  The estimate of free energy changes based on Bennett's acceptance ratio
method is examined in several limiting cases and compared with other estimates
based on the Jarzynski equality and on the Crooks relation. While the absolute
amount of dissipated work, defined as the surplus of average work over the free
energy difference, limits the practical applicability of Jarzynski's and
Crooks' methods, the reliability of Bennett's approach is restricted by the
difference of the dissipated works in the forward and the backward process. We
illustrate these points by considering a Gaussian chain and a hairpin chain
which both are extended during the forward and accordingly compressed during
the backward protocol. The reliability of the Crooks relation predominantly
depends on the sample size; for the Jarzynski estimator the slowness of the
work protocol is crucial, and the Bennett method is shown to give precise
estimates irrespective of the pulling speed and sample size as long as the
dissipated works are the same for the forward and the backward process as it is
the case for Gaussian work distributions. With an increasing dissipated work
difference the Bennett estimator also acquires a bias which increases roughly
in proportion to this difference. A substantial simplification of the Bennett
estimator is provided by the 1/2-formula which expresses the free energy
difference by the algebraic average of the Jarzynski estimates for the forward
and the backward processes. It agrees with the Bennett estimate in all cases
when the Jarzynski and the Crooks estimates fail to give reliable results.

Statistical Mechanics