1549380323-Statistical Mechanics Theory and Molecular Simulation

328 Free energy calculations

=

CN

QA(N,V,T)

∫

dx 0 e−βH(xτ(x^0 ),τ). (8.4.8)

Now let us change variables from x 0 to xτ(x 0 ) in the integral in eqn. (8.4.8). Since
the trajectory xt(x 0 ) is generated from Hamilton’s equations of motion, the mapping
of x 0 to xτ(x 0 ) is unique. Moreover, by Liouville’s theorem, the phase space measure
satisfies dxτ= dx 0. Therefore, we find that

〈

e−βWAB

〉

A=

CN

QA(N,V,T)

∫

dxτe−βHB(xτ)

=

QB(N,V,T)

QA(N,V,T)

= e−β∆AAB, (8.4.9)

which is Jarzynski’s equality. Note that by Jensen’s inequality,
〈
e−βWAB

〉

A≥e

−β〈WAB〉A, (8.4.10)

which implies that
e−β∆AAB≥e−β〈WAB〉A. (8.4.11)

Taking the natural log of both sides of eqn. (8.4.11) leads to the work–free–energy
inequality.
We now present a proof of Jarzynski’s equality that is relevant for finite systems
coupled to thermostats typically employed in molecular dynamics. Theoriginal version
of this derivation is due to Cuendet (2006) and was subsequently generalized by Sch ̈oll-
Paschinger and Dellago (2006). The proof does not depend criticallyon the particular
thermostatting mechanism as long as the scheme rigorously generates a canonical
distribution. For notational simplicity, we will employ the Nos ́e–Hoover scheme in
eqns. (4.8.19). Although we already know that the Nos ́e–Hoover equations have many
weaknesses which can be fixed using, for example, Nos ́e–Hoover chains, the former is
sufficient for our purpose here and keeps the notation simpler. We start, therefore,
with the equations of motion

r ̇i=

pi

mi

p ̇i=−

∂

∂ri

U(r,t)−

pη

Q

pi

η ̇=

pη

Q

p ̇η=

∑

i

p^2 i

mi

− 3 NkT, (8.4.12)

which are identical to eqns. (4.8.19) except for the time-dependent potentialU(r,t)
and the choice ofd= 3 for the dimensionality. As was discussed in Section 4.9, these