Jarzynski’s equality 331fixedvFig. 8.3Extension of deca-alanine by pulling via eqn. (8.4.22) (reprinted with permission
from Parket al.,J. Chem. Phys. 119 , 3559 (2003), copyright American Institute of Physics).
end-to-end distance|r 1 −rN|away from its equilibrium value at the folded statereq
by means of the time-dependent termvt, wherevis the pulling rate. In practice,
applying Jarzynski’s formula to such a problem requires generating an ensemble of
initial conditions x 0 and then performing the pulling “experiment” in order to obtain
a work valueWτ(x 0 ) for each chosen x 0. The final average of exp(−βWτ(x 0 )) then
leads to the free energy difference.
Several challenges arise in the use of Jarzynski’s formula. First, while it is an elegant
approach to free energy calculations, a potential bottleneck needs to be considered. The
work values that are generated from an ensemble of nonequilibrium processes have a
distributionP(Wτ). Thus, we could imagine calculating the average exp(−βWτ) using
this distribution according to
〈
e−βWτ〉
=
∫
dWτP(Wτ)e−βWτ. (8.4.23)However, as illustrated in Fig. 8.4,P(Wτ) andP(Wτ) exp(−βWτ) can peak at very
different locations, depending on the value ofβ. In Fig. 8.4, the average is dominated
by small values ofWτ, which lie predominantly in the left tail of the work probabil-