326 Free energy calculations
which states that if an amount of workWABtakes a system from stateAto stateB,
thenWAB≥∆AAB, where equality holds only if the work is performed reversibly.
WABis a thermodynamic quantity, which means that it can be expressed as an en-
semble average of a phase space function. Specifically,WABmust be an average of
the mechanicalWAB(x) performed on a single member of the ensemble to drive it
from a microstate ofAto a microstate ofB. However, we need to be careful about
how we define this ensemble average because, as we saw in Chapter 1(eqn. (1.2.6)),
the workWAB(x) is defined along a particular path or trajectory, while equilibrium
averages are performed over the microstates that describe a particular thermodynamic
state. This distinction is emphasized by the fact that the work couldbe carried out
irreversibly, such that the system is driven out of equilibrium.
To illustrate the use of the microscopic functionWAB(x), suppose we prepare an
initial distribution of microstates x 0 belonging toAand then initiate a trajectory from
each of these initial states. The ensemble average that determines the thermodynamic
workWABis an average ofWAB(x 0 ) over this initial ensemble, which we will take
to be a canonical ensemble. The trajectory xtalong which the work is computed is a
unique function of the initial condition x 0 , i.e. xt= xt(x 0 ). Thus, the workWAB(x 0 ) is
actually a functional of the pathWAB[xt]. However, since the trajectory xtis uniquely
determined by the initial condition x 0 ,WABis also determined by x 0 , and we have
WAB=〈WAB(x 0 )〉A=CN
QA(N,V,T)
∫
dx 0 e−βHA(x^0 )WAB(x 0 ). (8.4.1)Thus, the work–free–energy inequality can be stated as〈WAB(x 0 )〉A≥∆AAB.
From this inequality, it would seem that performing work on a system as a method
to compute free energy leads, at best, to an upper bound on the free energy. It turns
out, however, that irreversible work can be used to calculate freeenergy differences
by virtue of a connection between the two quantities first established by C. Jarzyn-
ski (1997) that is now referred to as theJarzynski equality. The equality states that if
instead of averagingWAB(x 0 ) over the initial canonical distribution (that of stateA),
an average of exp[−βWAB(x 0 )] is performed over the same distribution, the result is
exp[−β∆AAB], that is,
e−β∆AAB=〈
e−βWAB(x^0 )〉
A=
CN
QA(N,V,T)
∫
dx 0 e−βHA(x^0 )e−βWAB(x^0 ). (8.4.2)Hence, the free energy difference ∆AAB=−kTln〈exp(−βWAB(x 0 )〉A. This remark-
able result not only provides a foundation for the development of nonequilibrium free
energy methods but has important implications for thermodynamicsin general.^2 Since
its introduction, the Jarzynski equality has been the subject of both theoretical and
experimental investigation (Parket al., 2003; Liphardtet al., 2002).
The Jarzynski equality can be derived using different strategies, as we will now
show. Consider first a time-dependent Hamiltonian of the form
(^2) Jarzynski’s equality is actually implied by a more general theorem known as theCrooks fluctuation
theorem(Crooks, 1998,1999).