Forward and Backward Averages Crooks Relation

So far, we have only considered perturbations in the forward direction. As in conventional free energy calculations, powerful relations can be derived if forward and backward perturbations are combined. With the free energy being a state function, we can reverse the path direction. This leads to several useful relations derived originally by Crooks [16, 18, 19], In particular, we obtain immediately 

In combination with (5.6), this leads to an upper and lower bound for the free energy difference 

These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations 

For computer simulations, (5.35) leads to accurate estimates of free energies. It is also the basis for higher-order cumulant expansions [20] and applications of Bennett s optimal estimator [21-23], We note that Jarzynski s identity (5.8) follows from (5.35) simply by integration over w because the probability densities are normalized to 1  

Because of the normalization condition, the right-hand side is equal to exp( —f3AA), and Jarzynski s identity follows. 

---