Thermodynamic integration, perturbation theory, and $\lambda$-dynamics methods were applied to path integral molecular
dynamics calculations to investigate free energy differences due to ``alchemical'' transformations.
Several estimators were formulated to compute free energy differences in solvable model systems undergoing changes in mass and (or) potential.
Linear and non-linear alchemical interpolations were simulated for the thermodynamic integration.
We find improved convergence for the virial estimators, as well as for the thermodynamic integration over nonlinear interpolation paths.
Numerical results for the perturbative treatment of changes in mass and electric field strength in model systems are presented.
The performance of different free energy methods is discussed in the context of path integral molecular dynamics simulations.