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第12行： − − − − − <math>p_i \propto e^{-\frac{\varepsilon_i}{kT}}</math> − − [拉丁语] + − where {{mvar|p<sub>i</sub>}} is the probability of the system being in state {{mvar|i}}, {{mvar|ε<sub>i</sub>}} is the energy of that state, and a constant {{mvar|kT}} of the distribution is the product of [[Boltzmann's constant]] {{mvar|k}} and [[thermodynamic temperature]] {{mvar|T}}. The symbol <math display="inline">\propto</math> denotes [[proportionality (mathematics)|proportionality]] (see {{section link||The distribution}} for the proportionality constant). 第33行：
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第34行： − :<math>\frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}}</math>+ − − − [数学][数学] + − The Boltzmann distribution is named after [[Ludwig Boltzmann]] who first formulated it in 1868 during his studies of the [[statistical mechanics]] of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"<ref>http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf</ref> 第56行：
第44行： − <!-- − − <!-- − − <!-- − − It would be nice to have a citation here! The origin of the Boltzmann factor isn't entirely clear. According to some authors, Boltzmann's 1968 paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” − − It would be nice to have a citation here! The origin of the Boltzmann factor isn't entirely clear. According to some authors, Boltzmann's 1968 paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” − − 如果能在这里引用一下就好了！玻尔兹曼因子的起源并不完全清楚。根据一些作者的说法，Boltzmann 在1968年发表的论文《论热力学第二基本定理与热平衡条件的概率计算之间的关系》 − − "Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten" is the origin but I can't find this article at the moment, − − "Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten" is the origin but I can't find this article at the moment, − − "Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten" is the origin but I can't find this article at the moment, − − so I cannot confirm. − − so I cannot confirm. − − 所以我不能确定。 − − For example, this book says so, but uses suspiciously modern terminology − − For example, this book says so, but uses suspiciously modern terminology − − 例如，这本书是这么说的，但是使用了可疑的现代术语 − − http://books.google.es/books?id=u13KiGlz2zcC&lpg=PA92&ots=8H1DRURdxn&pg=PA93#v=onepage&f=false − − http://books.google.es/books?id=u13KiGlz2zcC&lpg=PA92&ots=8H1DRURdxn&pg=PA93#v=onepage&f=false − − Http://books.google.es/books?id=u13kiglz2zcc&lpg=pa92&ots=8h1drurdxn&pg=pa93#v=onepage&f=false − − On the other hand, Uffink's "Compendium of the foundations of classical statistical physics" does not seem to indicate quite this equation but rather that Boltzmann's 1968 distribution was the simple Maxwell–Boltzmann distribution (for a classical nonrelativistic gas), modified for particles in a potential. − − On the other hand, Uffink's "Compendium of the foundations of classical statistical physics" does not seem to indicate quite this equation but rather that Boltzmann's 1968 distribution was the simple Maxwell–Boltzmann distribution (for a classical nonrelativistic gas), modified for particles in a potential. − − 另一方面，Uffink 的“经典统计物理学基础纲要”似乎并没有完全表明这个方程，而是说 Boltzmann 在1968年的分布是简单的 Maxwell-波兹曼分布(用于经典的非气体) ，修改为势中的粒子。 − − --> − − --> − − --> 第110行：
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第88行： − 数学 − − </math> 第190行：
第124行： − − <nowiki>{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}</nowiki> − − <nowiki>{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}</nowiki> − − − − <nowiki></math></nowiki> − − 数学
无编辑摘要
在统计力学和数学中，波兹曼分布分布(也称为吉布斯分布<ref name="landau" />)是一个概率分布或机率量测，它给出了一个系统处于某种量子态的概率，这个概率是该状态的能量和系统温度的函数。分布情况以下列形式表示:
在统计力学和数学中，波兹曼分布分布(也称为吉布斯分布<ref name="landau" />)是一个概率分布或机率量测，它给出了一个系统处于某种量子态的概率，这个概率是该状态的能量和系统温度的函数。分布情况以下列形式表示:
:<math>p_i \propto e^{-\frac{\varepsilon_i}{kT}}</math>
:<math>p_i \propto e^{-\frac{\varepsilon_i}{kT}}</math>
where {{mvar|p<sub>i</sub>}} is the probability of the system being in state {{mvar|i}}, {{mvar|ε<sub>i</sub>}} is the energy of that state, and a constant {{mvar|kT}} of the distribution is the product of [[Boltzmann's constant]] {{mvar|k}} and [[thermodynamic temperature]] {{mvar|T}}. The symbol <math display="inline">\propto</math> denotes [[proportionality (mathematics)|proportionality]] (see {{section link||The distribution}} for the proportionality constant).
这里的术语系统含义广泛; 它可以从单个原子到宏观系统，如天然气储罐。正因为如此，波兹曼分布可以用来解决各种各样的问题。玻尔兹曼分布表明，能量较低的状态被占据的概率总是较高的。
这里的术语系统含义广泛; 它可以从单个原子到宏观系统，如天然气储罐。正因为如此，波兹曼分布可以用来解决各种各样的问题。玻尔兹曼分布表明，能量较低的状态被占据的概率总是较高的。
<math>\frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}}</math>
The Boltzmann distribution is named after [[Ludwig Boltzmann]] who first formulated it in 1868 during his studies of the [[statistical mechanics]] of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"<ref>http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf</ref>
The distribution was later investigated extensively, in its modern generic form, by [[Josiah Willard Gibbs]] in 1902.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]]
The distribution was later investigated extensively, in its modern generic form, by [[Josiah Willard Gibbs]] in 1902.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]]
这种分布后来在1902年由约西亚·威拉德·吉布斯进行了广泛的调查，以其现代通用形式。
这种分布后来在1902年由约西亚·威拉德·吉布斯进行了广泛的调查，以其现代通用形式。
|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>{{rp|Ch.IV}}、
|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>
在玻尔兹曼分布被发明之后，约西亚·华纳德·吉布森充分地研究了它并在1902年提出了它的一般形式。<ref name="gibbs" />
在玻尔兹曼分布被发明之后，约西亚·华纳德·吉布森充分地研究了它并在1902年提出了它的一般形式。<ref name="gibbs" />
值得一提的是，玻尔兹曼分布不应与麦克斯韦-玻尔兹曼分布混淆。前者给出了系统处于某种状态的概率，作为该状态能量的函数; 后者则是用于描述理想气体中的粒子速度。
值得一提的是，玻尔兹曼分布不应与麦克斯韦-玻尔兹曼分布混淆。前者给出了系统处于某种状态的概率，作为该状态能量的函数; 后者则是用于描述理想气体中的粒子速度。
==分布==
==分布==
</math>
</math>
where ''p<sub>i</sub>'' is the probability of state ''i'', ''ε<sub>i</sub>'' the energy of state ''i'', ''k'' the Boltzmann constant, ''T'' the temperature of the system and ''M'' is the number of all states accessible to the system of interest.<ref name="McQuarrie, A. 2000" /><ref name="Atkins, P. W. 2010" /> Implied parentheses around the denominator ''kT'' are omitted for brevity. The normalization denominator ''Q'' (denoted by some authors by ''Z'') is the [[canonical partition function]]
where ''p<sub>i</sub>'' is the probability of state ''i'', ''ε<sub>i</sub>'' the energy of state ''i'', ''k'' the Boltzmann constant, ''T'' the temperature of the system and ''M'' is the number of all states accessible to the system of interest.<ref name="McQuarrie, A. 2000" /><ref name="Atkins, P. W. 2010" /> Implied parentheses around the denominator ''kT'' are omitted for brevity. The normalization denominator ''Q'' (denoted by some authors by ''Z'') is the [[canonical partition function]]
</math>
</math>
where p<sub>i</sub> is the probability of state i, p<sub>j</sub> the probability of state j, and ε<sub>i</sub> and ε<sub>j</sub> are the energies of states i and j, respectively.
where p<sub>i</sub> is the probability of state i, p<sub>j</sub> the probability of state j, and ε<sub>i</sub> and ε<sub>j</sub> are the energies of states i and j, respectively.