# 玻尔兹曼分布

（重定向自波茲曼分布

${\displaystyle p_{i}\propto e^{-{\varepsilon _{i}}/{(kT)}}}$

${\displaystyle {\frac {p_{i}}{p_{j}}}=e^{{(\varepsilon _{j}-\varepsilon _{i})}/{(kT)}}}$

## 分布形式

${\displaystyle p_{i}={\frac {1}{Q}}}{e^{-{\varepsilon }_{i}/(kT)}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}$

${\displaystyle Q={\sum _{i=1}^{M}{e^{-{\varepsilon }_{i}/(kT)}}}}$

${\displaystyle H(p_{1},p_{2},\cdots ,p_{M})=-\sum _{i=1}^{M}p_{i}\log _{2}p_{i}}$

${\displaystyle {\frac {p_{i}}{p_{j}}}=e^{({\varepsilon }_{j}-{\varepsilon }_{i})/(kT)}}$

${\displaystyle p_{i}={\frac {N_{i}}{N}}}$

${\displaystyle {\frac {N_{i}}{N}}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}$

## 參考文獻

1. ^ Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich. Statistical Physics. Course of Theoretical Physics 5 3. Oxford: Pergamon Press. 1980 [1976]. ISBN 0-7506-3372-7. Translated by J.B. Sykes and M.J. Kearsley. See section 28
2. ^ Boltzmann, Ludwig. Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten [Studies on the balance of living force between moving material points]. Wiener Berichte. 1868, 58: 517–560.
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4. ^
5. ^ Gao, Xiang; Gallicchio, Emilio; Roitberg, Adrian. The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy. The Journal of Chemical Physics. 2019, 151 (3): 034113. PMID 31325924. . doi:10.1063/1.5111333.
6. Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York
7. McQuarrie, A. Statistical Mechanics. Sausalito, CA: University Science Books. 2000. ISBN 1-891389-15-7.
8. ^ NIST Atomic Spectra Database Levels Form页面存档备份，存于互联网档案馆） at nist.gov
9. ^ Atkins, P. W.; de Paula, J. Physical Chemistry 9th. Oxford: Oxford University Press. 2009. ISBN 978-0-19-954337-3.