value $A$ to a final value $B$. $W$ is the external work performed on the system during one realization; $\Delta F = F_B-F_A$ is the free energy difference between two equilibrium states of the system, corresponding to $\lambda=A$ and $B$; and $\beta$ is the inverse temperature of a heat reservoir with which the system is equilibrated prior to the start of each realization of the process. A sample of derivations of Eq. \[eq:nwt0\] can be found in Refs. [@Jarzynski97; @Crooks98; @Crooks99; @HummerSzabo01; @Neal01; @Sun03; @Evans03; @Oberhofer05; @Jarzynski04; @ImparatoPeliti05]; pedagogical and review treatments are given in Refs. [@Jarzynski.Ladek.02; @frenkelSmit; @Ritort03; @ParkSchulten04; @HummerSzabo05; @BustamanteLiphardtRitort05]; for experimental tests of this and closely related results, see Refs. [@Liphardt_etal02; @DouarcheCilibertoPetrosyanRabbiosi05; @DouarcheCilibertoPetrosyan05; @Collin05]; finally, Refs. [@Yukawa00; @Kurchan00; @Mukamel03; @deRoeck04; @Monnai05] discuss quantal versions of the theorem.

In principle, Eq.\[eq:nwt0\] implies that $\Delta F$ can be estimated using nonequilibrim experiments or numerical simulations. If we repeat the thermodynamic process $N$ times, and observe work values $W_1,\,W_2,\cdots,\,W_N$, then $$\label{eq:estDF}
\Delta F \approx -\beta^{-1}\ln
\Bigl[
\frac{1}{N} \sum_{n=1}^N
e^{-\beta W_n}
\Bigr],$$ where the approximation becomes an equality in the limit of infinitely many realizations, $N\rightarrow\infty$. In practice, the average of $e^{-\beta W}$ is often dominated by very rare realizations, leading 