  Recent theoretical predictions and experimental measurements have
demonstrated that equilibrium free energy differences can be obtained from
exponential averages of nonequilibrium work values. These results are similar
in structure, but not equivalent, to predictions derived nearly three decades
ago by Bochkov and Kuzovlev, which are also formulated in terms of exponential
averages but do not involve free energy differences. In the present paper the
relationship between these two sets of results is elucidated, then illustrated
with an undergraduate-level solvable model. The analysis also serves to clarify
the physical interpretation of different definitions of work that have been
used in the context of thermodynamic systems driven away from equilibrium.
