配分函数

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In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium.[citation needed] 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.

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、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。



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.


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。


Canonical partition function 标准配分函数

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 or continuous.[citation needed]

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。



Classical discrete system

经典离散系统


For a canonical ensemble that is classical and discrete, the canonical partition function is defined as

对于经典和离散的正则系综,典型的配分函数被定义为


[math]\displaystyle{ Z = \sum_{i} \mathrm{e}^{-\beta E_i}, }[/math]
[math]\displaystyle{  Z = \sum_{i} \mathrm{e}^{-\beta E_i},  }[/math]

where

在哪里


[math]\displaystyle{ i }[/math] is the index for the microstates of the system;
[math]\displaystyle{  i  }[/math] 是系统微观状态的指标;
[math]\displaystyle{ \mathrm{e} }[/math] is Euler's number;
[math]\displaystyle{  \mathrm{e}  }[/math] 是欧拉的数字;
[math]\displaystyle{ \beta }[/math] is the thermodynamic beta, defined as [math]\displaystyle{ \tfrac{1}{k_\text{B} T} }[/math];
[math]\displaystyle{  \beta  }[/math] 是热力学beta,定义为 [math]\displaystyle{  \tfrac{1}{k_\text{B} T}  }[/math];
[math]\displaystyle{ E_i }[/math] is the total energy of the system in the respective microstate.
[math]\displaystyle{  E_i  }[/math] 是系统在各自微观状态下的总能量。



The exponential factor [math]\displaystyle{ \mathrm{e}^{-\beta E_i} }[/math] is otherwise known as the Boltzmann factor.

指数因子 [math]\displaystyle{ \mathrm{e}^{-\beta E_i} }[/math] 也被称为玻尔兹曼因子。



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Classical continuous system

Classical continuous system

经典连续系统


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

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]\displaystyle{ Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, }[/math]
[math]\displaystyle{  Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p,  }[/math]

where

[math]\displaystyle{ h }[/math] is the Planck constant;
[math]\displaystyle{  h  }[/math] 是普朗克常数;
[math]\displaystyle{ \beta }[/math] is the thermodynamic beta, defined as [math]\displaystyle{ \tfrac{1}{k_\text{B} T} }[/math];
[math]\displaystyle{  \beta  }[/math] 是热力学beta,定义为 [math]\displaystyle{  \tfrac{1}{k_\text{B} T}  }[/math];
[math]\displaystyle{ H(q, p) }[/math] is the Hamiltonian of the system;
[math]\displaystyle{  H(q, p)  }[/math] 是系统的哈密顿函数;
[math]\displaystyle{ q }[/math] is the canonical position;
[math]\displaystyle{  q  }[/math] 是正则位置
[math]\displaystyle{ p }[/math] is the canonical momentum.
[math]\displaystyle{  p  }[/math] 是正则动量。



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).

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)。



Classical continuous system (multiple identical particles)

经典的连续系统(多全同粒子)


For a gas of [math]\displaystyle{ N }[/math] identical classical particles in three dimensions, the partition function is

对于一种三维空间中的全同经典粒子气体,配分函数是


[math]\displaystyle{ 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]\displaystyle{  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]

where



[math]\displaystyle{ h }[/math] is the Planck constant;
[math]\displaystyle{  h  }[/math] 是普朗克常数;
[math]\displaystyle{ \beta }[/math] is the thermodynamic beta, defined as [math]\displaystyle{ \tfrac{1}{k_\text{B} T} }[/math];
[math]\displaystyle{  \beta  }[/math] 是热力学beta,定义为  [math]\displaystyle{  \tfrac{1}{k_\text{B} T}  }[/math];
[math]\displaystyle{ i }[/math] is the index for the particles of the system;
[math]\displaystyle{  i  }[/math] 是系统粒子的指数
[math]\displaystyle{ H }[/math] is the Hamiltonian of a respective particle;
[math]\displaystyle{  H  }[/math] 是一个粒子的哈密顿量;
[math]\displaystyle{ q_i }[/math] is the canonical position of the respective particle;
[math]\displaystyle{  q_i  }[/math] 是各个粒子的正则位置;
[math]\displaystyle{ p_i }[/math] is the canonical momentum of the respective particle;
[math]\displaystyle{  p_i  }[/math] 是各个粒子的正则动量;
[math]\displaystyle{ \mathrm{d}^3 }[/math] is shorthand notation to indicate that [math]\displaystyle{ q_i }[/math] and [math]\displaystyle{ p_i }[/math] are vectors in three-dimensional space.
[math]\displaystyle{  \mathrm{d}^3  }[/math] 是一个简写符号,用来表示 [math]\displaystyle{  q_i  }[/math][math]\displaystyle{  p_i  }[/math] 是三维空间中的向量。

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 h3N (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h3N (h 通常被认为是普朗克常数)来除以它。



Quantum mechanical discrete system

量子力学离散系统


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:


[math]\displaystyle{ Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), }[/math]
[math]\displaystyle{  Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ),  }[/math]

where:



[math]\displaystyle{ \operatorname{tr} ( \circ ) }[/math] is the trace of a matrix;
[math]\displaystyle{  \operatorname{tr} ( \circ )  }[/math] 是矩阵的轨迹;
[math]\displaystyle{ \beta }[/math] is the thermodynamic beta, defined as [math]\displaystyle{ \tfrac{1}{k_\text{B} T} }[/math];
[math]\displaystyle{  \beta  }[/math] 是热力学beta,定义为  [math]\displaystyle{  \tfrac{1}{k_\text{B} T}  }[/math];
[math]\displaystyle{ \hat{H} }[/math] is the Hamiltonian operator.
[math]\displaystyle{  \hat{H}  }[/math] 是哈密尔顿算符。

The dimension of [math]\displaystyle{ \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]\displaystyle{ \mathrm{e}^{-\beta \hat{H}} }[/math] 是系统的能量本征态个数。



Quantum mechanical continuous system

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

对于一个量子力学的连续正则系综,标准配分函数被定义为


[math]\displaystyle{ Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \, \mathrm{d} q \, \mathrm{d} p, }[/math]
[math]\displaystyle{  Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \, \mathrm{d} q \, \mathrm{d} p,  }[/math]

where:



[math]\displaystyle{ h }[/math] is the Planck constant;
[math]\displaystyle{  h  }[/math] 是普朗克常数;
[math]\displaystyle{ \beta }[/math] is the thermodynamic beta, defined as [math]\displaystyle{ \tfrac{1}{k_\text{B} T} }[/math];
[math]\displaystyle{  \beta  }[/math] 是热力学beta,定义为 [math]\displaystyle{  \tfrac{1}{k_\text{B} T}  }[/math];
[math]\displaystyle{ \hat{H} }[/math] is the Hamiltonian operator;
[math]\displaystyle{  \hat{H}  }[/math] 是哈密尔顿算符;
[math]\displaystyle{ q }[/math] is the canonical position;
[math]\displaystyle{  q  }[/math]是正则位置;
[math]\displaystyle{ p }[/math] is the canonical momentum.
[math]\displaystyle{  p  }[/math] 是正则动量。



In systems with multiple quantum states s sharing the same energy Es, 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Es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:



[math]\displaystyle{ Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j}, }[/math]
[math]\displaystyle{  Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j}, }[/math]



where gj is the degeneracy factor, or number of quantum states s that have the same energy level defined by Ej = Es.

其中gj是简并因子,或者是由 Ej = Es 定义的具有相同能级的量子态 s 的数目。



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) :



[math]\displaystyle{ Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), }[/math]
[math]\displaystyle{ Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), }[/math]

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。



The classical form of Z is recovered when the trace is expressed in terms of coherent states[1]

are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:

经典形式的Z 可以用迹的相干态[1]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 uncertainties in the position and momentum of a particle

[math]\displaystyle{ 《数学》 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 } , :\lt math\gt }[/math]

\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 是一个正态高斯波包被

</math>

position x and momentum p. Thus

位置 x 和动量 p包围

where |x, p模板:Rangle is a normalised Gaussian wavepacket centered at

[math]\displaystyle{ 《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 } :\lt math\gt = \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]

  = \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]\displaystyle{ \hat{x} }[/math] and [math]\displaystyle{ \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[math]\displaystyle{ \hat{x} }[/math][math]\displaystyle{ \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Δx 和 Δp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。



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.

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 pi denote the probability that the system S is in a particular microstate, i, with energy Ei. According to the fundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probability pi will be proportional to the number of microstates of the total closed system (S, B) in which S is in microstate i with energy Ei. Equivalently, pi will be proportional to the number of microstates of the heat bath B with energy EEi:

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。pi 表示系统 S 处于特定微观状态的概率 I,有能量Ei。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) ,pi 的概率将与总封闭系统 (S, B) 中 S 处于能量为 Ei 的微观状态的数量成正比。等价地,pi 将与热浴 B 中能量 E − Ei 的微观状态数成正比:

[math]\displaystyle{ 《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)}. :\lt math\gt }[/math]

p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.

Assuming that the heat bath's internal energy is much larger than the energy of S (E ≫ Ei), we can Taylor-expand [math]\displaystyle{ \Omega_B }[/math] to first order in Ei and use the thermodynamic relation [math]\displaystyle{ \partial S_B/\partial E = 1/T }[/math], where here [math]\displaystyle{ S_B }[/math], [math]\displaystyle{ 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S (EEi) ,我们可以对Ei 进行一阶泰勒展开 [math]\displaystyle{ \Omega_B }[/math] ,并利用热力学关系式 [math]\displaystyle{ \partial S_B/\partial E = 1/T }[/math],这里[math]\displaystyle{ S_B }[/math], [math]\displaystyle{ T }[/math] 分别是热水池的熵和温度:

[math]\displaystyle{ 《数学》 \begin{align} 开始{ align } :\lt math\gt 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]

\end{align}

</math>

Thus

因此得到

[math]\displaystyle{ 《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 }. :\lt math\gt }[/math]

p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}.

Since the total probability to find the system in some microstate (the sum of all pi) 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 pi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]\displaystyle{ 《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)}. :\lt math\gt }[/math]

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

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:

[math]\displaystyle{ \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 }

: \lt math\gt \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]

or, equivalently,

或者,等价地说,

[math]\displaystyle{ \langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}. }[/math]
[math]\displaystyle{ \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

如果微态能量依赖于参数 λ 的方式

[math]\displaystyle{ E_s = E_s^{(0)} + \lambda A_s \qquad \mbox{for all}\; s  }[/math]
[math]\displaystyle{ 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

[math]\displaystyle{ \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]
[math]\displaystyle{ \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.[citation needed]

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

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。


As we have already seen, the thermodynamic energy is

正如我们已经看到的,热力学能

[math]\displaystyle{ \langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}. }[/math]
[math]\displaystyle{ \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

[math]\displaystyle{ \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]
[math]\displaystyle{ \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

[math]\displaystyle{ C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle. }[/math]
[math]\displaystyle{ 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:

[math]\displaystyle{ \langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}. }[/math]
[math]\displaystyle{ \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

[math]\displaystyle{ \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]
[math]\displaystyle{ \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

的特殊情况下,熵是由

[math]\displaystyle{ 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]\displaystyle{ 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 = UTS, where U = 模板:LangleE模板: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

[math]\displaystyle{ A = \langle E\rangle -TS= - k_B T \ln Z. }[/math]
[math]\displaystyle{ A = \langle E\rangle -TS= - k_B T \ln Z. }[/math]


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 ζ1, ζ2, ..., ζN, 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 ζ1, ζ2, ..., ζN,那么整个系统的配分函数就是单个配分函数的乘积:

[math]\displaystyle{ Z =\prod_{j=1}^{N} \zeta_j. }[/math]
[math]\displaystyle{ Z =\prod_{j=1}^{N} \zeta_j. }[/math]

If the sub-systems have the same physical properties, then their partition functions are equal, ζ1 = ζ2 = ... = 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,ζ1 = ζ2 = ... = ζ,在这种情况下

[math]\displaystyle{ Z = \zeta^N. }[/math]
[math]\displaystyle{ 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) :

[math]\displaystyle{ Z = \frac{\zeta^N}{N!}. }[/math]
[math]\displaystyle{ 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

Meaning and significance 影响

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 E1, E2, E3, 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 E1, E2, E3, 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 Ps 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Ps

[math]\displaystyle{ P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}.  }[/math]


[math]\displaystyle{ 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:

[math]\displaystyle{ \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]
[math]\displaystyle{ \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 巨正则配分函数

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]\displaystyle{ \mathcal{Z} }[/math],是微状态上的和

[math]\displaystyle{ \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]\displaystyle{ i }[/math], and has total particle number [math]\displaystyle{ N_i }[/math] and total energy [math]\displaystyle{ E_i }[/math]. This partition function is closely related to the grand potential, [math]\displaystyle{ \Phi_{\rm G} }[/math], by the relation

每个微观状态都用[math]\displaystyle{ i }[/math]标记,并且有总粒子数 [math]\displaystyle{ N_i }[/math] 和总能量 [math]\displaystyle{ E_i }[/math]。这种配分函数与大位能密切相关, [math]\displaystyle{ \Phi_{\rm G} }[/math], 通过这种关系

[math]\displaystyle{ \mathcal{Z}(\mu, V, T) = \sum_{i} \exp\left(\frac{N_i\mu - E_i}{k_B T} \right). }[/math]

[math]\displaystyle{ -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.

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]\displaystyle{ 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]\displaystyle{ i }[/math]有关:

[math]\displaystyle{ 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。

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[2]来表示的

[math]\displaystyle{ \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), }[/math]

其中, [math]\displaystyle{ z \equiv \exp(\mu/kT) }[/math] 被称为绝对活度(或逸度) ,而 [math]\displaystyle{ Z(N_i, V, T) }[/math]则是典范配分函数。

[math]\displaystyle{ \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), }[/math]


相关阅读


参考文献

  1. 1.0 1.1 Klauder, John R.; Skagerstam The classical form of Z is recovered when the trace is expressed in terms of coherent states 当轨迹用相干态表示时,恢复了 z 的经典形式, Bo-Sture (1985). 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. World Scientific. pp. 71–73. ISBN 978-9971-966-52-2. 
  2. Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc.. ISBN 9780120831807. 
  • Huang, Kerson, "Statistical Mechanics", John Wiley & Sons, New York, 1967.
  • A. Isihara, "Statistical Physics", Academic Press, New York, 1971.
  • L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996.

模板:Statistical mechanics topics

Category:Concepts in physics

分类: 物理概念



This page was moved from wikipedia:en:Partition function (statistical mechanics). Its edit history can be viewed at 配分函数/edithistory