 ===
=== 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.
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.
正如我们在上面定义的那样，为什么配分函数是一个重要的量可能看起来并不明显。首先，考虑其中的内容。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。微态能量由其他热力学变量决定，例如粒子数和体积，以及质量等微观量的组成粒子。这种对微观变量的依赖是统计力学的中心点。有了系统微观成分的模型，我们可以计算微观状态能量，从而计算配分函数，然后我们就可以计算系统的所有其他热力学特性。
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
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
配分函数与热力学性质有关，因此它具有非常重要的统计意义。系统处于微观状态 ''s'' 的概率''P<sub>s</sub>''为
<math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math>
<math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math>
: <math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math>
: <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:
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:
<math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
<math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
= 1. </math>
= 1. </math>
: <math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
: <math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
= 1. </math>
= 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 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.
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.
这就是把 ''Z'' 称为“配分函数”的原因: 它表达了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]]拉普拉斯逆变换配分函数重新要求能量的态密度函数。
==Grand canonical partition function==
==Grand canonical partition function 巨正则配分函数==
{{Main|Grand canonical ensemble}}
{{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''.
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正则系综的巨正则配分函数，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。
巨正则配分函数，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>
<math> \mathcal{Z}(\mu, V, T) = \sum_{i} \exp\left(\frac{N_i\mu - E_i}{k_B T} \right). </math>
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
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
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[[大位能]]密切相关, <math>\Phi_{\rm G}</math>, 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> \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>
<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]].
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正则配分函数相对照，后者与亥姆霍兹自由能相关。
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>:
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 <math>i</math>有关:
<math> p_i = \frac{1}{\mathcal Z} \exp\left(\frac{N_i\mu - E_i}{k_B T}\right).</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.
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。
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。
The grand partition function is sometimes written (equivalently) in terms of alternate variables as
The grand partition function is sometimes written (equivalently) in terms of alternate variables as
巨配分函数有时候是用交替变量<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>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} Z(N_i, V, T), </math>
其中， <math>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>
:<math> \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), </math>