Jarzynski Equality

The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two equilibrium states and non-equilibrium processes. It is named after the physicist Christopher Jarzynski (then at Los Alamos National Laboratory) who derived it in 1997.

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:

,

the equality happening 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 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. (For example, in the textbook case of a gas compressed by a piston, the gas is equilibrated at piston position A and compressed to piston position B; in the Jarzynski equality, the final state of the gas does not need to be equilibrated at this new piston position). 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). 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.