1549380323-Statistical Mechanics Theory and Molecular Simulation

332 Free energy calculations

Wt

P(Wt )e

• Wt

P(Wt )e P(Wt )

-3Wt

Fig. 8.4Shift in a Gaussian work distribution as a result of the multiplication by exp(−βWτ)
for various values ofβ.

ity distributionP(Wτ) and occur only rarely. Consequently, many “fast-switching”
trajectories with high pulling ratesvare needed in order to sample the tail of the
work distribution adequately. Alternatively, trajectories with very slow pulling rates
(“slow-switching” trajectories) could be used, in which case a smaller ensemble can
be employed; however, the method will then have an efficiency comparable to equilib-
rium methods (Oberhoferet al., 2005). It is worth noting that if the work distribution
P(Wτ) is Gaussian or nearly Gaussian, then the exponential average canbe computed
reliably using a truncated cumulant expansion (see eqns. (4.7.21) and (4.7.23)):

ln

〈

e−βWτ

〉

≈−β〈Wτ〉+

β^2

2

(

〈Wτ^2 〉−〈Wτ〉^2

)

, (8.4.24)

which eliminates the problem of poor overlap betweenP(Wτ) andP(Wτ) exp(−βWτ)
in the exponential average.
A second challenge raises a more fundamental question concerningthe proof of the
equality. In the proof we have presented, it is tacitly assumed thatafter the transfor-
mation x 0 to xτis made, the integral that results constitutes an actual equilibrium
partition function (see eqn. (8.4.9)). That is, we assume an equilibrium distribution
of phase space points xτ, but if the the driving force is too strong, this might not be
valid in a finite system. Certainly, if the driving force is sufficiently mild tomaintain
the system close to equilibrium along the time-dependent path between statesAand
B, then the assumption is valid, and indeed, in this limit, the Jarzynski equality seems
to be most effective in actual applications (Oberhoferet al., 2005). Away from this
limit, it might be necessary to allow the final states xτ(x 0 ) to relax to an equilibrium