# 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 $\Delta F = F_B - F_A$ between two states A and B is connected to the work W done on the system through the inequality:

$\Delta F \leq W$,

which 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:

$e^ { -\Delta F / k T} = \overline{ e^{ -W/kT } }.$

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 $\Delta F = W$ (see above). In general, however, W depends upon the specific initial microstate of the system, though its average can still be related to $\Delta F$ through an application of Jensen's inequality in the JE, viz.

$\Delta F \leq \overline{W},$

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 (see Bibliography) proposed a generalized version of the Fluctuation-Dissipation relations 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 (see references below).

Another similar statement to the Jarzynski equality is the nonequilibrium partition identity, which can be traced back to Yamada and Kawasaki. (The Nonequilibrium Partition Identity is the Jarzynski equality applied to two systems whose free energy difference is zero - like straining a fluid.) However, these early statements are very limited in their application. Both Bochkov and Kuzovlev as well as Yamada and Kawasaki consider a deterministic time reversible Hamiltonian 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 thermostated systems in contact with heat baths.

## Bibliography

For earlier results dealing with the statistics of work in adiabatic (i.e. Hamiltonian) nonequilibrium processes, see:

• Bochkov, G. N.; Kuzovlev, Yu. E. (1977), Zh. Eksp. Teor. Fiz. 72: 238; op. cit. 76, 1071 (1979)
• Bochkov, G. N.; Kuzovlev, Yu. E. (1981), "Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem", Physica A 106: 443, doi:10.1016/0378-4371(81)90122-9; op. cit. 106A, 480 (1981)
• Kawasaki, K.; Gunton, J.D. (1973), "Theory of Nonlinear Transport Processes: Nonlinear Shear Viscosity and Normal Stress Effects", Phys. Rev. A 8: 2048, doi:10.1103/PhysRevA.8.2048
• Yamada, T.; Kawasaki, K. (1967), "Nonlinear Effects in the Shear Viscosity of Critical Mixtures", Prog. Theo. Phys. 38: 1031, doi:10.1143/PTP.38.1031

For a comparison of such results, see: