In 2003 Sun published \cite{Sun1} a method to achieve better statistics in free energy simulations using Jarzynski's theorem.
This method, based on a kind of thermodynamic integration \cite{Kirkwood1} for Jarzynski's equation, will be described in the
following.

Setting out from Jarzynski's equality (see section \ref{jarzgeneral}),
\begin{equation}
	\label{effjarzint}
	e^{-\beta \Delta F} = \frac{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t) e^{-\beta W(t)}}{\int d\mathbf{z}_0 \, \rho_0
	(\mathbf{z}_0)} ,
\end{equation}
we will sketch the derivation of Sun's average. Defining
\begin{equation}
	\label{sunf}
	f(\alpha)=\frac{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t) e^{-\alpha \beta W(t)}}{\int d\mathbf{z}_0 \, \rho_0(\mathbf{z}_0)} ,
\end{equation}
we see, that $\Delta F$ is obtained for$\alpha = 1$. The derivation of $f$ with respect to $\alpha$ is then:
\begin{equation}
	\label{fabl}
	\frac{\partial f}{\partial \alpha} = -\beta \frac{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t) W(t) e^{-\alpha \beta W(t)}}
	{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t) e^{-\alpha \beta W(t)}} \frac{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t)
	e^{-\alpha \beta W(t)}}{\int d\mathbf{z}_0 \, \rho_0(\mathbf{z}_0)} .
\end{equation}
Denoting
\begin{equation}
	\label{Walpha}
	\langle W(t)\rangle_\alpha = -\beta \frac{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t) W(t) e^{-\alpha \beta W(t)}}
	{\int d\mathbf{Z}_t \, P(\mathbf{Z}_t) e^{-\alpha \beta W(t)}} ,
\end{equation}
we can rewrite Eq. (\ref{fabl}) as ordinary differential equation
\begin{equation}
	\label{fabl2}
	\frac{\partial f}{\partial \alpha} = -\beta \langle W(t)\rangle_\alpha f(\alpha) ,
\end{equation}
with the solution
\begin{equation}
	\label{fabl2sol}
	f(\alpha)= \exp\left\{-\beta \int_{0}^{\alpha} d\alpha^{\prime} \, \langle W(t) \rangle_{\alpha^{\prime}} \right\} .
\end{equation}
Remembering that $f(1)=e^{-\beta \Delta F}$ we get from Eq. (\ref{fabl2sol}):
\begin{equation}
	\label{sunsint}
	\Delta F = \int_0^1 d\alpha \, \langle W(t) \rangle_\alpha .
\end{equation}
Eq. (\ref{sunsint}) gives the free energy difference as an ordinary average over the work with respect to a so called work
weighted ensemble $P(\mathbf{Z}_t)e^{-\alpha \beta W(t)}$, thus eliminating the problem of bad overlapping functions (see
section \ref{jarzgeneral}). The sampling of trajectories with appropriate probabibility, namely $P(\mathbf{Z}_t)
e^{-\alpha \beta W(t)}$ can be achieved by using the transition path sampling technique described later.

Sun also showed \cite{Sun1} that in general his averaging method is less efficient than traditional sampling methods, such as
for example thermodynamic integration.