# 熵 (统计物理学)

## 吉布斯熵公式

${\displaystyle S=-k_{\text{B}}\,\sum _{i}p_{i}\ln \,p_{i}}$

${\displaystyle dS=-k_{\text{B}}\,\sum _{i}dp_{i}\ln p_{i}}$
${\displaystyle \,\,\,=-k_{\text{B}}\,\sum _{i}dp_{i}(-E_{i}/k_{\text{B}}T-\ln Z)}$
${\displaystyle \,\,\,=\sum _{i}E_{i}dp_{i}/T}$
${\displaystyle \,\,\,=\sum _{i}[d(E_{i}p_{i})-(dE_{i})p_{i}]/T}$

${\displaystyle \sum _{i}dp_{i}=0}$

${\displaystyle \sum _{i}d(E_{i}p_{i})}$

${\displaystyle \sum _{i}p_{i}dE_{i}}$

${\displaystyle dE=\delta w+\delta q}$

${\displaystyle dS={\frac {\delta \langle q_{\text{rev}}\rangle }{T}}}$

## 玻尔兹曼原理

${\displaystyle S=k_{\text{B}}\ln \Omega }$

${\displaystyle k_{\text{B}}}$玻尔兹曼常数，以及
${\displaystyle \Omega }$ 为给定宏观态对应的微观态的数量。

### 系综

${\displaystyle S=k_{\text{B}}\ln \Omega _{\rm {mic}}=k_{\text{B}}(\ln Z_{\rm {can}}+\beta {\bar {E}})=k_{\text{B}}(\ln {\mathcal {Z}}_{\rm {gr}}+\beta ({\bar {E}}-\mu {\bar {N}}))}$

${\displaystyle \Omega _{\rm {mic}}}$微正则配分函数
${\displaystyle Z_{\rm {can}}}$正则配分函数
${\displaystyle {\mathcal {Z}}_{\rm {gr}}}$巨正则配分函数

## 认识的匮乏和热力学第二定律

“任何孤立的热力学系统的总熵都会随着时间的推移而增加，接近最大值。”

## 微观态的计数

${\displaystyle S=-k_{\rm {B}}H_{\rm {B}}:=-k_{\rm {B}}\int f(q_{i},p_{i})\,\ln f(q_{i},p_{i})\,dq_{1}dp_{1}\cdots dq_{N}dp_{N}}$

${\displaystyle E_{\nu }=h\nu _{0}(n+{\begin{matrix}{\frac {1}{2}}\end{matrix}})}$

## 参考文献

1. E.T. Jaynes; Gibbs vs Boltzmann Entropies; American Journal of Physics, 391, 1965
2. ^ E. T. Jaynes, Gibbs vs Boltzmann Entropies, American Journal of Physics 33, 391 (1965); https://doi.org/10.1119/1.1971557
• Boltzmann, Ludwig (1896, 1898). Vorlesungen über Gastheorie : 2 Volumes - Leipzig 1895/98 UB: O 5262-6. English version: Lectures on gas theory. Translated by Stephen G. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5