In this section, the relationship between work and free energy
will be explored in greater detail. We have already introduced
the inequality in eqn. (16), which
states that if an amount of work
is performed on a system,
taking from state to state , then
Here, equality holds only if the work is performed reversibly.
The work referred to here is thermodynamic quantity and,
as such, must be regarded as an ensemble average. In statistical
mechanics, we can also introduce the mechanical or microscopic
performed on one member of the ensemble to
drive it from state to state . Then,
is simply an ensemble average of
. However, we
need to be somewhat careful about how we define this ensemble
the work is defined along a particular path or trajectory which
takes the system from state to state , and equilibrium
averages do not refer not to paths but to microstates. This distinction
is emphasized by the fact that the work could be carried out
irreversibly, such that the system is driven out of equilibrium.
Thus, the proper definition of the ensemble average follows along the
lines already discussed in the context of the free-energy perturbation
approach, namely, averaging over the canonical distribution for
the state . In this case, since we will be discussing actual
paths , we let the initial condition be the phase
space vector for the system in the (initial) state .
is a unique function of the
initial conditions. Then
From such an inequality, it would seem that using the work
as a method for calculating the free energy is of limited
utility, since the work necessarily must be performed reversibly,
otherwise one obtains only upper bound on the free energy.
It turns out, however, that irreversible work can be used to
calculate free energy differences by virtue of a connection
between the two quantities first discovered in 1997 by
C. Jarzynski that as come to be known as the
Jarzynski equality. This equality
states that if, instead of averaging
initial canonical distribution (that of state ), an
is performed over the
same distribution, the result is
The Jarzynski equality be proved using different strategies.
Here, however, we will present a proof that is most relevant
for the finite-sized systems and techniques employed in
molecular dynamics calculations.
Consider a time-dependent
Hamiltonian of the form
With the above condition, we can write the microscopic work as
Note that due to Jensen's inequality: