Comparison of work fluctuation relations

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.