330 Free energy calculations
〈
e−βWAB〉
A=
1
ZT(0)
∫
dxτe^3 Nητe−βWτ(x^0 (xτ))×δ(H′(xτ,τ)−Wτ(x 0 (xτ))−C). (8.4.18)The integration overητcan now be performed by noting that theδ-function requires
ητ=1
3 NkT[
C−H(rτ,pτ)−p^2 η,τ
2 Q+Wτ(x 0 (xτ))]
. (8.4.19)
Using eqn. (8.4.19), together with eqn. (4.8.6), to perform the integration overητ
causes the exponential factors ofWτ(x 0 (xτ)) to cancel, yielding
〈
e−βWAB〉
A=
1
ZT(0)
eβC
3 NkT∫
dpη,τe−βp(^2) η,τ/ 2 Q
∫
dNpτdNrτe−βH(rτ,pτ)∝
1
ZT(0)
QB(N,V,T), (8.4.20)
where, since the integration is performed using the phase space vector and Hamiltonian
att=τ, the result is proportional to the canonical partition function in stateB. In
fact, when eqn. (8.4.16) forZT(0) is substituted into eqn. (8.4.20), the prefactors
cancel, yielding simply
〈
e−βWAB〉
A=
QB
QA
= e−β∆AAB, (8.4.21)which, again, is Jarzynski’s equality.
Jarzynski’s equality has an intriguing connection with mechanical pulling experi-
ments involving laser trapping (Liphardtet al., 2002) as suggested by Hummer and
Szabo (2001), or atomic force microscopy (Binniget al., 1986) applied, for example, to
biomolecules (Fernandez and Li, 2004). Experiments such as thesecan be mimicked in
molecular dynamics calculations (Parket al., 2003; Park and Schulten, 2004). For in-
stance, suppose we wish to unfold a protein or polypeptide by “pulling” on the ends as
illustrated in Fig. 8.3. Within the Jarzynski framework, we could perform nonequilib-
rium calculations and obtain the free energy change ∆Aassociated with the unfolding
process. This could be accomplished by adding a time-dependent term to the potential
that “drives” the distance between the two ends of the molecule from its (small) value
in the folded state to a final (large) value in the unfolded state. Forconcreteness, let
us designate the atomic coordinates at the ends of the molecule byr 1 andrN. (In
practice,r 1 andrNcould be nitrogen atoms at the N- and C-termini of a protein or
polypeptide.) The time-dependent potential would then take the form
U(r 1 ,...,rN,t) =U 0 (r 1 ,...,rN) +1
2
κ(|r 1 −rN|−req−vt)^2 , (8.4.22)whereU 0 is the internal potential described, for example, by a force field. The second
term in eqn. (8.4.22) is a harmonic potential with force constantκthat drives the