# 自由能微扰

${\displaystyle \Delta F(A\rightarrow B)=F_{B}-F_{A}=-k_{B}T\ln \left\langle \exp \left(-{\frac {H_{B}-H_{A}}{k_{B}T}}\right)\right\rangle _{A}}$

## 原理

${\displaystyle Q_{A}=\int d{\textbf {r}}^{N}{\textbf {p}}^{N}e^{-\beta H_{A}({\textbf {r}}^{N},{\textbf {p}}^{N})},}$

{\displaystyle {\begin{aligned}Q_{B}&=\int d{\textbf {r}}^{N}{\textbf {p}}^{N}e^{-\beta H_{B}({\textbf {r}}^{N},{\textbf {p}}^{N})}\\&=\int d{\textbf {r}}^{N}{\textbf {p}}^{N}e^{-\beta H_{A}({\textbf {r}}^{N},{\textbf {p}}^{N})}e^{-\beta [H_{B}({\textbf {r}}^{N},{\textbf {p}}^{N})-H_{A}({\textbf {r}}^{N},{\textbf {p}}^{N})]}\\&=Q_{A}\int d{\textbf {r}}^{N}{\textbf {p}}^{N}{\frac {e^{-\beta [H_{B}({\textbf {r}}^{N},{\textbf {p}}^{N})-H_{A}({\textbf {r}}^{N},{\textbf {p}}^{N})]}}{Q_{A}}}\\&=Q_{A}\langle e^{-\beta [H_{B}-H_{A}]}\rangle _{A}\end{aligned}}}

${\displaystyle {\frac {Q_{B}}{Q_{A}}}=\langle e^{-\beta [H_{B}-H_{A}]}\rangle _{A}}$.

{\displaystyle {\begin{aligned}\Delta F(A\rightarrow B)&=F_{B}-F_{A}\\&=-(k_{B}T\ln Q_{B}-k_{B}T\ln Q_{A})\\&=-k_{B}T\ln {\frac {Q_{B}}{Q_{A}}}\end{aligned}}}

${\displaystyle \Delta F(A\rightarrow B)=F_{B}-F_{A}=-k_{B}T\ln \left\langle \exp \left(-{\frac {H_{B}-H_{A}}{k_{B}T}}\right)\right\rangle _{A}}$

## 参考资料

1. ^ Zwanzig, R. W. J. Chem. Phys. 1954, 22, 1420-1426. doi:10.1063/1.1740409
2. ^ http://www.ambermd.org
3. ^ Pohorille A, Jarzynski C, Chipot C J Phys Chem B. 2010 Aug 19;114(32):10235-53. doi: 10.1021/jp102971x.