# 配分函数

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.

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.

## 正则配分函数

#### 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]

#### 经典离散系统

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

$\displaystyle{ Z = \sum_{i} \mathrm{e}^{-\beta E_i}, }$

where

$\displaystyle{ i }$ 是系统微观状态的指标；$\displaystyle{ \mathrm{e} }$ 是欧拉的数字;$\displaystyle{ \beta }$ 是热力学beta，定义为 $\displaystyle{ \tfrac{1}{k_\text{B} T} }$; $\displaystyle{ E_i }$ 是系统在各自微观状态下的总能量。

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#### 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

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

where

$\displaystyle{ h }$ 是普朗克常数; $\displaystyle{ \beta }$ 是热力学beta，定义为 $\displaystyle{ \tfrac{1}{k_\text{B} T} }$; $\displaystyle{ H(q, p) }$ 是系统的哈密顿函数; $\displaystyle{ q }$ 是正则位置; $\displaystyle{ p }$ 是正则动量。

#### 经典的连续系统(多全同粒子)

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

$\displaystyle{ Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N }$

where

$\displaystyle{ h }$ 是普朗克常数; $\displaystyle{ \beta }$ 是热力学beta，定义为 $\displaystyle{ \tfrac{1}{k_\text{B} T} }$; $\displaystyle{ i }$ 是系统粒子的指数; $\displaystyle{ H }$ 是一个粒子的哈密顿量; $\displaystyle{ q_i }$ 是各个粒子的正则位置; $\displaystyle{ p_i }$ 是各个粒子的正则动量;

$\displaystyle{ \mathrm{d}^3 }$ 是一个简写符号，用来表示 $\displaystyle{ q_i }$$\displaystyle{ p_i }$ 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).

#### 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:

$\displaystyle{ Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), }$

where:

$\displaystyle{ \operatorname{tr} ( \circ ) }$ 是矩阵的轨迹; $\displaystyle{ \beta }$ 是热力学beta，定义为 $\displaystyle{ \tfrac{1}{k_\text{B} T} }$; $\displaystyle{ \hat{H} }$ 是哈密尔顿算符。

#### 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

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

where:

$\displaystyle{ h }$ 是普朗克常数; $\displaystyle{ \beta }$ 是热力学beta，定义为 $\displaystyle{ \tfrac{1}{k_\text{B} T} }$; $\displaystyle{ \hat{H} }$ 是哈密尔顿算符；$\displaystyle{ q }$是正则位置；$\displaystyle{ p }$ 是正则动量。

$\displaystyle{ Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j}, }$

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):

$\displaystyle{ Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), }$

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。

$\displaystyle{ \boldsymbol{1} = \int |x, p\rangle \langle x,p| \frac{dx \,dp}{h}, 1} = int | x，p rangle langle x，p | frac { dx，dp }{ h } , }$

$\displaystyle{ Z = \int \operatorname{tr} \left( \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \langle x, p| \right) \frac{dx \,dp}{h} = \int \langle x,p| \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \frac{dx \,dp}{h}. }$

## 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.

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:

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

\begin{align}

\displaystyle{ 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} &\approx -\frac{E_i}{T} + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)} \end{align} </math\gt }

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

$\displaystyle{ Z = \sum_i e^{-\beta E_i} = \frac{\Omega_{(S,B)}(E)}{\Omega_B(E)}. }$

## 计算热力学总能

$\displaystyle{ \langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s e^{- \beta E_s} = - \frac{1}{Z} \frac{\partial}{\partial \beta} Z(\beta, E_1, E_2, \cdots) = - \frac{\partial \ln Z}{\partial \beta} }$

$\displaystyle{ \langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}. }$

$\displaystyle{ E_s = E_s^{(0)} + \lambda A_s \qquad \mbox{for all}\; s }$

$\displaystyle{ \langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta} \frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda). }$

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]

## 与热力学变量的关联

$\displaystyle{ \langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}. }$

$\displaystyle{ \langle (\Delta E)^2 \rangle \equiv \langle (E - \langle E\rangle)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2}. }$

$\displaystyle{ C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle. }$

$\displaystyle{ \langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}. }$

$\displaystyle{ \langle (\Delta X)^2 \rangle \equiv \langle (X - \langle X\rangle)^2 \rangle = \frac{\partial \langle X \rangle}{\partial \beta Y} = \frac{\partial^2 \ln Z}{\partial (\beta Y)^2}. }$

$\displaystyle{ S \equiv -k_B\sum_s P_s\ln P_s= k_B (\ln Z + \beta \langle E\rangle)=\frac{\partial}{\partial T}(k_B T \ln Z) =-\frac{\partial A}{\partial T} }$

$\displaystyle{ A = \langle E\rangle -TS= - k_B T \ln Z. }$

## 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:

$\displaystyle{ Z =\prod_{j=1}^{N} \zeta_j. }$

$\displaystyle{ Z = \zeta^N. }$

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):

$\displaystyle{ Z = \frac{\zeta^N}{N!}. }$

## 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.

$\displaystyle{ P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. }$

$\displaystyle{ \sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z = 1. }$

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.

## 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.

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

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

$\displaystyle{ -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. }$

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 $\displaystyle{ i }$:

$\displaystyle{ p_i = \frac{1}{\mathcal Z} \exp\left(\frac{N_i\mu - E_i}{k_B T}\right). }$

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.

$\displaystyle{ \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), }$

$\displaystyle{ \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), }$

## 参考文献

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.

Category:Concepts in physics

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