Jarzynski equality

The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two states and the irreversible work along an ensemble of trajectories joining the same states. It is named after the physicist Christopher Jarzynski (then at the University of Washington and Los Alamos National Laboratory, currently at the University of Maryland) who derived it in 1996. Fundamentally, the Jarzynski equality points to the fact that the fluctuations in the work satisfy certain constraints separately from the average value of the work that occurs in some process. In thermodynamics, the free energy difference between two states A and B is connected to the work W 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 (such that all intermediate states are in thermodynamic equilibrium). In contrast to the thermodynamic statement above, the JE remains valid no matter how fast the process happens. The JE states: Here k is the Boltzmann constant and T is the temperature of the system in the equilibrium state A 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. This average over possible realizations is an average over different possible fluctuations that could occur during the process (due to Brownian motion, for example), each of which will cause a slightly different value for the work done on the system. In the limit of an infinitely slow process, the work W performed on the system in each realization is numerically the same, so the average becomes irrelevant and the Jarzynski equality reduces to the thermodynamic equality (see above).