In thermodynamics, the free energy difference between two states A and B is connected to the workW done on the system through the inequality: with equality holding only in the case of a quasistatic process, i.e. when one takes the system from A to B infinitely slowly. In contrast to the thermodynamic statement above, the JE remains valid no matter how fast the process happens. The equality itself can be straightforwardly derived from the Crooks fluctuation theorem. The JE states: Here k is the Boltzmann constant and T is the temperature of the system in the equilibrium stateA or, equivalently, the temperature of the heat reservoir with which the system was thermalized before the process took place. The over-line indicates an average over all possible realizations of an external process that takes the system from the equilibrium state A to a new, generally nonequilibrium state under the same external conditions as that of the equilibrium state B.. In the limit of an infinitely slow process, the work Wperformed on the system in each realization is numerically the same, so the average becomes irrelevant and the Jarzynski equality reduces to the thermodynamic equality . In general, however, W depends upon the specific initial microstate of the system, though its average can still be related to through an application of Jensen's inequality in the JE, viz. in accordance with the second law of thermodynamics. Since its original derivation, the Jarzynski equality has been verified in a variety of contexts, ranging from experiments with biomolecules to numerical simulations. Many other theoretical derivations have also appeared, lending further confidence to its generality.
History
A question has been raised about who gave the earliest statement of the Jarzynski equality. For example, in 1977 the Russian physicists G.N. Bochkov and Yu. E. Kuzovlev proposed a generalized version of the Fluctuation-dissipation theorem which holds in the presence of arbitrary external time-dependent forces. Despite its close similarity to the JE, the Bochkov-Kuzovlev result does not relate free energy differences to work measurements, as discussed by Jarzynski himself in 2007. Another similar statement to the Jarzynski equality is the nonequilibrium partition identity, which can be traced back to Yamada and Kawasaki. However, these early statements are very limited in their application. Both Bochkov and Kuzovlev as well as Yamada and Kawasaki consider a deterministic timereversibleHamiltonian system. As Kawasaki himself noted this precludes any treatment of nonequilibrium steady states. The fact that these nonequilibrium systems heat up forever because of the lack of any thermostatting mechanism leads to divergent integrals etc. No purely Hamiltonian description is capable of treating the experiments carried out to verify the Crooks fluctuation theorem, Jarzynski equality and the Fluctuation theorem. These experiments involve thermostatted systems in contact with heat baths.
