# 自由能微扰

$\Delta F(A \rightarrow B) = F_B - F_A = -k_B T \ln \left \langle \exp \left ( - \frac{H_B - H_A}{k_B T} \right ) \right \rangle _A$

## 原理

$Q_A = \int d\textbf{r}^N \textbf{p}^N e^{-\beta H_A(\textbf{r}^N,\textbf{p}^N)},$

\begin{align} 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{align}

$\frac{Q_B}{Q_A} = \langle e^{-\beta [H_B- H_A]}\rangle_A$.

\begin{align} \Delta F(A \rightarrow B) &= F_B - F_A \\ & = - (k_BT \ln Q_B - k_BT\ln Q_A) \\ & = -k_BT \ln \frac{Q_B}{Q_A} \end{align}

$\Delta F(A \rightarrow B) = F_B - F_A = -k_B T \ln \left \langle \exp \left ( - \frac{H_B - H_A}{k_B T} \right ) \right \rangle _A$

## 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.