  We compare two predictions regarding the microscopic fluctuations of a system
that is driven away from equilibrium: one due to Crooks [J. Stat. Phys. 90,
1481 (1998)] which has gained recent attention in the context of nonequilibrium
work and fluctuation theorems, and an earlier, analogous result obtained by
Bochkov and Kuzovlev [Zh. Eksp. Teor. Fiz. 72(1), 238247 (1977)]. Both results
quantify irreversible behavior by comparing probabilities of observing
particular microscopic trajectories during thermodynamic processes related by
time-reversal, and both are expressed in terms of the work performed when
driving the system away from equilibrium. By deriving these two predictions
within a single, Hamiltonian framework, we clarify the precise relationship
between them, and discuss how the different definitions of work used by the two
sets of authors gives rise to different physical interpretations. We then
obtain a extended fluctuation relation that contains both the Crooks and the
Bochkov-Kuzovlev results as special cases.
