  We present a detailed comparison of computational efficiency and precision
for several free energy difference ($\Delta F$) methods. The analysis includes
both equilibrium and non-equilibrium approaches, and distinguishes between
uni-directional and bi-directional methodologies. We are primarily interested
in comparing two recently proposed approaches, adaptive integration and
single-ensemble path sampling, to more established methodologies. As test
cases, we study relative solvation free energies, of large changes to the size
or charge of a Lennard-Jones particle in explicit water. The results show that,
for the systems used in this study, both adaptive integration and path sampling
offer unique advantages over the more traditional approaches. Specifically,
adaptive integration is found to provide very precise long-simulation $\Delta
F$ estimates as compared to other methods used in this report, while also
offering rapid estimation of $\Delta F$. The results demonstrate that the
adaptive integration approach is the best overall method for the systems
studied here. The single-ensemble path sampling approach is found to be
superior to ordinary Jarzynski averaging for the uni-directional,
``fast-growth'' non-equilibrium case. Closer examination of the path sampling
approach on a two-dimensional system suggests it may be the overall method of
choice when conformational sampling barriers are high. However, it appears that
the free energy landscapes for the systems used in this study have rather
modest configurational sampling barriers.
