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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ 第15行：
第15行： − 在物理学中，配分函数描述了热力学平衡中一个系统的统计特性。配分函数是热力学状态变量的函数，比如温度和体积。体系中的大多数热力学变量，如总能量、自由能、熵和压力，都可以用配分函数或其衍生物来表示。配分函数是无量纲的，它是一个纯数。+ 第23行：
第23行： − + 第55行：
第55行： − :<math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math> − <math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math> − 数学，数学，数学，数学+ + + + + + + 第67行：
第71行： − :<math> i </math> is the index for the [[Microstate (statistical mechanics)|microstates]] of the system, − <math> i </math> is the index for the microstates of the system, − Math i / math 是系统微观状态的索引,+ − :<math> \mathrm{e} </math> is [[e (mathematical constant)|Euler's number]],+ − <math> \mathrm{e} </math> is Euler's number,+ − + − :<math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>,+ − <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>,+ − + − :<math> E_i </math> is the total energy of the system in the respective [[Microstate (statistical mechanics)|microstate]].+ − + − + + + + + 第97行：
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第111行： − + 第125行：
第131行： − + − + − 有多种方法来推导配分函数。下面的推导遵循强大的和一般的信息理论 Jaynesian 最大熵方法。+ 第137行：
第143行： − 根据热力学第二定律的研究，一个系统在热力学平衡时会呈现出最大熵的状态。我们寻求一个概率分布的状态数学，它能最大化离散吉布斯熵+ 第145行：
第151行： − + 第157行：
第163行： − + − + − + 第167行：
第173行： − 数学+ 第173行：
第179行： − 什么意思。+ 第183行：
第189行： − + − + − + 第193行：
第199行： − 数学+ − + − + − 三角形 e rangle sum i rho i e i = u。+ 第209行：
第215行： − Using the method of [[Lagrange multipliers]], we write the Lagrangian (or Lagrange function) <math> \mathcal{L} </math> as+ − Using the method of Lagrange multipliers, we write the Lagrangian (or Lagrange function) <math> \mathcal{L} </math> as+ − 利用拉格兰奇乘数法，我们将拉格朗日函数(或拉格兰奇函数)的数学表达式写成+ 第220行：
第226行： + + + + + + + + + + + + − \mathcal{L} = \left( -k_\text{B} \sum_i \rho_i \ln \rho_i \right) + \lambda_1 \left( \sum_i \rho_i - 1 \right) + \lambda_2 \left( \sum_i \rho_i E_i - U \right). − \mathcal{L} = \left( -k_\text{B} \sum_i \rho_i \ln \rho_i \right) + \lambda_1 \left( \sum_i \rho_i - 1 \right) + \lambda_2 \left( \sum_i \rho_i E_i - U \right). − 左(- k text { b } sum i rho ln i rho right) + λ 1( sum i rho-1 right) + λ 2( sum i rho i-u right)。+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 第237行：
第307行： − Differentiating and extremizing <math> \mathcal{L} </math> with respect to <math> \rho_i </math> leads to+ − Differentiating and extremizing <math> \mathcal{L} </math> with respect to <math> \rho_i </math> leads to+ − 数学的微分和极值化+ 第249行：
第319行： − 数学+ − + − + − + 第269行：
第339行： − 数学 rho i / 数学产额的孤立+ 第277行：
第347行： − 数学+ − + − + − + 第297行：
第367行： − + 第305行：
第375行： − 数学+ 第311行：
第381行： − Begin { align }+ 第317行：
第387行： − + − + − + − + 第329行：
第399行： − end { align }+ 第343行：
第413行： − 其中 math z / math 是一个定义为正则系综 / 配分函数的常数:+ 第351行：
第421行： − 数学+ − + − + − + 第371行：
第441行： − 隔离数学 lambda 1 / math 产生数学 lambda 1-k b ln (z) + k b / math。+ 第379行：
第449行： − 用数学 z / math 来重写数学 rho i / math+ 第387行：
第457行： − 数学+ − + − + − 向左(向右)打开文本。+ 第407行：
第477行： − 用数学 z / math 来重写数学 s / math+ 第415行：
第485行： − 数学+ 第421行：
第491行： − Begin { align }+ 第427行：
第497行： − + 第433行：
第503行： − &-k text { b } sum i rho i ( frac { lambda 2}{ k text { b } e i- ln (z)右)+ 第441行：
第511行： − + − + − + 第451行：
第521行： − End { align }+ 第465行：
第535行： − 为了得到 lambda 数学，我们对数学 s / math 与平均能量数学 u / math 进行了区分，并应用能量守恒定律，数学 dU t dS-p dV / math:+ 第473行：
第543行： − 数学+ − + − + − Frac { dS }{ dU } lambda 2 equiv frac {1}{ t }.+ 第493行：
第563行： − 因此，规范的配分函数数学 z / math 变成了+ 第501行：
第571行： − 数学+ − + − + − + 第521行：
第591行： − + 第529行：
第599行： − 数学+ 第535行：
第605行： − Begin { align }+ − + − + − 我和我的朋友们都很高兴见到你们+ − + − + − S & frac { t } + k text { b } ln z.+ 第553行：
第623行： − End { align }+ 第583行：
第653行： − 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术语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> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \,d^3 q \,d^3 p, </math> − <math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \,d^3 q \,d^3 p, </math> − 数学 z frac {1}{ h ^ 3} int mathrm { e ^ {- beta h (q，p)} ，d ^ 3 q ，d ^ 3 p，/ math+ + + + + + + 第597行：
第671行： − : <math> h </math> is the [[Planck constant]], − <math> h </math> is the Planck constant, − 数学 h 是普朗克常数,+ − : <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>,+ − <math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>,+ − + − : <math> H(q, p) </math> is the [[Hamiltonian mechanics|Hamiltonian]] of the system,+ − <math> H(q, p) </math> is the Hamiltonian of the system,+ − 数学 h (q，p) / math 是系统的哈密顿量,+ − : <math> q </math> is the [[Canonical coordinates|canonical position]],+ − <math> q </math> is the canonical position,+ − 数学 q / 数学是标准位置,+ + + + + 第625行：
第701行： − 数学 p / 数学是正则动量。+ 第645行：
第721行： − + − + − 对于一种由 n 个经典粒子组成的气体，在三维空间中，配分函数是+ − + − + − + 第669行：
第745行： − + + + + + + + + + + + + + + + − p<sub>i</sub> indicate particle momenta,+ − P 次 i / 次表示粒子动量,+ − : ''x<sub>i</sub>'' indicate particle positions,+ − x<sub>i</sub> indicate particle positions,+ − 子 i / 子表示粒子的位置,+ − : ''d''<sup>3</sup> is a shorthand notation serving as a reminder that the ''p<sub>i</sub>'' and ''x<sub>i</sub>'' are vectors in three-dimensional space,+ − d<sup>3</sup> is a shorthand notation serving as a reminder that the p<sub>i</sub> and x<sub>i</sub> are vectors in three-dimensional space,+ − d sup 3 / sup 是一个简写符号，提醒我们 p / sub i / sub 和 x / sub i / sub 是三维空间中的向量,+ − : ''H'' is the classical [[Hamiltonian mechanics|Hamiltonian]] of a single particle.+ − H is the classical Hamiltonian of a single particle.+ − h 是单粒子的经典哈密顿量。+ + + + + 第699行：
第793行： − + 第715行：
第809行： − 对于量子力学和离散的正则系综，标准配分函数被定义为玻尔兹曼因子的轨迹:+ + + + + + + + + + + − + − <math> Z = \operatorname{tr} (\mathrm{e}^{-\beta \hat{H}}), </math>+ − 数学运算器名称{ tr }( mathrum { e } ^ {- beta hat { h }) ，/ math+ − where − where − 在哪里+ + + + + − + − + − Math beta / math 是热力学beta，定义为 math tfrac {1}{ k text { b } t } / math,+ 第739行：
第845行： − math [ h ] / math 是哈密顿算符。+ + + 第745行：
第853行： − 数学数学的维数是系统能量本征态的个数。+ 第763行：
第871行： − : <math> Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \,dq \,dp, </math> − <math> Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \,dq \,dp, </math> − 数学 zfrac {1}{ h } int langle q，p | mathrm { e } ^-beta { h } | q，p rangle ，dq ，dp，/ math+ + + + + − where − where − + + + + + + + − + − + − 数学 h 是普朗克常数,+ − + − + − math beta / math 是热力学beta，定义为 math tfrac {1}{ k text { b } t } / math,+ − + − + − 数学是哈密尔顿算符,+ − + − + − 数学 q / 数学是标准位置,+ 第803行：
第917行： − 数学 p / 数学是正则动量。+ 第811行：
第925行： − 在多个量子态共享同一能量 e 子 / 子的系统中，系统的能级是简并的。在简并能级的情况下，我们可以用能级的贡献来表示配分函数，如下:+ 第819行：
第933行： − 数学 z sum j j cdot mathrm { e } ^ {- beta e j } ，/ math+ 第827行：
第941行： − + 第835行：
第949行： − 上面的处理适用于量子统计力学，在有限大小的盒子里的物理系统通常会有一组离散的能量本征态，我们可以用它作为上面的状态。在量子力学中，配分函数可以更正式地写成状态空间上的跟踪(这与基的选择无关) :+ 第843行：
第957行： − 数学运算器名称{ tr }( mathrum { e } ^ {- beta hat { h }) ，/ math+ 第857行：
第971行： − + − − 当轨迹用相干态表示时，z 的经典形式被恢复了，相干态为{ cite book | first1 John r | last1 Klauder | first2 Bo-Sture | last2 Skagerstam − |title=Coherent States: Applications in Physics and Mathematical Physics+ − − 相干态: 在物理学和数学物理学中的应用 − − |publisher=World Scientific |date=1985 |pages=71–73 |isbn=978-9971-966-52-2 }}</ref> − − | 出版商 World Scientific | 日期1985 | 第71-73页 | isbn 978-9971-966-52-2} / ref − − 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 − − 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。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: − − <math> − − 数学 − − \boldsymbol{1} = \int |x, p\rangle \langle x,p| \frac{dx \,dp}{h}, − − 符号{1} int | x，p rangle langle x，p | frac { dx ，dp } , − − </math> − − 数学 − − 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 − − position ''x'' and momentum ''p''. Thus − − position x and momentum p. Thus − − 位置 x 和动量 p − − :<math> − − <math> − − 数学 − − 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} − − Z int operatorname { tr } left ( mathrum { e } ^-beta hat { h } | x，p rangle langle x，p | right) frac { dx ，dp } − − = \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}. − − 整形 x，p | mathrm { e } ^-beta { h } | x，p rangle frac { dx ，dp }{ h }。 − − </math> − − </math> − − 数学 − − 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. − − 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 math 和 math 的近似本征态，因此也是 Hamiltonian 的近似本征态，误差为不确定性的大小。如果 x 和 p 可以看作为零，则经典哈密顿量将其作用减少为乘法，而 z 则减少为经典构型积分。 − − − − === 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. − − 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形式。 − − − − Consider a system ''S'' embedded into a [[heat bath]] ''B''. Let the total [[energy]] of both systems be ''E''. Let ''p<sub>i</sub>'' denote the [[probability]] that the [[Microstate (statistical mechanics)|microstate]] that system ''S'' is in has 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>'': − − 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 microstate that system S is in has 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。设系统的总能量为 e，p 子 i / 子表示系统 s 处于能量 e 子 i / 子的微观状态的概率。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 子 i / 子的概率将与 s 处于能量 e 子 i / 子的微观状态的总封闭系统(s，b)的微观状态数成正比。等价地，p 分子 i / 分子将成正比的微观状态数的热浴 b 与能量 e-e 分子 i / 分子: − − − − :<math> − − <math> − − 数学 − − p_i = \frac{\Omega_B(E - E_i)}{\Omega_B(E)}. − − p_i = \frac{\Omega_B(E - E_i)}{\Omega_B(E)}. − − 欧米茄 b (e-e)}{ Omega b (e)}。 − − </math> − − </math> − − 数学 − − − − 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: − − 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: − − 假设热池的内能远大于 s (e something e 子 i / 子)的能量，我们可以在 e 子 i / 子中将数学 · 欧米加 · b / 数学展开为一阶，并利用热力学关系数学 · 偏 s b / 偏 e 1 / t / 数学，其中数学 · s b / 数学、数学 · t / 数学分别是热池的熵和温度: − − − − :<math> − − <math> − − 数学 − − \begin{align} − − \begin{align} − − Begin { align } − − k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_B(E) \\[5pt] − − k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_B(E) \\[5pt] − − K ln p i & k ln Omega b (e-ei)-k ln Omega b (e)[5 pt ] − − &\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i \\[5pt] − − &\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i \\[5pt] − − 大约部分大(kln Omega b (e) big)}部分 e 开头[5 pt ] − − &\approx -\frac{\partial S_B}{\partial E} E_i \\[5pt] − − &\approx -\frac{\partial S_B}{\partial E} E_i \\[5pt] − − [5 pt ] ，[5 pt ] − − &\approx -\frac{E_i}{T} − − &\approx -\frac{E_i}{T} − − 和 frac { e i }{ t } − − \end{align} − − \end{align} − − End { align } − − </math> − − </math> − − 数学 − − − − Thus − − Thus − − 因此 − − :<math> − − <math> − − 数学 − − p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}. − − p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}. − − P i propto e ^ {-e i / (kT)} e ^ {- beta e i }. − − </math> − − </math> − − 数学 − − − − 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 can define the partition function as the [[Normalizing constant|normalization constant]]: − − 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 can define the partition function as the normalization constant: − − 由于发现系统处于某种微观状态的总概率(所有 p 子 i / sub 的和)必须等于1，我们可以将配分函数归一化常数定义为: − − − − :<math> − − <math> − − 数学 − − Z = \sum_i e^{-\beta E_i}. − − Z = \sum_i e^{-\beta E_i}. − − Z sum i e ^ {- beta e i }. − − </math> − − </math> − − 数学 − − − − === 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: − − 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>\langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s − − <math>\langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s − − 数学长角 e-rangle-sum s-e-s-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}{\partial \beta} − − E ^ {- beta e s }- frac {1}{ z } frac { partial }{ partial beta } − − 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} − − Z ( beta，e1，e2， cdots)-frac { partial ln z }{ partial beta } − − </math> − − </math> − − 数学 − − − − or, equivalently, − − or, equivalently, − − 或者，相当于, − − − − : <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math> − − <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math> − − 数学方程式，数学方程式，数学方程式 − − − − Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner − − Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner − − 顺便说一句，我们应该注意到，如果微态能量依赖于一个参数的方式 − − − − : <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> − − 数学 e s ^ {(0)} + lambda a s qquad mbox { for all } ; s / math − − − − then the expected value of ''A'' is − − then the expected value of A is − − 那么 a 的期望值是 − − − − : <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta} − − <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta} − − 数学角度 a rangle sum s s s frac {1}{ beta } − − \frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math> − − \frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math> − − ( 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|date=December 2015}} − − This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set λ to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory. − − 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 === − − − − In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations. − − In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations. − − 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 − − As we have already seen, the thermodynamic energy is − − 正如我们已经看到的，热力学能 − − − − : <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math> − − <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math> − − 4. math langle e rangle frac { partial beta } . / math − − − − The [[variance]] in the energy (or "energy fluctuation") is − − The variance in the energy (or "energy fluctuation") is − − 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 − − 数学 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> − − E-rangle) ^ 2 rangle frac { partial ^ 2 ln z }{ partial beta ^ 2} . / math − − − − The [[heat capacity]] is − − The heat capacity is − − 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> − − <math>C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle.</math> − − 数学 c 语言部分语言部分语言部分语言部分语言部分语言部分语言部分语言部分语言部分语言部分语言部分语言 − − − − 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> − − <math>\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math> − − 数学，数学，数学，数学，数学，数学 − − − − 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 − − − − : <math>\langle (\Delta X)^2 \rangle \equiv \langle (X - \langle − − <math>\langle (\Delta X)^2 \rangle \equiv \langle (X - \langle − − (x- langle) ^ 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> − − 部分贝塔部分贝塔部分贝塔部分贝塔部分贝塔部分贝塔部分贝塔部分贝塔 − − − − 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> − − <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> − − 数学 s-equiv-k b-sum s p s ln p s k b ( ln z + beta langle e rangle) frac { partial t }(k b t ln z)-frac { partial a } / 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 = 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> − − <math>A = \langle E\rangle -TS= - k_B T \ln Z.</math> − − 数学 a langle e rangle-TS-k b 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: − − 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，我们可以假设粒子基本上是不相互作用的。如果子系统的配分函数是子1 / 子，子2 / 子，... ，子 n / 子，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> − − <math>Z =\prod_{j=1}^{N} \zeta_j.</math> − − 数学，数学，数学，数学 − − − − 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相等，子1 / 子2 / 子... ，在这种情况下 − − − − : <math>Z = \zeta^N.</math> − − <math>Z = \zeta^N.</math> − − 数学 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]]): − − 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，即使在原则上也不可能区分，那么总的配分函数必须除以 n！(n 阶乘) : − − − − : <math>Z = \frac{\zeta^N}{N!}.</math> − − <math>Z = \frac{\zeta^N}{N!}.</math> − − 我不知道你在说什么！数学 − − − − 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 === − − − − It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature ''T'' and the microstate energies ''E''<sub>1</sub>, ''E''<sub>2</sub>, ''E''<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system. − − It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature T and the microstate energies E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system. − − 为什么我们上面定义的配分函数是一个重要的数量，这可能并不明显。首先，考虑一下其中包含的内容。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。通过一个系统的微观组分模型，我们可以计算出微观能量，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。 − − − − 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 − − − − : <math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math> − − <math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math> − − 数学 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: − − 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 − − 数学总和 s s s frac {1}{ z }和 s mathrm { e } ^ {- beta e s } frac {1}{ z } − − = 1. </math> − − = 1. </math> − − = 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 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 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: 它编码概率是如何根据各自的能量在不同的微观状态之间分配的。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。找到配分函数也相当于从能量域到能量域执行一个拉普拉斯变换的态密度函数，而拉普拉斯逆变换配分函数的态密度函数则重新得到能量的态密度函数。 − − − − ==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. − − 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. − − 我们可以定义一个巨典型配分函数，它描述了一个恒定体积系统的统计巨正则系综，这个系统可以同时与一个蓄热器交换热量和粒子。 − − The reservoir has a constant temperature ''T'', and a [[chemical potential]] ''μ''. − − The reservoir has a constant temperature T, and a chemical potential μ. − − 储层具有恒定的温度 t 和化学势。 − − − − The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over [[microstate (statistical mechanics)|microstates]] − − The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over microstates − − 巨正则配分函数，由 math mathcal { z } / math 表示，是微状态上的下列和 − − :<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> − − 数学{ z }( mu，v，t) sum { i } exp 左( frac { ni mu-e i }{ kt }右)。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>. − − 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>. − − 在这里，每个微观状态都被数学 i / math 标记，并且有总粒子数、数学 i / math 和总能量数学 ei / math。 − − This partition function is closely related to the [[Grand potential]], <math>\Phi_{\rm G}</math>, by the relation − − This partition function is closely related to the Grand potential, <math>\Phi_{\rm G}</math>, by the relation − − 通过这个关系式，这个配分函数与大势、数学 / 数学密切相关 − − :<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> − − 数学-k-b-t-ln-z-z-φ-g-langle e-rangle-TS-mu-langle n-rangle。数学 − − 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权威配分函数相对照，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, − − It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, − − 值得注意的是，巨正则系综的微观状态数量可能远远大于正则系综, − − since here we consider not only variations in energy but also in particle number. − − since here we consider not only variations in energy but also in particle number. − − 因此我们不仅要考虑能量的变化，还要考虑粒子数的变化。 − − 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>: − − 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>: − − 同样，巨典型配分函数的效用在于它与系统处于状态数学 i / math 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> − − <math> p_i = \frac{1}{\mathcal Z} \exp\left(\frac{N_i\mu - E_i}{k_B T}\right).</math> − − 数学公式{1}{数学公式 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. − − 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. − − 巨正则系综的一个重要应用是精确地导出非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，它更普遍地适用于。 − − The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases. − − The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases. − − 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<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> − − 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} Z(N_i, V, T), </math> − − 数学{ z }(z，v，t)和{ 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. − − 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 math z equiv exp ( mu / kt) / math 称为绝对活动(或逸度) ，math z (ni，v，t) / math 称为配分函数。 − − − − ==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]] 第1,607行：
第995行： − [[Category:Partition functions| ]]+
Moved page from wikipedia:en:Partition function (statistical mechanics) (history)
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{{About|statistical mechanics|other uses|partition function (disambiguation)}}
{{About|statistical mechanics|other uses|partition function (disambiguation)}}
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number.
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number.
在物理学中，一个配分函数描述了一个热力学平衡系统的统计特性。配分函数是热力学状态变量的函数，比如温度和体积。体系中的大多数热力学变量，如总能量、自由能、熵和压力，都可以用配分函数或其衍生物来表示。配分函数是无量纲的，它是一个纯数。
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.
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(反过来，对应一个特定的自由能)。最常见的统计集合称为配分函数。典型的配分函数适用于正则系综，系统允许在固定的温度、体积和粒子数量下与环境进行热交换。巨正则配分函数适用于巨正则系综，系统可以在固定的温度、体积和化学势下同时与环境交换热量和粒子。其他类型的配分函数可以根据不同的情况来定义; 参见配分函数(数学)的一般化。配分函数有许多物理意义，正如在意义和重要性中所讨论的那样。
每个配分函数都代表一个特定的系综(反过来，对应一个特定的自由能)。最常见的统计集合称为配分函数。典型的配分函数适用于正则系综，系统允许在固定的温度、体积和粒子数量下与环境进行热交换。巨正则配分函数适用于巨正则系综，系统可以在固定的温度、体积和化学势下同时与环境交换热量和粒子。其他类型的配分函数可以根据不同的情况来定义; 有关一般化，请参阅数学配分函数。配分函数有许多物理意义，正如在意义和重要性中讨论的那样。
对于经典和离散的正则系综，典型的配分函数被定义为
对于经典和离散的正则系综，典型的配分函数被定义为
: <math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math>
<math> Z = \sum_{i} \mathrm{e}^{-\beta E_i}, </math>
[数学] z = sum { i } mathrm { e } ^ {-beta e _ i } ，[数学]
where
where
在哪里
在哪里
: <math> i </math> is the index for the [[Microstate (statistical mechanics)|microstates]] of the system;
<math> i </math> is the index for the microstates of the system;
是系统微观状态的指标
math 数学是欧拉数,
: <math> \mathrm{e} </math> is [[e (mathematical constant)|Euler's number]];
<math> \mathrm{e} </math> is Euler's number;
数学是欧拉的数字;
math beta / math 是热力学beta，定义为 math tfrac {1}{ k text { b } t } / math,
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
<math> E_i </math> is the total energy of the system in the respective microstate.
是热力学beta，定义为 < math > tfrac
math 是系统在各自微观状态下的总能量。
: <math> E_i </math> is the total energy of the system in the respective [[Microstate (statistical mechanics)|microstate]].
<math> E_i </math> is the total energy of the system in the respective microstate.
是系统在各自微观状态下的总能量。
The exponential factor <math> \mathrm{e}^{-\beta E_i} </math> is otherwise known as the Boltzmann factor.
The exponential factor <math> \mathrm{e}^{-\beta E_i} </math> is otherwise known as the Boltzmann factor.
指数因子 math mathrm { e } ^ {- beta e i } / math 又称玻尔兹曼因子。
指数因子 < math > mathrm { e } ^ {-beta e _ i } </math > 也被称为玻尔兹曼因子。
{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
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! Derivation of canonical partition function (classical, discrete)
! Derivation of canonical partition function (classical, discrete)
|
|
There are multiple approaches to deriving the partition function. The following derivation follows the powerful and general [[information theory|information-theoretic]] [[Edwin Thompson Jaynes|Jaynesian]] [[maximum entropy thermodynamics|maximum entropy]] approach.
There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general [[information theory|information-theoretic]] [[Edwin Thompson Jaynes|Jaynesian]] [[maximum entropy thermodynamics|maximum entropy]] approach.
There are multiple approaches to deriving the partition function. The following derivation follows the powerful and general information-theoretic Jaynesian maximum entropy approach.
There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general information-theoretic Jaynesian maximum entropy approach.
有多种方法来推导配分函数。下面的推导遵循更强大和一般的信息理论 Jaynesian 最大熵方法。
According to the second law of thermodynamics, a system assumes a configuration of maximum entropy at thermodynamic equilibrium. We seek a probability distribution of states <math> \rho_i </math> that maximizes the discrete Gibbs entropy
According to the second law of thermodynamics, a system assumes a configuration of maximum entropy at thermodynamic equilibrium. We seek a probability distribution of states <math> \rho_i </math> that maximizes the discrete Gibbs entropy
根据热力学第二定律的研究，一个系统在热力学平衡时会呈现出最大熵的状态。我们在寻找一个最大化离散吉布斯熵的状态概率分布
<math> S = - k_\text{B} \sum_i \rho_i \ln \rho_i </math>
<math> S = - k_\text{B} \sum_i \rho_i \ln \rho_i </math>
数学 s-k text { b } sum i rho i / math
[数学] s =-k text { b } sum i rho _ i ln rho _ i
1. The probabilities of all states add to unity:
1. The probabilities of all states add to unity ([[Probability axioms#Second axiom|second axiom of probability]]):
1. The probabilities of all states add to unity:
1. The probabilities of all states add to unity (second axiom of probability):
1.所有国家的可能性加强了团结:
1.所有状态的概率加上统一(概率的第二公理) :
:<math>
:<math>
<math>
<math>
《数学》
\sum_i \rho_i = 1.
\sum_i \rho_i = 1.
\sum_i \rho_i = 1.
\sum_i \rho_i = 1.
总和 i rho = 1。
</math>
</math>
2. In the [[canonical ensemble]], the average energy is fixed:
2. In the [[canonical ensemble]], the average energy is fixed ([[conservation of energy]]):
2. In the canonical ensemble, the average energy is fixed:
2. In the canonical ensemble, the average energy is fixed (conservation of energy):
2.在正则系综，平均能量是固定的:
2.在正则系综中，平均能量是固定的(能量守恒) :
:<math>
:<math>
<math>
<math>
《数学》
\langle E \rangle = \sum_i \rho_i E_i \equiv U.
\langle E \rangle = \sum_i \rho_i E_i \equiv U .
\langle E \rangle = \sum_i \rho_i E_i \equiv U.
\langle E \rangle = \sum_i \rho_i E_i \equiv U .
长角 e rangle = sum i rho i e i equiv u。
</math>
</math>
Applying [[calculus of variations|variational calculus]] with constraints (analogous to the method of [[Lagrange multipliers]]), we write the Lagrangian (or Lagrange function) <math> \mathcal{L} </math> as
Applying variational calculus with constraints (analogous to the method of Lagrange multipliers), we write the Lagrangian (or Lagrange function) <math> \mathcal{L} </math> as
应用带约束的变分演算(类似于拉格兰奇乘子法) ，我们将拉格朗日函数(或拉格兰奇函数) < math > mathcal { l } </math > 写成
<math>
<math>
《数学》
\mathcal{L} = \left( -k_\text{B} \sum_i \rho_i \ln \rho_i \right) + \lambda_1 \left( 1 - \sum_i \rho_i \right) + \lambda_2 \left( U - \sum_i \rho_i E_i \right) .
\mathcal{L} = \left( -k_\text{B} \sum_i \rho_i \ln \rho_i \right) + \lambda_1 \left( 1 - \sum_i \rho_i \right) + \lambda_2 \left( U - \sum_i \rho_i E_i \right) .
数学{ l } = 左(- k _ text { b } sum _ i rho _ i ln rho _ i 右) + lambda _ 1左(1-sum _ i rho _ i 右) + lambda _ 2左(u-sum _ i rho _ i e _ i 右)。
</math>
</math>
数学
数学
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 >
:<math>
<math>
《数学》
\begin{align}
\begin{align}
开始{ align }
0 & \equiv \delta \mathcal{L} \\
0 & \equiv \delta \mathcal{L} \\
0 & equiv delta mathcal { l }
&= \delta \left( - \sum_i k_\text{B} \rho_i \ln \rho_i \right) + \delta \left( \lambda_1 - \sum_i \lambda_1 \rho_i \right) + \delta \left( \lambda_2 U - \sum_i \lambda_2 \rho_i E_i \right) \\
&= \delta \left( - \sum_i k_\text{B} \rho_i \ln \rho_i \right) + \delta \left( \lambda_1 - \sum_i \lambda_1 \rho_i \right) + \delta \left( \lambda_2 U - \sum_i \lambda_2 \rho_i E_i \right) \\
& = delta left (- sum _ i k _ text { b } rho _ i ln rho _ i 右) + delta left (lambda _ 1-sum _ i lambda _ 1 rho _ i 右) + delta left (lambda _ 2 u-sum _ i lambda _ 2 rho _ i 右)
&= \sum_i \bigg[ \delta \Big( - k_\text{B} \rho_i \ln \rho_i \Big) + \delta \Big( \lambda_1 \rho_i \Big) + \delta \Big( \lambda_2 E_i \rho_i \Big) \bigg] \\
&= \sum_i \bigg[ \delta \Big( - k_\text{B} \rho_i \ln \rho_i \Big) + \delta \Big( \lambda_1 \rho_i \Big) + \delta \Big( \lambda_2 E_i \rho_i \Big) \bigg] \\
和 = sum _ i bigg [ delta Big (- k _ text { b } rho _ i ln rho _ i Big) + delta Big (lambda _ 1 rho _ i Big) + delta Big (lambda _ 2 e _ i rho _ i Big)]
&= \sum_i \left[ \frac{\partial}{\partial \rho_i } \Big( - k_\text{B} \rho_i \ln \rho_i \Big) \, \delta ( \rho_i ) + \frac{\partial}{\partial \rho_i } \Big( \lambda_1 \rho_i \Big) \, \delta ( \rho_i ) + \frac{\partial}{\partial \rho_i } \Big( \lambda_2 E_i \rho_i \Big) \, \delta ( \rho_i ) \right] \\
&= \sum_i \left[ \frac{\partial}{\partial \rho_i } \Big( - k_\text{B} \rho_i \ln \rho_i \Big) \, \delta ( \rho_i ) + \frac{\partial}{\partial \rho_i } \Big( \lambda_1 \rho_i \Big) \, \delta ( \rho_i ) + \frac{\partial}{\partial \rho_i } \Big( \lambda_2 E_i \rho_i \Big) \, \delta ( \rho_i ) \right] \\
我离开[ frac { partial } rho _ i } Big (- k text { b } rho _ i ln rho _ i Big) ，delta (rho _ i) + frac { partial } Big (lambda _ 1 rho _ i Big) ，delta (rho _ i) + frac { partial }{ partial rho _ i } Big (da _ 2 e _ i rho _ i Big) ，delta (rho _ i) right ]
&= \sum_i \bigg[ -k_\text{B} \ln \rho_i - k_\text{B} + \lambda_1 + \lambda_2 E_i \bigg] \, \delta ( \rho_i ) .
&= \sum_i \bigg[ -k_\text{B} \ln \rho_i - k_\text{B} + \lambda_1 + \lambda_2 E_i \bigg] \, \delta ( \rho_i ) .
和 = sum _ i bigg [-k _ text { b } ln rho _ i-k _ text { b } + lambda _ 1 + lambda _ 2 e _ i bigg ] ，delta (rho _ i)。
\end{align}
\end{align}
结束{ align }
</math>
</math>
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
由于这个方程对任何变化都适用
<math>
<math>
《数学》
\frac{\partial \mathcal{L}}{\partial \rho_i} = -k_\text{B} \ln \rho_i - k_\text{B} + \lambda_1 + \lambda_2 E_i \equiv 0.
0 \equiv -k_\text{B} \ln \rho_i - k_\text{B} + \lambda_1 + \lambda_2 E_i .
\frac{\partial \mathcal{L}}{\partial \rho_i} = -k_\text{B} \ln \rho_i - k_\text{B} + \lambda_1 + \lambda_2 E_i \equiv 0.
0 \equiv -k_\text{B} \ln \rho_i - k_\text{B} + \lambda_1 + \lambda_2 E_i .
1 + 2 e i equiv 0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.
0 equiv-k text { b } ln rho i-k text { b } + lambda 1 + lambda 2 e i.
</math>
</math>
Isolating for <math> \rho_i </math> yields
Isolating for <math> \rho_i </math> yields
孤立的 < math > rho _ i </math > 产量
<math>
<math>
《数学》
\rho_i = \exp \left( \frac{-k_\text{B} + \lambda_1 + \lambda_2 E_i}{k_\text{B}} \right).
\rho_i = \exp \left( \frac{-k_\text{B} + \lambda_1 + \lambda_2 E_i}{k_\text{B}} \right) .
\rho_i = \exp \left( \frac{-k_\text{B} + \lambda_1 + \lambda_2 E_i}{k_\text{B}} \right).
\rho_i = \exp \left( \frac{-k_\text{B} + \lambda_1 + \lambda_2 E_i}{k_\text{B}} \right) .
( frac {-k text { b } + lambda 1 + lambda 2 e i }{ k text { b }右)。
Rho _ i = exp left (frac {-k _ text { b } + lambda _ 1 + lambda _ 2 e _ i }{ k _ text { b }} right).
</math>
</math>
To obtain <math> \lambda_1 </math>, one substitutes the probability into the first constraint:
To obtain <math> \lambda_1 </math>, one substitutes the probability into the first constraint:
为了获得 math lambda 1 / math，可以将概率替换为第一个约束:
为了获得 < math > lambda _ 1 </math > ，我们用概率代替第一个约束:
<math>
<math>
《数学》
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
1 &= \sum_i \rho_i \\
1 &= \sum_i \rho_i \\
1 &= \sum_i \rho_i \\
1 &= \sum_i \rho_i \\
1 & sum i rho
1 & = sum i rho i
&= \exp \left( \frac{-k_\text{B} + \lambda_1}{k_\text{B}} \right) Z,
&= \exp \left( \frac{-k_\text{B} + \lambda_1}{k_\text{B}} \right) Z ,
&= \exp \left( \frac{-k_\text{B} + \lambda_1}{k_\text{B}} \right) Z,
&= \exp \left( \frac{-k_\text{B} + \lambda_1}{k_\text{B}} \right) Z ,
( frac {-k text { b } + lambda 1}{ k text { b }右) z,
和 = exp left (frac {-k _ text { b } + lambda _ 1}{ k _ text { b }右) z,
\end{align}
\end{align}
\end{align}
\end{align}
结束{ align }
</math>
</math>
where <math> Z </math> is a constant number defined as the canonical ensemble partition function:
where <math> Z </math> is a constant number defined as the canonical ensemble partition function:
其中，z </math > 是一个常数，定义为正则系综配分函数:
<math>
<math>
《数学》
Z \equiv \sum_i \exp \left( \frac{\lambda_2}{k_\text{B}} E_i \right).
Z \equiv \sum_i \exp \left( \frac{\lambda_2}{k_\text{B}} E_i \right) .
Z \equiv \sum_i \exp \left( \frac{\lambda_2}{k_\text{B}} E_i \right).
Z \equiv \sum_i \exp \left( \frac{\lambda_2}{k_\text{B}} E_i \right) .
Z 等价和 i 左(frac { lambda 2}{ k { b } e 右)。
Z equiv sum i exp left (frac { lambda _ 2}{ k _ text { b } e _ i right) .
</math>
</math>
Isolating for <math> \lambda_1 </math> yields <math> \lambda_1 = - k_B \ln(Z) + k_B </math>.
Isolating for <math> \lambda_1 </math> yields <math> \lambda_1 = - k_B \ln(Z) + k_B </math>.
分离出 < math > > lambda _ 1 </math > 产生 < math > > lambda _ 1 =-k _ b ln (z) + k _ b </math > 。
Rewriting <math> \rho_i </math> in terms of <math> Z </math> gives
Rewriting <math> \rho_i </math> in terms of <math> Z </math> gives
根据《数学》 ，重写数学
<math>
<math>
《数学》
\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) .
\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(frac { λ _ 2}{ k _ text { b }} e _ i right)。
</math>
</math>
Rewriting <math> S </math> in terms of <math> Z </math> gives
Rewriting <math> S </math> in terms of <math> Z </math> gives
根据《数学》重写《数学》 s </math >
<math>
<math>
《数学》
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
S &= - k_\text{B} \sum_i \rho_i \ln \rho_i \\
S &= - k_\text{B} \sum_i \rho_i \ln \rho_i \\
S &= - k_\text{B} \sum_i \rho_i \ln \rho_i \\
S &= - k_\text{B} \sum_i \rho_i \ln \rho_i \\
S & k { b } sum i rho ln i ho i
S & =-k text { b } sum i rho _ i ln rho _ i
&= - k_\text{B} \sum_i \rho_i \left( \frac{\lambda_2}{k_\text{B}} E_i - \ln(Z) \right) \\
&= - k_\text{B} \sum_i \rho_i \left( \frac{\lambda_2}{k_\text{B}} E_i - \ln(Z) \right) \\
&= - k_\text{B} \sum_i \rho_i \left( \frac{\lambda_2}{k_\text{B}} E_i - \ln(Z) \right) \\
&= - k_\text{B} \sum_i \rho_i \left( \frac{\lambda_2}{k_\text{B}} E_i - \ln(Z) \right) \\
左(frac { lambda _ 2}{ k _ text { b } e _ i-ln (z) right))
&= - \lambda_2 \sum_i \rho_i E_i + k_\text{B} \ln(Z) \sum_i \rho_i \\
&= - \lambda_2 \sum_i \rho_i E_i + k_\text{B} \ln(Z) \sum_i \rho_i \\
&= - \lambda_2 \sum_i \rho_i E_i + k_\text{B} \ln(Z) \sum_i \rho_i \\
&= - \lambda_2 \sum_i \rho_i E_i + k_\text{B} \ln(Z) \sum_i \rho_i \\
&= - \lambda_2 U + k_\text{B} \ln(Z).
&= - \lambda_2 U + k_\text{B} \ln(Z) .
&= - \lambda_2 U + k_\text{B} \ln(Z).
&= - \lambda_2 U + k_\text{B} \ln(Z) .
&- lambda 2 u + k text { b } ln (z).
& =-lambda _ 2 u + k _ text { b } ln (z).
\end{align}
\end{align}
\end{align}
\end{align}
结束{ align }
</math>
</math>
To obtain <math> \lambda_2 </math>, we differentiate <math> S </math> with respect to the average energy <math> U </math> and apply the first law of thermodynamics, <math> dU = T dS - P dV </math>:
To obtain <math> \lambda_2 </math>, we differentiate <math> S </math> with respect to the average energy <math> U </math> and apply the first law of thermodynamics, <math> dU = T dS - P dV </math>:
为了获得 < math > lambda 2 </math > ，我们根据平均能量 < math > u </math > 来区分 < math > s </math > 并应用能量守恒定律，< math > dU = t dS-p math </math > :
<math>
<math>
《数学》
\frac{dS}{dU} = -\lambda_2 \equiv \frac{1}{T}.
\frac{dS}{dU} = -\lambda_2 \equiv \frac{1}{T} .
\frac{dS}{dU} = -\lambda_2 \equiv \frac{1}{T}.
\frac{dS}{dU} = -\lambda_2 \equiv \frac{1}{T} .
2 equiv frac {1}{ t }.
</math>
</math>
Thus the canonical partition function <math> Z </math> becomes
Thus the canonical partition function <math> Z </math> becomes
因此，规范的配分函数/数学变成了
<math>
<math>
《数学》
Z \equiv \sum_i \mathrm{e}^{-\beta E_i},
Z \equiv \sum_i \mathrm{e}^{-\beta E_i} ,
Z \equiv \sum_i \mathrm{e}^{-\beta E_i},
Z \equiv \sum_i \mathrm{e}^{-\beta E_i} ,
Z \equiv \sum_i \mathrm{e}^{-\beta E_i},
Z \equiv \sum_i \mathrm{e}^{-\beta E_i} ,
</math>
</math>
where <math> \beta \equiv 1/(k_\text{B} T) </math> is defined as the thermodynamic beta. Finally, the probability distribution <math> \rho_i </math> and entropy <math> S </math> are respectively
where <math> \beta \equiv 1/(k_\text{B} T) </math> is defined as the thermodynamic beta. Finally, the probability distribution <math> \rho_i </math> and entropy <math> S </math> are respectively
其中 math beta equiv 1 / (k text { b } t) / math 被定义为热力学beta。最后，分别给出了概率分布数学和熵数学 s / math
其中 < math > beta equiv 1/(k text { b } t) </math > 被定义为热力学beta。最后，概率分布和熵分别是
<math>
<math>
《数学》
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\rho_i & = \frac{1}{Z} \mathrm{e}^{-\beta E_i}, \\
\rho_i & = \frac{1}{Z} \mathrm{e}^{-\beta E_i} , \\
\rho_i & = \frac{1}{Z} \mathrm{e}^{-\beta E_i}, \\
\rho_i & = \frac{1}{Z} \mathrm{e}^{-\beta E_i} , \\
1}{ z } mathrm { e } ^ {-beta e _ i } ,
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.
S & = \frac{U}{T} + k_\text{B} \ln Z .
2. s & = frac { u }{ t } + k text { b } ln z.
\end{align}
\end{align}
\end{align}
\end{align}
结束{ align }
</math>
</math>
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，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
: <math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math>
<math> Z = \frac{1}{h^3} \int \mathrm{e}^{-\beta H(q, p)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math>
< math > z = frac {1}{ h ^ 3} int mathrm { e } ^ {-beta h (q，p)} ，mathrm { d } ^ 3 q，mathrm { d } ^ 3 p，</math >
where
where
在哪里
在哪里
: <math> h </math> is the [[Planck constant]];
<math> h </math> is the Planck constant;
是普朗克常数;
math beta / math 是热力学beta，定义为 math tfrac {1}{ k text { b } t } / math,
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
是热力学beta，定义为 < math > tfrac
: <math> H(q, p) </math> is the [[Hamiltonian mechanics|Hamiltonian]] of the system;
<math> H(q, p) </math> is the Hamiltonian of the system;
H (q，p) </math > 是系统的哈密顿函数;
: <math> q </math> is the [[Canonical coordinates|canonical position]];
<math> q </math> is the canonical position;
Q </math > 是典型的位置
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
<math> p </math> is the canonical momentum.
<math> p </math> is the canonical momentum.
P 是典型的动量。
For a gas of ''N'' identical classical particles in three dimensions, the partition function is
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is
For a gas of N identical classical particles in three dimensions, the partition function is
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is
对于一种三维空间中的全同经典粒子气体，配分函数是
: <math>Z=\frac{1}{N!h^{3N}} \int \, \exp\biggl(-\beta \sum_{a=1}^N H(\textbf p_a, \textbf x_a)\biggr) \; \mathrm{d}^3p_1 \cdots \mathrm{d}^3p_N \, \mathrm{d}^3x_1 \cdots \mathrm{d}^3x_N </math>
: <math> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N </math>
<math>Z=\frac{1}{N!h^{3N}} \int \, \exp\biggl(-\beta \sum_{a=1}^N H(\textbf p_a, \textbf x_a)\biggr) \; \mathrm{d}^3p_1 \cdots \mathrm{d}^3p_N \, \mathrm{d}^3x_1 \cdots \mathrm{d}^3x_N </math>
<math> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N </math>
数学 zfrac {1}{ n! h ^ {3N } int， exp biggl (- beta sum { a 1} ^ n h ( textbf p a， textbf x a) biggr) ; mathrm { d ^ 3p1 cturm { d ^ 3p n， mathrm { d ^ 3x1 cturm { d } ^ 3x n / math
[数学] z = frac {1}{ n! h ^ {3N } int，exp left (- beta sum { i = 1} ^ n h (textbf q _ i，textbf p _ i) right) ; mathrm { d } ^ 3 q _ 1 cdots mathrm { d } ^ 3 q _ n，mathrm { d } ^ 3 p _ 1 cdots mathrm { d } ^ 3 p _ n </math >
: ''p<sub>i</sub>'' indicate particle momenta,
: <math> h </math> is the [[Planck constant]];
<math> h </math> is the Planck constant;
是普朗克常数;
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
是热力学beta，定义为 < math > tfrac
: <math> i </math> is the index for the particles of the system;
<math> i </math> is the index for the particles of the system;
是系统粒子的指数
: <math> H </math> is the [[Hamiltonian mechanics|Hamiltonian]] of a respective particle;
<math> H </math> is the Hamiltonian of a respective particle;
是一个粒子的哈密顿量;
: <math> q_i </math> is the [[Canonical coordinates|canonical position]] of the respective particle;
<math> q_i </math> is the canonical position of the respective particle;
Q _ i </math > 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 momentum of the respective particle;
是各个粒子的正则动量;
: <math> \mathrm{d}^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
<math> \mathrm{d}^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
是一个简写符号，用来表示 < math > q _ i </math > 和 < math > p _ i </math > 是三维空间中的向量。
The reason for the factorial factor N! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h<sup>3N</sup> (where h is usually taken to be Planck's constant).
The reason for the factorial factor N! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h<sup>3N</sup> (where h is usually taken to be Planck's constant).
阶乘因子 n 的原因！将在下面讨论。在分母中引入了额外的常数因子，因为与离散形式不同，上面显示的连续形式不是无量纲的。正如前面的章节所说，为了使它成为一个无量纲量，我们必须将它除以 h sup 3N / sup (h 通常被认为是普朗克常数)。
阶乘因子 n 的原因！下面将讨论。在分母中引入了额外的常数因子，因为与离散形式不同，上面显示的连续形式不是无量纲的。正如前面的章节所说，为了使它成为一个无量纲量，我们必须用 h < sup > 3N </sup > (h 通常被认为是普朗克常数)来除以它。
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:
对于量子力学和离散的正则系综，典型的配分函数被定义为玻尔兹曼因子的轨迹:
: <math> Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), </math>
<math> Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ), </math>
[ math > z = operatorname { tr }(mathrm { e } ^ {-beta hat { h }) ，</math >
: <math> Z = \operatorname{tr} (\mathrm{e}^{-\beta \hat{H}}), </math>
where:
where:
在哪里:
: <math> \operatorname{tr} ( \circ ) </math> is the [[trace (linear algebra)|trace]] of a matrix;
<math> \operatorname{tr} ( \circ ) </math> is the trace of a matrix;
是矩阵的轨迹;
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>,
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>,
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
是热力学beta，定义为 < math > tfrac
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]].
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]].
<math> \hat{H} </math> is the Hamiltonian operator.
<math> \hat{H} </math> is the Hamiltonian operator.
是哈密尔顿算符。
The [[dimension]] of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of [[energy eigenstates]] of the system.
The [[dimension]] of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of [[energy eigenstates]] of the system.
The dimension of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of energy eigenstates of the system.
The dimension of <math> \mathrm{e}^{-\beta \hat{H}} </math> is the number of energy eigenstates of the system.
系统的能量本征态个数是系统的能量本征态个数。
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
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> Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \, \mathrm{d} q \, \mathrm{d} p, </math>
<math> Z = \frac{1}{h} \int \langle q, p | \mathrm{e}^{-\beta \hat{H}} | q, p \rangle \, \mathrm{d} q \, \mathrm{d} p, </math>
1}{ h } int langle q，p | mathrm { e } ^ {-beta hat { h } | q，p rangle，mathrm { d } q，mathrm { d } p，</math >
在哪里
where:
where:
在哪里:
: <math> h </math> is the [[Planck constant]],
: <math> h </math> is the [[Planck constant]];
<math> h </math> is the Planck constant,
<math> h </math> is the Planck constant;
是普朗克常数;
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>,
: <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>,
<math> \beta </math> is the thermodynamic beta, defined as <math> \tfrac{1}{k_\text{B} T} </math>;
是热力学beta，定义为 < math > tfrac
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]],
: <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]];
<math> \hat{H} </math> is the Hamiltonian operator,
<math> \hat{H} </math> is the Hamiltonian operator;
哈密尔顿算符是哈密尔顿算符
: <math> q </math> is the [[Canonical coordinates|canonical position]],
: <math> q </math> is the [[Canonical coordinates|canonical position]];
<math> q </math> is the canonical position,
<math> q </math> is the canonical position;
Q </math > 是典型的位置
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
: <math> p </math> is the [[Canonical coordinates|canonical momentum]].
<math> p </math> is the canonical momentum.
<math> p </math> is the canonical momentum.
是典型的动量。
In systems with multiple quantum states s sharing the same energy E<sub>s</sub>, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows:
In systems with multiple quantum states s sharing the same energy E<sub>s</sub>, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows:
在具有多个量子态共享相同能量的系统中，系统的能级是简并的。在简并能级的情况下，我们可以用能级的贡献来表示配分函数，如下:
<math> Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j},</math>
<math> Z = \sum_j g_j \cdot \mathrm{e}^{-\beta E_j},</math>
[数学，数学]
where g<sub>j</sub> is the degeneracy factor, or number of quantum states s that have the same energy level defined by E<sub>j</sub> = E<sub>s</sub>.
where g<sub>j</sub> is the degeneracy factor, or number of quantum states s that have the same energy level defined by E<sub>j</sub> = E<sub>s</sub>.
其中 g 子 j / sub 是简并因子，或者具有 e 子 j / sub e 子 s / sub 定义的相同能级的量子态的数目。
其中 g < sub > j </sub > 是简并因子，或者 e < sub > j </sub > = e < sub > s </sub > 定义的具有相同能级的量子态的数目。
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):
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跟踪(这与基的选择无关) :
<math>Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ),</math>
<math>Z = \operatorname{tr} ( \mathrm{e}^{-\beta \hat{H}} ),</math>
[ math > z = operatorname { tr }(mathrm { e } ^ {-beta hat { h }) ，</math >
The classical form of ''Z'' is recovered when the trace is expressed in terms of [[coherent state]]s<ref>{{cite book |first1=John R. |last1=Klauder |first2=Bo-Sture |last2=Skagerstam
The classical form of ''Z'' is recovered when the trace is expressed in terms of [[coherent state]]s<ref>{{cite book |first1=John R. |last1=Klauder |first2=Bo-Sture |last2=Skagerstam
The classical form of Z is recovered when the trace is expressed in terms of coherent states<ref>{{cite book |first1=John R. |last1=Klauder |first2=Bo-Sture |last2=Skagerstam
The classical form of Z is recovered when the trace is expressed in terms of coherent states
当轨迹用相干态表示时，恢复了 z 的经典形式
|title=Coherent States: Applications in Physics and Mathematical Physics
|title=Coherent States: Applications in Physics and Mathematical Physics
|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>
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
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:
:<math>
:<math>
\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},
</math>
</math>
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
Category:Concepts in physics
Category:Concepts in physics
分类: 物理概念
分类: 物理概念
position ''x'' and momentum ''p''. Thus
<noinclude>
<noinclude>