Adiabatic switching and thermodynamic integration

The free-energy perturbation approach evokes a physical picture in which configurations sampled from the canonical distribution of state are immediately ``switched'' to the state by simply changing the potential from to . Such ``instantaneous'' switching clearly represents an unphysical path from one state to the other, but we need not concern ourselves with this because the free energy is a state function and, therefore, independent of the path connecting the states. Nevertheless, we showed that the free-energy perturbation theory formula, eqn. (6), is only useful if the states and do not differ vastly from one another, thus naturally raising the question of what can be done if the states are very different.

The use of a series of intermediate states, by which
eqn. (7) is derived, exploits
the fact that any path between the states can be
employed to obtain the free energy difference.
In this section, we will discuss an alternative
approach in which the system is switched slowly or *adiabatically* from
one state to the other, allowing the system to fully relax
at each point along a chosen path from state
to state , rather than instantaneously switching the
system between intermediate states, as occurs in eqn. (7).
In order to effect the switching from
one state to the other, we will employ a common
trick in the form of an ``external'' switching
parameter, . This paramter is introduced by defining
a new potential energy function

In order to see how eqn. (8) can be used
to compute the free energy difference
, consider the
canonical partition function of a system described by
the potential of eqn. (8) for a
particular choice of :

Recall, however, that the derivatives of the free energy with repsect to , and and lead to the chemical potential, pressure and entropy, respectively. What does the derivative of the free energy with respect to represent? According to eqn. (10)

The reader should check that the expressions involving and are equivalent. Computing the derivative of with respect to , we find

Now, the free energy difference can be obtained trivially from the relation

Substituting eqns. (11) and (12) into eqn. (13), we obtain the free energy difference as

where denotes an average over the canonical ensemble described by the distribution with fixed at a particular value. The special choice of and has a simple interpretation. For this choice, eqn. (14) becomes

The content of eqn. (15) can be understood by recalling the relationship between work and free energy from the second law of thermodynamics. If, in transforming the system from state to state , an amount of work is performed on the system, then

where equality holds

In practice, the thermodynamic integration formula is implemented as follows: A set of values of is chosen from the interval , and at each chosen value , a full molecular dynamics or Monte Carlo calculation is carried out in order to generate the average . The resulting values of , are then substituted into eqn. (14), and the resulted is integrated numerically to produce the free energy difference . Thus, we see that the selected values can be evenly spaced, for example, or they could be a set of Gaussian quadrature nodes, depending on how is expected to vary with for the chosen and .

As with free-energy perturbation theory, the thermodynamic integration approach can be implemented very easily. An immediately obvious disadvantage of the method, however, is the same one that applies to eqn. (7): In order to perform the numerical integration, it is necessary to perform many simulations of a system at physically uninteresting intermediate values of where the potential is, itself, unphysical. Only correspond to actual physical states and ultimately, we can only attach physical meaning to the free energy difference . Nevertheless, the intermediate averages must be accurately calculated in order for the integration to yield a correct result. The approach to be presented in the next section attempts to reduce the time spent in such unphysical intermediate states and focuses the sampling in the important regions .